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This article is cited in 2 scientific papers (total in 2 papers) On multipliers for Fourier series in Sobolev orthogonal polynomials B. P. Osilenker Moscow, Russia
Russian version:
Matematicheskii Sbornik, 2022, Volume 213, Number 8, DOI: https://doi.org/10.4213/sm9556 
Bibliographic databases:
    § 1. Statement of the problemLet $\theta(x)$ be a finite positive Borel measure with support on the interval $[-1, 1]$ with infinite number of points of increase in its support, and let $a_k$, $-1\leqslant a_k \leqslant 1$, $k=1,2,\dots,m$, be some points. Given $L^2_{\theta}$-functions $f$ and $g$ that are differentiable at the points $a_k$, consider the inner product
$$
\begin{equation}
\langle f, g\rangle =\int^1_{-1} f(x)g(x)\,d\theta(x)+\sum^m_{k=1}\sum^{N_k}_{i=0}M_{k, i}f^{(i)}(a_k)g^{(i)}(a_k),
\end{equation}
\tag{1.1}
$$
where $M_{k,i}>0$, $i=0,1,\dots,N_k$, $k=1,2,\dots,m$,
$$
\begin{equation}
\theta(\{a_k\})=0, \quad k=1,2,\dots,m,\quad\text{and} \quad \theta'(x)> 0 \quad\text{almost everywhere}.
\end{equation}
\tag{1.2}
$$
If the measure $d\theta(x)$ is absolutely continuous and ${d\theta(x)}/{dx}=\omega(x)$, then $\omega(x)$ is known as a weight function (a weight). Linear spaces equipped with the inner product (1.1) are called continuous-discrete Sobolev spaces with measure (or weighed Sobolev spaces). The loaded spaces (or discrete loaded spaces) with the inner product
$$
\begin{equation*}
\langle f, g\rangle=\int^1_{-1} f(x)g(x)\,d\theta(x)+\sum^m_{k=1} M_kf(a_k)g(a_k), \qquad M_k>0, \quad k=1,\dots,m,
\end{equation*}
\notag
$$
constitute a particular class of continuous-discrete Sobolev spaces. Let $\{\widehat{q}_n(x)\}$, $n\in\mathbb{Z}_+$, $\mathbb{Z}_+=\{0,1,2,\dots \}$ be a system of polynomials of degree $n$ that are orthonormal with respect to the inner product (1.1):
$$
\begin{equation}
\begin{gathered} \, \notag \widehat{q}_n(x)=k(\widehat{q}_n)x^n+r(\widehat{q}_n)x^{n-1}+\dotsb, \qquad k(\widehat{q}_n)>0, \quad n\in\mathbb{Z}_+, \\ \begin{split} \langle \widehat{q}_n, \widehat{q}_m\rangle &=\int^1_{-1}\widehat{q}_n(x)\widehat{q}_m(x)\,d\theta(x) \\ &\qquad+\sum^m_{k=1}\sum^{N_k}_{i=0}M_{k,i}\widehat{q}_n^{\,(i)}(a_k)\widehat{q}_m^{\,(i)}(a_k) =\delta_{n,m},\qquad n,m\in\mathbb{Z}_+. \end{split} \end{gathered}
\end{equation}
\tag{1.3}
$$
The polynomials $\widehat{q}_n(x)$, $n\in\mathbb{Z}_+$, are called Sobolev polynomials (or Sobolev type polynomials). Such systems (and their differential analogues) appeared in the classical book [1] in the study of boundary-value problems for second-order differential operators, in the problem of classification of eigenfunctions of fourth-order linear differential operators (see [2] and [3]), and in the problem of best polynomial approximation in discrete Sobolev spaces (see [4]). Orthogonal systems in Sobolev spaces have been studied extensively in recent years (see the survey [5], and a series of papers by Sharapudinov [6]–[8] with the references given there). The inner product (1.1) and the corresponding orthogonal systems (and their differential analogues) play an important role in many problems of the theory of functions, functional analysis, quantum mechanics, mathematical physics and numerical mathematics (see [9]–[18]). Given $p$, $1\leqslant p<\infty$, consider the set of functions
$$
\begin{equation*}
\mathfrak{R}_p=\begin{Bmatrix} f\colon \displaystyle\int_{-1}^1|f(x)|^p\,d\theta(x)<\infty\text{ and } f^{(i)}(a_k)\text{ exists for }i=0,1,2,\dots,N_k, \\ \text{where } -1\leqslant a_k\leqslant 1, \qquad k=1,2,\dots,m \end{Bmatrix}.
\end{equation*}
\notag
$$
In particular,
$$
\begin{equation*}
\mathfrak{R}_1\equiv \mathfrak{R}=\begin{Bmatrix} f\colon \displaystyle\int_{-1}^1|f(x)|\,d\theta(x)<\infty\text{ and } f^{(i)}(a_k)\text{ exists for }i=0,1,2,\dots,N_k, \\ \text{where } -1\leqslant a_k\leqslant 1, \qquad k=1,2,\dots,m \end{Bmatrix}.
\end{equation*}
\notag
$$
In view of the inclusion
$$
\begin{equation*}
\mathfrak{R} \supset\mathfrak{R}_p, \qquad 1<p<\infty,
\end{equation*}
\notag
$$
the proofs (and statements) of the results that follow are mostly given for $p=1$. With each function $f\in\mathfrak{R}$, we associate the Fourier-Sobolev series
$$
\begin{equation}
f(x) \sim\sum^\infty_{k=0}c_k(f)\widehat{q}_k(x), \qquad x\in[-1,1],
\end{equation}
\tag{1.4}
$$
where
$$
\begin{equation}
c_k(f)=\langle f,\widehat{q}_k\rangle=\int^1_{-1} f(x)\widehat{q}_k(x)\,d\theta(x)+\sum^m_{s=1}\sum^{N_s}_{i=0} M_{s,i}f^{(i)}(a_s)\widehat{q}_k^{\,(i)}(a_s), \quad k\in\mathbb{Z}_+.
\end{equation}
\tag{1.5}
$$
Consider a sequence of real numbers
$$
\begin{equation}
\Phi=\bigl\{\phi_k,\ k\in\mathbb{Z}_+;\ \phi_0=1,\ \{\phi_k\}\in1^\infty\bigr\}.
\end{equation}
\tag{1.6}
$$
For each function $f\in\mathfrak{R}$ and its Fourier series (1.4), (1.5) we introduce the linear mapping $T$ defined by
$$
\begin{equation}
f(x)\sim\sum^\infty_{k=0}c_k(f)\widehat{q}_k(x) \quad\Longrightarrow\quad T(f;x;\Phi)=\sum^\infty_{k=0}\phi_kc_k(f)\widehat{q}_k(x), \quad x\in[-1,1].
\end{equation}
\tag{1.7}
$$
The mapping $T$ is called a multiplier operator, the sequence $\Phi=\{\phi_k\}^\infty_{k=0}$ is known as a multiplier (a multiplier sequence), and the series (1.7) is called a multiplier series. Here we are concerned with the problem of finding conditions on a system $\{\widehat{q}_n(x)\}$ and elements of a multiplier sequence (1.6) for which the definition of a multiplier operator is consistent and the norm of the multiplier operator in continuous-discrete Sobolev spaces can be estimated. It is known that, according to the generalized Favard theorem for Sobolev spaces (see [19]), an orthogonal system of polynomials can equivalently be defined via a measure or with the help of recurrence relations. In this paper we consider the approach based on recurrence relations (see (3.7), (3.36), (4.8)). It would be interesting to study the above problem in the case when a system of orthogonal polynomials is defined via a measure (an inner product). Here we only note that estimate (3.7) can be tested with the help of Lemma 3.2 from [20]. It is worth pointing out that, unlike the investigations in [6]–[8], here we are concentred with different systems of orthogonal polynomials and we examine different problems. Some results obtained below were announced in [21] and [22]. § 2. Auxiliary resultsLet $N^*_k$ be the natural number defined by
$$
\begin{equation*}
N^*_k=\begin{cases} N_k+1&\text{if }N_k\text{ is odd}, \\ N_k+2&\text{if }N_k\text{ is even}. \end{cases}
\end{equation*}
\notag
$$
We also set
$$
\begin{equation}
w_N(x)=\prod^m_{k=1} (x-a_k)^{N^*_k}, \qquad N=\sum^m_{k=1}N^*_k.
\end{equation}
\tag{2.1}
$$
Lemma 2.1 (see [20], Theorem 2.3). The orthonormal polynomials $\widehat{q}_n(x)$ satisfy the recurrence relation Furthermore, if $\theta'(x)> 0$ almost everywhere, then We set
$$
\begin{equation}
\varepsilon_m:=(-1,1)\setminus\bigcup^m_{s=1}\{a_s\}.
\end{equation}
\tag{2.3}
$$
For a proof of (2.4), see [20], Lemma 3.1. From (2.4) we obtain
$$
\begin{equation}
\lim_{n\to\infty} \widehat{q}^{\,(i)}_n(a_s)=0, \qquad i=0,1,\dots,N_s, \quad s=1,2,\dots,m.
\end{equation}
\tag{2.5}
$$
Let $(a,b)$ be a fixed (open or closed) interval and let be $\rho$ be an absolutely continuous positive Borel measure on $(a,b)$. Given a function $f\in L^1_\rho((a,b))$, the Hardy-Littlewood maximal function $M_\rho f$ is defined by
$$
\begin{equation}
M_\rho f(x)=\sup\frac{1}{\rho(J)}\int_J |f(x)|\,d\rho(x),
\end{equation}
\tag{2.6}
$$
where the supremum is taken over the family $\{J\}$ of all open intervals $J$ with centre $x\in(a,b)$. Given $f\in L^1_\rho((a,b))$, for each $n\in\mathbb{Z}_+$ consider the operator
$$
\begin{equation}
I_nf(x)=I_n(f):=\int^b_a f(t)H_n(t,x)\,d\rho(t), \qquad n\in\mathbb{Z}_+, \quad x\in(a,b).
\end{equation}
\tag{2.7}
$$
A nonnegative function $H^*_n(t,x)$, $n\in\mathbb{Z}_+$, $x\in(a,b)$, is called a humpbacked majorant for the sequence $H_n(t,x)$ (of kernels of the integrals) with respect to $t$ at the point $x\in(a,b)$ if:
If a humpbacked majorant $H^*_n(t,x)$ satisfies where the positive constant $C$ is independent of $n\in\mathbb{Z}_+$ and $x\in(a,b)$, then $H^*_n(t,x)$ is called an integrable humpbacked majorant of the function $H_n(t,x)$ on $(a,b)$.Lemma 2.3 (see [23], [24], Ch. 6.3, p. 249, and [25]). Let $\rho$ be an absolutely continuous positive Borel measure on $(a,b)$ and let the function $H_n(t,x)$ have an integrable humpbacked majorant $H^*_n(t,x)$ Then the following results hold for the integral $I_n(f;x)$ in (2.7): 1) if $f\in L^1_\rho((a,b))$, then
$$
\begin{equation}
\sup_{n\in\mathbb{Z}_+}|I_nf(x)|\leqslant CM_\rho f(x),\qquad x\in(a,b),
\end{equation}
\tag{2.8}
$$
where the positive constant $C$ is independent of $f$ and $x\in(a,b)$; 2) if $f\in L^p_\rho((a,b))$, $1<p<\infty$, then where $C_p>0$ is independent of $f$ andRemark 2.1. In the definition of $H^*_n(t,x)$ and the results that follow, in place of $x\in(a,b)$ we can consider $x$ in a set $ F\subseteq(a,b)$ (with appropriate modifications). Lemma 2.4 (Fatou; see [26], Ch. 3, § 16, and [27], Ch. III, § 19, Exercise 35). If a sequence of nonnegative measurable functions $f_1(x),f_2(x),\dots$ converges almost everywhere on $E$ to a function $F(x)$, then § 3. The behaviour of partial sums of a Fourier-Sobolev seriesLet
$$
\begin{equation*}
S_nf(x)=\sum^n_{k=0} c_k(f)\widehat{q}_k(x), \qquad n\in\mathbb{Z}_+, \quad x\in[-1,1],
\end{equation*}
\notag
$$
denote the partial sums of the Fourier-Sobolev series (1.4), (1.5). Let us study the behaviour of the $S_nf(x)$. We have
$$
\begin{equation}
\begin{aligned} \, S_nf(x) &=\int^1_{-1} f(t)D_n(t;x)\,d\theta(t) \nonumber \\ &\qquad +\sum^m_{s=1}\sum^{N_s}_{i=0} M_{s,i}f^{(i)}(a_s)\frac{\partial^i D_n}{\partial t^i}(a_s; x),\qquad n\in\mathbb{Z}_+,\quad x\in[-1,1], \end{aligned}
\end{equation}
\tag{3.1}
$$
where $D_n(t,x)$ is the Dirichlet kernel of the system $\{\widehat{q}_n\}^\infty_{n=0}$:
$$
\begin{equation*}
D_n(t,x)=\sum^n_{l=0}\widehat{q}_l(t)\widehat{q}_l(x), \qquad t,x\in[-1,1], \quad n\in\mathbb{Z}_+.
\end{equation*}
\notag
$$
By (1.3) we have
$$
\begin{equation}
\int^1_{-1} D_n(t,x)\,d\theta(t)+\sum^m_{s=1} M_{s,0}D_n(a_s, x)=1, \qquad n\in\mathbb{Z}_+, \quad x\in[-1,1].
\end{equation}
\tag{3.2}
$$
If $x_0\in[-1,1]$, then the polynomial $w_N(x)-w_N(x_0)$ can have more than one zero on $[-1, 1]$, which is not convenient for further analysis. So instead of $w_N(x)$ we consider the polynomial $\displaystyle\pi_{N+1}(x)=\int^x_{-1} w_N(t)\,dt$. Now, since $w_N(x)$ is positive for $x_0\neq a_s$, $s=1,2,\dots,m$, the function $\pi_{N+1}(x)-\pi_{N+1}(x_0)$ has the unique zero $x_0$ on $[-1,1]$. The derivatives $\pi_{N+1}(x)$ vanish at the points $a_s$, and so we have $\langle \pi_{N+1}\widehat{q}_n, \widehat{q}_m\rangle=\langle \widehat{q}_n, \pi_{N+1}\widehat{q}_m\rangle$. Therefore, the polynomials $\widehat{q}_n(x)$ satisfy the recurrence relation (see [20])
$$
\begin{equation}
\begin{gathered} \, \pi_{N+1}(x)\widehat{q}_n(x)=\sum^{N+1}_{j=0} d_{n+j,j}\widehat{q}_{n+j}(x)+\sum^{N+1}_{j=1} d_{n,j}\widehat{q}_{n-j}(x), \\ n\,{\in}\,\mathbb{Z}_+, \qquad \widehat{q}_{-j}(x)\,{=}\,0, \quad j\,{=}\,1,2,\dots, \qquad d_{n,j}\,{=}\,0, \quad j\,{>}\,n, \qquad x\,{\in}\,[-1,1]; \end{gathered}
\end{equation}
\tag{3.3}
$$
moreover, the coefficients $d_{n,j}$ are bounded for all $j$ and $n$: (see [20]). We have $\{\pi_{N+1}(t)\}'=w_N(t)$, where $w_N(t)$ has the form (2.1), hence at each point $x\in \varepsilon_m$ (see (2.3))
$$
\begin{equation}
\frac{|t-x|}{|\pi_{N+1}(t)-\pi_{N+1}(x)|}\leqslant C_x, \qquad t\in(-1,1), \quad x\in\varepsilon_m,
\end{equation}
\tag{3.5}
$$
and estimate (3.5) is uniform in $x$ on compact subsets $K$ of $\varepsilon_m$:
$$
\begin{equation}
\frac{|t-x|}{|\pi_{N+1}(t)-\pi_{N+1}(x)|}\leqslant C, \qquad t\in(-1,1), \quad x\in K,
\end{equation}
\tag{3.6}
$$
where the constant $C$ is independent of $t\in(-1,1)$ and $x\in K$. Remark 3.1. If the polynomial $w_N(x)-w_N(x_0)$ has at most one zero on $[-1, 1]$, then recurrence relation (2.2) is sufficient for our further analysis; in this case it is only required to verify (3.5), (3.6) and check the further conditions imposed on the system $\widehat{q}_n(x)$, $n\in\mathbb{Z}_+$. Lemma 3.1. The following analogue of the Christoffel-Darboux formula holds for the Dirichlet kernel $D_n(t,x)$ of an orthonormal system of polynomials $\{\widehat{q}_n\}^\infty_{n=0}$: The proof of the Christoffel-Darboux formula follows directly from recurrence relation (3.3) (see Lemma 3.3 in [20], in which the kernel is represented in a different form); in the limiting case l’Hôpital’s rule should be applied. Lemma 3.2. Let there exist a positive continuous function $h(x)$ that is $\theta$-integrable on $\varepsilon_m$ and satisfies Then for any function $f\in\mathfrak{R}_p$, $1\leqslant p <\infty$, at each point $x\in\varepsilon_m$ where the convergence is uniform on all compact subsets $K$ of $\varepsilon_m$. Proof. We have
It is clear from the Christoffel-Darboux formula that
$$
\begin{equation*}
\sum^m_{s=1}\sum^{N_s}_{i=0} M_{s,i}f^{(i)}(a_s)\frac{\partial^i D_n}{\partial t^i}(a_s,x)
\end{equation*}
\notag
$$
is a sum of a finite number (depending on $N$) of terms of the form
$$
\begin{equation*}
\begin{gathered} \, M_{s,i}f^{(i)}(a_s) d_{l+j,j}\frac{\widehat{q}^{\,(i)}_{l+j}(a_s)\widehat{q}_l(x)- \widehat{q}_l(a_s)\widehat{q}^{\,(i)}_{l+j}(x)}{\pi_{N+1}(a_s)-\pi_{N+1}(x)}, \\ s=1,2,\dots,m, \qquad i=0,1,\dots,N_s, \qquad j=1,2,\dots,N+1, \\ l=n-j+1,n-j+2,\dots, n. \end{gathered}
\end{equation*}
\notag
$$
Since the coefficients $d_{l, j}$, $M_{s,i}$ are bounded, it follows that $|\widehat{q}_n(x)|\leqslant h(x)$, $x\in\varepsilon_m$, where $h(x)$ is continuous on $\varepsilon_m$, uniformly bounded on compact subsets $K$ of $\varepsilon_m$, and $\lim_{n\to\infty}\widehat{q}_n^{\,(i)}(a_s)=0$ (see (2.5)). This proves Lemma 3.2. Remark 3.2. Note that in the actual fact we have also proved that, under the hypotheses of Lemma 3.2, at each point $x\in\varepsilon_m $ (and uniformly on $K$), Lemma 3.3. Let condition (3.7) be met, the measure $d\theta(x)$ be absolutely continuous on $\varepsilon_m$, let ${d\theta(x)}/{dx}=\omega(x)$, and let $\omega(x)$ be a continuous positive integrable function on $\varepsilon_m$: Then the function has a humpbacked majorant $\widetilde{D}^*_n(t,x)$ satisfying where the positive constant $C$ is independent of $n\in\mathbb{Z}_+$ and $x\in K$, and $K$ is an arbitrary compact subset of $\varepsilon_m$. Proof. By the Christoffel-Darboux formula, (3.4), (3.6) and (3.7) we have and
Hence
$$
\begin{equation}
|\widetilde{D}_n(t,x)|\leqslant C(n+1)\quad\text{for }\ t\in\varepsilon_m, \ \ x\in K,\ \ 0\leqslant |t-x|\leqslant \frac{1}{n+1},
\end{equation}
\tag{3.13}
$$
and
$$
\begin{equation}
|\widetilde{D}_n(t,x)|\leqslant C\frac{1}{|t-x|} \quad\text{for }\ t\in\varepsilon_m, \ \ x\in K,\ \ \frac{1}{n+1}<|t-x|,
\end{equation}
\tag{3.14}
$$
where the positive constant $C$ is independent of $n\in\mathbb{Z}_+$, $t\in(-1,1)$, $x\in K$.
1. We claim that the kernel $\widetilde{D}_n(t,x)$ has a humpbacked majorant
$$
\begin{equation*}
\widetilde{D}^*_n(t,x)=\frac{2C(n+1)}{1+(n+1)|t-x|}, \qquad n\in\mathbb{Z}_+, \quad t\in\varepsilon_m, \quad x\in K.
\end{equation*}
\notag
$$
Indeed,
Using (3.13) and (3.14) we find that $|\widetilde{D}_n(t,x)|\leqslant \widetilde{D}^*_n(t,x)$. 2. The monotonicity of the majorant (in the definition of a humpbacked majorant) follows from the estimates
$$
\begin{equation*}
\frac{\partial}{\partial t}\widetilde{D}^*_n(t,x)=\frac{\partial}{\partial t}\biggl\{\frac{2C(n+1)}{1+(n+1)(x-t)}\biggr\} > 0, \qquad t\leqslant x,
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\frac{\partial}{\partial t}\biggl\{\frac{2C(n+1)}{1+(n+1)(x-t)}\biggr\} < 0, \qquad t> x.
\end{equation*}
\notag
$$
3. Let us prove (3.10). We set
$$
\begin{equation}
\varepsilon_m=\bigcup^m_{k=0}(a_k, a_{k+1}), \qquad k=0,1,\dots,m, \qquad a_0=-1, \quad a_{m+1}=1.
\end{equation}
\tag{3.15}
$$
Let $K$ be a compact subset of $\varepsilon_m$; this set can be covered by a finite number of compact subsets of the intervals $(a_{k},a_{k+1})$, $k=0,1,\dots,m$ (some of which are possibly empty). Let us show that this estimate holds on any compact subset $[a_k+h^{(k)}, {a_{k+1}+h^{(k+1)}}]$ of $(a_k,a_{k+1})$. Assume that $x\in K\cap (a_{k_0},a_{k_0+1})$ for some $k_0$, $0\leqslant k_0\leqslant m$. Let $h$, $0<h<(a_{k_0+1}-a_{k_0})/2$, be such that $x\in[a_{k_0}+h, a_{k_0+1}-h]$ (here we consider the case when $h^{(k_0)}=h^{(k_{0}+1)}=h$; this does not change the proof). We have
$$
\begin{equation*}
\begin{aligned} \, I_n(x) &=\frac{1}{\ln(n+2)}\int^1_{-1} \widetilde{D}^*_n(t,x)\,d\theta(t) \\ &=\frac{1}{\ln(n+2)} \int^{a_{k_0+1}}_{a_{k_0}} \widetilde{D}^*_n(t,x)\,d\theta(t)+\frac{1}{\ln(n+2)} \int_{\substack{ t\notin(a_{k_0}, a_{k_0+1})\\ t\in\varepsilon_m}} \widetilde{D}^*_n(t,x)\,d\theta(t) \\ &=\widetilde{I}^{(1)}_n(x)+\widetilde{I}^{(2)}_n(x). \end{aligned}
\end{equation*}
\notag
$$
Let $n$ be a natural number such that
$$
\begin{equation*}
\biggl[x-\frac{1}{n+1}, x+\frac{1}{n+1}\biggr] \subset\biggl(a_{k_0}+\frac h2, a_{k_0+1}-\frac h2\biggr);
\end{equation*}
\notag
$$
by symmetry it suffices to consider only the case when $t\geqslant x$. Setting
$$
\begin{equation*}
\delta_1=\biggl[x, x+\frac{1}{n+1}\biggr],\quad \delta_2=\biggl[x+\frac{1}{n+1}, a_{k_0+1}-\frac{h}{2}\biggr] \quad \text{and}\quad \delta_3=\biggl[a_{k_0+1}-\frac{h}{2}, a_{k_0+1}\biggr]
\end{equation*}
\notag
$$
we have
$$
\begin{equation*}
\widetilde{I}^{(1)}_n(x)=\frac{1}{\ln(n+2)} \int_{\delta_1}+\frac{1}{\ln(n+2)} \int_{\delta_2}+\frac{1}{\ln(n+2)} \int_{\delta_3}=I^{(1)}_n(x)+I^{(2)}_n(x)+I^{(3)}_n(x).
\end{equation*}
\notag
$$
To estimate the first two terms we use the fact that the weight function (3.8) is bounded. We have
$$
\begin{equation*}
\begin{aligned} \, I_n^{(1)}(x) &=\frac{1}{\ln(n+2)}\int_{\delta_1} \widetilde{D}^*_n(t,x)\,d\theta(t) \\ &=\frac{1}{\ln(n+2)}\int^{x+1/(n+1)}_x \frac{2C(n+1)}{1+(n+1)(t-x)}\omega(t)\,dt \\ &\leqslant \frac{1}{\ln(n+2)}\int^{x+1/(n+1)}_x \frac{2C(n+1)}{1+(n+1)(t-x)}\,dt=O(1) \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, I_n^{(2)}(x) &=\frac{1}{\ln(n+2)}\int_{\delta_2} \widetilde{D}^*_n(t,x)\,d\theta(t) \\ &=\frac{1}{\ln(n+2)}\int^{a_{k_0+1}-h/2}_{x+1/(n+1)} \frac{2C(n+1)}{1+(n+1)(t-x)}\omega(t)\,dt \\ &\leqslant \frac{2C(n+1)}{\ln(n+2)}\int^{a_{k_0+1}-h/2}_{x+1/(n+1)}\frac{dt}{1+(n+1)(t-x)} \\ & =\frac{C}{\ln(n+2)}\ln[1+(n+1)(t-x)]\Big|^{a_{k_0+1}-h/2}_{x+1/(n+1)}\leqslant C(h). \end{aligned}
\end{equation*}
\notag
$$
For $t\in\delta_3$ and $x\in[a_{k_0}+h, a_{k_0+1} -h]$, we have $t-x\geqslant h/2$, $1+(n+1)(t-x)\geqslant1+(n+1)h/2$, hence
$$
\begin{equation*}
\begin{aligned} \, I_n^{(3)}(x) &=\frac{1}{\ln(n+2)}\int_{\delta_3} \widetilde{D}^*_n(t,x)\,d\theta(t) \\ &=\frac{1}{\ln(n+2)}\int^{a_{k_0+1}}_{a_{k_0+1}-h/2} \frac{2C(n+1)}{1+(n+1)(t-x)}\omega(t)\,dt \\ &\leqslant\frac{C(h)}{\ln(n+2)} \int^1_{-1} d\theta(t) \leqslant C(h). \end{aligned}
\end{equation*}
\notag
$$
So where the constant $C(h)>0$ is independent of $n\in\mathbb{Z}_+$ and $x\in[a_{k_0}+h, a_{k_0+1}-h]$.To estimate the last term $\widetilde{I}^{(2)}_n(x)$ we take the inequality $|t-x|\geqslant h$ for $t\notin(a_{k_0}, a_{k_0+1})$ and $x\in[a_{k_0}+h, a_{k_0+1}-h]$ into account. We have
$$
\begin{equation*}
\begin{aligned} \, \widetilde{I}^{(2)}_n(x) &=\frac{1}{\ln(n+2)}\int_{\substack{t\notin(a_{k_0}, a_{k_0+1})\\ t\in\varepsilon_m}} \widetilde{D}^*_n(t,x)\,d\theta(t) \\ &=\frac{1}{\ln(n+2)} \int_{\substack{t\notin(a_{k_0}, a_{k_0+1})\\ t\in\varepsilon_m}} \frac{2C(n+1)}{1+(n+1)|t-x|}\omega(t)\,dt \\ &\leqslant \frac{C(h)}{\ln(n+2)} \int^1_{-1} \omega(t)\,dt \leqslant C(h), \end{aligned}
\end{equation*}
\notag
$$
which completes the proof of Lemma 3.3. Corollary 3.1. Under the hypotheses of Lemma 3.3 the estimate for the Lebesgue function holds uniformly on all compact subsets $K$ of $\varepsilon_m$. Indeed, the Lebesgue function is estimated as follows:
$$
\begin{equation*}
\begin{aligned} \, L_n(x) &=\int^1_{-1} |D_n(t,x)|\,d\theta(t)= h(x)\int^1_{-1} |\widetilde{D}_n(t,x)| h(t)\,d\theta(t) \\ &\leqslant C\int^1_{-1} |\widetilde{D}_n(t,x)| h(t)\,d\theta(t), \qquad n\in\mathbb{Z}_+, \quad x\in K. \end{aligned}
\end{equation*}
\notag
$$
The measure $\theta$ satisfies (1.2) and (3.8), that is, $\theta\{a_k\}=0$, $k=1,2,\dots,m$, and $d\theta(x)=\omega(x)\,dx$, where $\omega(x)$ is a positive continuous integrable function. The majorant $h(x)$ is a positive continuous $\omega$-integrable function (see (3.7)): Hence the integral $\displaystyle\int^1_{-1}|\widetilde{D}_n(t,x)|h(t)\omega(t)\,dt$ can be estimated similarly to $\displaystyle\int^1_{-1}\!\widetilde{D}^*_n(t,x)\,d\theta(t)$, $n\in\mathbb{Z}_+$, $x\in K$:
$$
\begin{equation*}
\int^1_{-1}|\widetilde{D}_n(t,x)|h(t)\omega(t)\,dt\leqslant C \ln(n+2), \qquad n\in\mathbb{Z}_+, \quad x\in K,
\end{equation*}
\notag
$$
which proves (3.16). Recall that $x\in(-1,1)$ is called a Lebesgue point of a function $f\in L^p_\rho([-1,1])$ if
$$
\begin{equation}
\int_{x-h}^{x+h} |f(t)-f(x)|^p\,d\rho(t)=o(h), \qquad h\to0.
\end{equation}
\tag{3.17}
$$
It is well known (see [28], Ch. IV, pp. 142–145, and [24], Ch. 1, p. 11) that $\rho$-almost all points in $[-1,1]$ are Lebesgue points of $f$. Theorem 3.1. Let conditions (3.7) be met, let $f\in\mathfrak{R}_p$ for some $p$, $1\leqslant p <\infty$, and let where $h(t)$ is the majorant $\widehat{q}_n(t)$. Then the following results hold: (i) at each Lebesgue point $x\in\varepsilon_m$ of $f$ the partial sums $S_nf(x)$ of the Fourier-Sobolev series (1.4), (1.5) satisfy (ii) if $f$ is continuous on $[-1,1]$ and the measure $d\theta(x)$ satisfies (3.8) on $\varepsilon_m$, then uniformly on compact subsets $K$ of $\varepsilon_m$;(iii) if $f\in L^2_\theta ([-1,1])$, then Proof. Assertion (iii) was proved in [16], Corollary 3.1. We verify assertion (i) for ${p=1}$; the general case is considered similarly. We have (see (3.1) and (3.2))
In view of Lemma 3.2 (see also Remark 3.2), from (3.21) we obtain
$$
\begin{equation}
S_nf(x)-f(x)=\int^1_{-1}[f(t)-f(x) ]D_n(t,x)\,d\theta(t)+o_x(1), \qquad n\to\infty.
\end{equation}
\tag{3.22}
$$
Let $x\in\varepsilon_m$ be a Lebesgue point (3.17) and let $p=1$. As above (see (3.15)), we set
$$
\begin{equation*}
\varepsilon_m=\bigcup^m_{k=0}(a_k,a_{k+1}), \qquad a_0=-1, \quad a_{m+1}=1.
\end{equation*}
\notag
$$
Then $x\in(a_{k_0}, a_{k_0+1})$ for some $k_0$, $0\leqslant k_0\leqslant m$.
Let $n$ be a natural number such that $[x-1/(n+1), x+1/(n+1)]\subset (a_{k_0}, a_{k_0+1})$. We write the integral in the form
$$
\begin{equation}
\begin{aligned} \, \notag &\int^1_{-1} [f(t)-f(x)]D_n(t,x)\,d\theta(t) =\int_{\substack{|t-x|\leqslant1/(n+1)\\ t\in(a_{k_0}, a_{k_0+1})}} [f(t)-f(x)]D_n(t,x)\,d\theta(t) \\ \notag &\qquad+\int_{\substack{1/(n+1)<|t-x|\\ t\in(a_{k_0}, a_{k_0+1})}} [f(t)-f(x)]D_n(t,x)\,d\theta(t) \\ &\qquad+\int_{t\in\varepsilon_m\setminus (a_{k_0}, a_{k_0+1})} [f(t)-f(x)]D_n(t,x)\,d\theta(t) =I^{(1)}_n(x)+I^{(2)}_n(x)+I^{(3)}_n(x). \end{aligned}
\end{equation}
\tag{3.23}
$$
From the definition of a Lebesgue point (3.17) for $p=1$, using (3.11) and (3.18) we obtain
$$
\begin{equation}
|I^{(1)}_n(x) |\leqslant C(n+1)h(x) \int_{\substack{|t-x|\leqslant1/(n+1)\\ t\in(a_{k_0}, a_{k_0+1})}} |f(t)-f(x)|h(t)\,d\theta(t)=o_x(1), \qquad n\to\infty.
\end{equation}
\tag{3.24}
$$
Next, employing the Christoffel-Darboux formula and (3.4), (3.5) and (3.7) we have
$$
\begin{equation}
|I^{(2)}_n(x) |\leqslant C_x \int_{\substack{1/(n+1)<|t-x|\\ t\in(a_{k_0}, a_{k_0+1})}} \frac{|f(t)-f(x)|}{|t-x|}h(t)\,d\theta(t).
\end{equation}
\tag{3.25}
$$
To estimate the integral in (3.25) we define
$$
\begin{equation*}
\widetilde{I}^{(2)}_n(x)=C_x \int^{a_{k_0+1}}_{x+1/(n+1)} \frac{|f(t)-f(x)|}{t-x}h(t)\,d\theta(t).
\end{equation*}
\notag
$$
(The integral $\displaystyle\int^{x-1/(n+1)}_{a_{k_0}} \frac{|f(t)-f(x)|}{x-t}h(t)\,d\theta(t)$ is estimated similarly.)
We set By the definition of a Lebesgue point (3.17) and in view of conditions (3.18) for ${p=1}$, given $\epsilon>0$, there exists $\delta>0$ such that $\Phi_x(x+h)\leqslant \epsilon h$ for $h\leqslant \delta$.Let $1/(n+1)\leqslant \delta$. Then
$$
\begin{equation*}
\widetilde{I}^{(2)}_n(x)=C_x \int^{x+\delta}_{x+1/(n+1)}\frac{|f(t)-f(x)|}{t-x}h(t)\,d\theta(t) +C_x \int^{a_{k_0+1}}_{x+\delta} \frac{|f(t)-f(x)|}{t-x}h(t)\,d\theta(t).
\end{equation*}
\notag
$$
In the second integral we have $|t-x|\geqslant \delta$, hence
$$
\begin{equation}
\int^{a_{k_0+1}}_{x+\delta} \frac{|f(t)-f(x)|}{t-x}h(t)\,d\theta(t)=O_x(1).
\end{equation}
\tag{3.26}
$$
Integration by parts in the first integral shows that
$$
\begin{equation*}
\begin{aligned} \, &\int^{x+\delta}_{x+1/(n+1)} \frac{|f(t)-f(x)|}{t-x}h(t)\,d\theta(t) \\ &\qquad=\frac{1}{t-x}\biggl\{\int^t_0|f(u)-f(x)|h(u)\,d\theta (u)\biggr\}\Bigr|_{x+1/(n+1)}^{x+\delta} \\ &\qquad\qquad +\int^{x+\delta}_{x+1/(n+1)}\frac{1}{(t-x)^2} \biggl\{\int^t_0|f(u)-f(x)|h(u)\,d\theta(u)\biggr\}\,dt. \end{aligned}
\end{equation*}
\notag
$$
By the definition of the function $\Phi_x(t)$,
$$
\begin{equation*}
\begin{aligned} \, &\int^{x+\delta}_{x+1/(n+1)} \frac{|f(t)-f(x)|}{t-x}h(t)\,d\theta(t) \\ &\qquad =\frac {1}{\delta}\Phi_x(x+\delta) -(n+1)\Phi_x\biggl(x+\frac{1}{n+1}\biggr) +\int^{x+\delta}_{x+1/(n+1)}\frac{\Phi_x(t)}{(t-x)^2}\,dt \\ &\qquad=O_x(1)+\epsilon+\epsilon \int^{x+\delta}_{x+1/(n+1)}\frac{dt}{t-x} =O_x(1)+\epsilon\ln(n+1){\delta}. \end{aligned}
\end{equation*}
\notag
$$
Combining the last relation with (3.26), this gives
$$
\begin{equation*}
\int^{x+\delta}_{x+1/(n+1)} \frac{|f(t)-f(x)|}{t-x}h(t)\,d\theta(t)=o_x(1)\ln(n+2), \qquad n \to\infty.
\end{equation*}
\notag
$$
As a result, An appeal to (3.12) shows that
$$
\begin{equation*}
\begin{aligned} \, |I^{(3)}_n(x)| &\leqslant C_x\int_{t\in\varepsilon_m\setminus (a_{k_0}, a_{k_0+1})} \frac{|f(t)-f(x)|}{|t-x|}h(t)\,d\theta(t) \\ &=O_x(1)\int^1_{-1} |f(t)-f(x)|h(t)\,d\theta(t) \end{aligned}
\end{equation*}
\notag
$$
and therefore, using conditions (3.18) for $p=1$ we obtain
Now estimate (3.19) follows from (3.22)–(3.24), (3.27) and (3.28). (ii) Let us verify the uniform estimate (3.20). The function $f$ is continuous on ${[-1, 1]}$, and therefore it is uniformly continuous on $[-1,1]$. Given $\varepsilon>0$, we can find $\delta >0$ such that $|f(t)-f(x)|<\varepsilon$ for $|t-x|<\delta$. For some $k_0$, $0\leqslant k_0\leqslant m$, let $x\in K_0=K\cap (a_{k_0},a_{k_0+1})$ and $\delta>0$ be such that the interval $|t-x|<\delta$ lies in $ K_0$ (if $x$ is a boundary point, then we take the one-sided $\delta$-neighbourhood of $x$). As above, we consider only the integral We have
$$
\begin{equation}
\begin{aligned} \, \notag &\int^1_{-1} [f(t)-f(x)]D_n(t,x)\,d\theta(t) \\ \notag &\qquad =\int_{|t-x|<\delta}[f(t)-f(x)]D_n(t,x)\,d\theta(t) +\int_{|t-x|\geqslant\delta}[f(t)-f(x)]D_n(t,x)\,d\theta(t) \\ &\qquad =R_n^{(1)}(x)+R_n^{(2)}(x). \end{aligned}
\end{equation}
\tag{3.29}
$$
From estimate (3.16) we have
$$
\begin{equation}
|R^{(1)}_n(x)| \leqslant\varepsilon L_n(x)\leqslant C\varepsilon \ln(n+2).
\end{equation}
\tag{3.30}
$$
Using (3.8) and (3.12) we obtain in succession
$$
\begin{equation*}
\begin{aligned} \, |R^{(2)}_n(x)| &=\biggl|\int_{|t-x|\geqslant\delta}[f(t)-f(x)]D_n(t,x)\,d\theta(t)\biggr| \\ &\leqslant Ch(x)\int_{|t-x|\geqslant\delta}\frac{|f(t)-f(x)|}{|t-x|}h(t)\,d\theta(t) \\ &=O(1) \int_{|t-x|\geqslant\delta}|f(t)-f(x)|h(t)\,d\theta(t). \end{aligned}
\end{equation*}
\notag
$$
Now, employing conditions (3.18) for $p=1$, we have uniformly in $x\in K$ and $n\in\mathbb{Z}_+$.
Substituting (3.30) and (3.31) into (3.29) we arrive at (3.20). This completes the proof of Theorem 3.1. The space $W^p_\theta ([-1,1])$, $1\leqslant p <\infty$, will be considered as the set of $\mathfrak{R}_p$-functions $f$ equipped with a norm:
$$
\begin{equation*}
\begin{aligned} \, W^p_\theta ([-1,1]) &=\biggl\{f\colon \|f\|_{W^p_\theta([-1,1])}<+\infty, \\ &\qquad\text{where }\|f\|^p_{W^p_\theta([-1,1])}=\|f\|^p_{L^p_\theta([-1,1])} +\sum^m_{k=1}\sum^{N_k}_{i=0} M_{k,i}|f^{(i)}(a_k)|^p \biggr\}. \end{aligned}
\end{equation*}
\notag
$$
One can similarly define the space $W^p_\theta(F)$ for a subset $F$ of $[-1,1]$:
$$
\begin{equation}
\begin{aligned} \, \notag W^p_\theta (F) &=\biggl\{f\colon \|f\|_{W^p_\theta(F)}<+\infty, \\ &\qquad\text{where }\|f\|^p_{W^p_\theta(F)}=\|f\|^p_{L^p_\theta(F)} +\sum^m_{k=1}\sum^{N_k}_{i=0} M_{k,i}|f^{(i)}(a_k)|^p \biggr\} \end{aligned}
\end{equation}
\tag{3.32}
$$
(for a theory of these spaces, see [29]–[31] and the references given there). Note that the space $W^p_\theta ([-1,1])$, $1\leqslant p<\infty$, is not complete. With each function $f\in W^p_\theta ([-1,1])$, for some $p$, $1\leqslant p < \infty$, we associate the Fourier-Sobolev series (1.4), (1.5) and estimate the norm of a majorant of the partial sums $S_nf(x)$. We set
$$
\begin{equation}
G_nf(x)=\frac{1}{\ln(n+2)}\int^1_{-1} f(t)D_n(t,x)\,d\theta(t), \qquad n\in\mathbb{Z}_+, \quad x\in\varepsilon_m,
\end{equation}
\tag{3.33}
$$
and define
$$
\begin{equation}
G_*f(x) :=\sup_{n\in\mathbb{Z}_+}|G_nf(x)|, \qquad x\in\varepsilon_m.
\end{equation}
\tag{3.34}
$$
Lemma 3.4. Let the polynomial system $\{\widehat{q}_n(x)\}^\infty_{n=0}$ and measure $d\theta(x)$ satisfy conditions (3.7) and (3.8), respectively. Then the following assertions hold. (i) If a function $f\in\mathfrak{R}_p$, $1<p<\infty$, satisfies (3.18), then for any compact subset $K$ of $\varepsilon_m$
$$
\begin{equation}
\|G_*f\|_{L^p_\theta(K)}\leqslant C_p\|f\|_{L^p_\theta(K)},
\end{equation}
\tag{3.35}
$$
where the positive constant $C_p$ is independent of the function $f\in L^p_\theta(K)$. (ii) If, in addition,
$$
\begin{equation}
\sum^{\infty}_{j=0}|\widehat{q}^{\,(i)}_j(a_s)|<\infty, \qquad i=0,1,\dots,N_s, \quad s=1,2,\dots,m,
\end{equation}
\tag{3.36}
$$
then
$$
\begin{equation}
\biggl\| \sup_{n\in\mathbb{Z}_+}\biggl|\sum^m_{s=1}\sum^{N_s}_{i=0}M_{s,i} f^{(i)}(a_s)\, \frac{\partial^iD_n}{\partial t^i}(a_s, x)\biggr| \biggr\|_{L^p_\theta(K)} \leqslant C_p\biggl\{\sum^m_{s=1}\sum^{N_s}_{i=0} M_{s,i}|f^{(i)}(a_s)|^p\biggr\}^{1/p},
\end{equation}
\tag{3.37}
$$
where $C_p$ is independent of the function $f$. Proof. (i) It is easily checked that
From Lemmas 2.3 and 3.3 and conditions (3.18), using estimate (2.8) we obtain (see (3.34))
$$
\begin{equation*}
G_*f(x)\leqslant CM_\theta[fh](x)h(x), \qquad x\in K,
\end{equation*}
\notag
$$
where the maximal function $M_\theta$ is defined in (2.6) and $C > 0$ is independent of the function $f\in L^p_\theta(K)$. Since $h(x)$ is bounded on $K$, by Lemma 2.3 we have
$$
\begin{equation*}
\begin{aligned} \, \|G_*f\|_{L^p_\theta(K)} &=\biggl(\int_K[G_*f(x)]^p\,d\theta(x)\biggr)^{1/p} \leqslant C_p\biggl(\int_K\{M_\theta[fh](x)\}^ph^p(x)\,d\theta(x)\biggr)^{1/p} \\ &\leqslant C_p\biggl(\int_K|f(x)|^ph^{2p}(x)\,d\theta(x)\biggr)^{1/p} \leqslant C_p\biggl(\int_K|f(x)|^p\,d\theta(x)\biggr)^{1/p}, \end{aligned}
\end{equation*}
\notag
$$
which proves (i).
(ii) It is easily checked that by (3.7) and (3.36)
$$
\begin{equation}
\biggl|\frac{\partial^i D_n}{\partial t^i}(a_s,x)\biggr| =\biggl|\sum^n_{j=0}\widehat{q}^{\,(i)}_j(a_s)\widehat{q}_j(x)\biggr| \leqslant Ch(x), \qquad x\in\varepsilon_m, \quad n\in\mathbb{Z}_+.
\end{equation}
\tag{3.39}
$$
As a result,
$$
\begin{equation*}
\begin{aligned} \, &\biggl\|\sup_{n\in\mathbb{Z}_+}\biggl|\sum^m_{s=1}\sum^{N_s}_{i=0} M_{s,i}f^{(i)}(a_s)\, \frac{\partial^i D_n}{\partial t^i}(a_s, x)\biggr|\biggr\|^p_{L^p_\theta(K)} \\ &\qquad\leqslant C_p\biggl\|\sum^m_{s=1}\sum^{N_s}_{i=0} M_{s,i}|f^{(i)}(a_s)|h(x)\biggr\|^p_{L^p_\theta(K)} \\ &\qquad\leqslant C_p\biggl\{\sum^m_{s=1}\sum^{N_s}_{i=0} M_{s,i}|f^{(i)}(a_s)|\biggr\}^p \biggl\{\int_K h^p(x)\,d\theta(x)\biggr\}. \end{aligned}
\end{equation*}
\notag
$$
If $1/p+1/q=1$, $1<p<\infty$, then from Hölder’s inequality we obtain
$$
\begin{equation}
\begin{aligned} \, \notag \biggl\{\sum^m_{s=1}\sum^{N_s}_{i=0} M_{s,i}|f^{(i)}(a_s)|\biggr\}^p &\leqslant \biggl\{\sum^m_{s=1}\sum^{N_s}_{i=0} M_{s,i}\biggr\}^{p/q} \sum^m_{s=1}\sum^{N_s}_{i=0} M_{s,i}|f^{(i)}(a_s)|^p \\ &\leqslant C_p\sum^m_{s=1}\sum^{N_s}_{i=0} M_{s,i}|f^{(i)}(a_s)|^p. \end{aligned}
\end{equation}
\tag{3.40}
$$
Since the function $h(x)$ is bounded on $K$, we have estimate (3.37). Lemma 3.4 is proved. Lemma 3.5. Let assumptions (3.7), (3.8) and (3.36) of Lemma 3.4 be met, let (3.18) hold for some $p$, $1<p<\infty$, and let Then for any function $f\in W^p_\theta([-1,1])$ where the positive constant $C_p$ is independent of $f$. Proof. By (3.1) we have
$$
\begin{equation*}
\begin{aligned} \, (S_n f)^{(j)}(a_k) &=\int^1_{-1} f(t)\biggl[\sum^n_{l=0}\widehat{q}_l(t)\widehat{q}^{\,(j)}_l(a_k)\biggr]\,d\theta(t) \\ &\qquad +\sum^m_{s=1}\sum^{N_s}_{i=0} M_{s,i}f^{(i)}(a_s)\biggl[\sum^n_{l=0}\widehat{q}^{\,(i)}_l(a_s)\widehat{q}^{\,(j)}_l(a_k)\biggr]. \end{aligned}
\end{equation*}
\notag
$$
Hence
$$
\begin{equation*}
\begin{aligned} \, |(S_n f(x))^{(j)}(a_k)| &\leqslant\int^1_{-1}|f(t)|\sup_{n\in\mathbb{Z}_+}\biggl|\sum^n_{l=0} \widehat{q}_l(t)\widehat{q}^{\,(j)}_l(a_k)\biggr|\,d\theta(t) \\ &\qquad+\sum^m_{s=1}\sum^{N_s}_{i=0} M_{s,i}|f^{(i)}(a_s)| \sup_{n\in\mathbb{Z}_+}\biggl|\sum^n_{l=0}\widehat{q}^{\,(i)}_l(a_s)\widehat{q}^{\,(j)}_l(a_k)\biggr|. \end{aligned}
\end{equation*}
\notag
$$
Applying inequalities (3.7) and (3.36) we obtain the estimates
$$
\begin{equation}
\begin{gathered} \, \biggl|\sum^n_{l=0}\widehat{q}^{\,(i)}_l(a_s)\widehat{q}^{\,(j)}_l(a_k)\biggr| \leqslant C \quad\text{and}\quad \biggl|\sum^n_{l=0}\widehat{q}_l(t)\widehat{q}^{\,(j)}_l(a_k)\biggr| \leqslant Ch(t), \\ s,k=1,\dots,m, \qquad i=0,1,\dots,N_s, \qquad j=0,1,\dots,N_k, \end{gathered}
\end{equation}
\tag{3.43}
$$
and from (3.41), (3.43) and the integral Hölder inequality we obtain
$$
\begin{equation*}
\begin{aligned} \, &\int^1_{-1}|f(t)|\sup_{n\in\mathbb{Z}_+}\biggl|\sum^n_{l=0} \widehat{q}_l(t)\widehat{q}^{\,(j)}_l(a_k)\biggr|\,d\theta(t) \leqslant C_p\int^1_{-1}|f(t)|h(t)\,d\theta(t) \\ &\qquad \leqslant C_p\{\|f\|_{L^p_\theta([-1,1])}\|h\|_{L^q_\theta([-1,1])}\} \leqslant C_p\|f\|_{L^p_\theta([-1,1])}. \end{aligned}
\end{equation*}
\notag
$$
An appeal to (3.43) shows that
$$
\begin{equation*}
\begin{aligned} \, &\sum^{m}_{k=1}\sum^{N_k}_{j=0} M_{k,j} \Bigl[\sup_{n\in\mathbb{Z}_+}|(S_n f)^{(j)}(a_k)|\Bigr]^p \\ &\qquad \leqslant C_p\sum^m_{k=1}\sum^{N_k}_{j=0} M_{k,j}\biggl\{\|f\|_{L^p_\theta([-1,1])} +\sum^m_{s=1}\sum^{N_s}_{i=0} M_{s,i}|f^{(i)}(a_s)|\biggr\}^p \\ &\qquad \leqslant C_p\biggl\{\|f\|_{L^p_\theta([-1,1])}+\sum^m_{s=1}\sum^{N_s}_{i=0} M_{s,i}|f^{(i)}(a_s)|\biggr\}^p. \end{aligned}
\end{equation*}
\notag
$$
Now estimate (3.42) follows from Minkowski’s inequality, the definition of the norm (3.32) and relation (3.40).
Lemma 3.5 is proved. § 4. Fejér means of Fourier-Sobolev seriesConsider the Fejér means of the Fourier-Sobolev series (1.4), (1.5) of a function $f\in\mathfrak{R}_p$, $1\leqslant p<\infty$:
$$
\begin{equation}
\sigma_nf(x)=\frac{1}{n+1}\sum^n_{k=0}S_kf(x), \qquad n\in\mathbb{Z}_+, \quad x\in[-1,1],
\end{equation}
\tag{4.1}
$$
where the $S_k(f;x)$ are partial sums of the Fourier-Sobolev series. The Fejér kernel of a polynomial system $\{\widehat{q}_n(x)\}^\infty_{n=0}$ is defined by
$$
\begin{equation}
F_n(t,x)=\frac{1}{n+1}\sum^n_{i=0}D_i(t,x), \qquad n\in\mathbb{Z}_+, \quad t,x \in[-1,1],
\end{equation}
\tag{4.2}
$$
where $D_i(t,x)$ is the Dirichlet kernel. From (3.1) and (4.1) we obtain
$$
\begin{equation}
\begin{gathered} \, \sigma_nf(x)=\int^1_{-1} f(t)F_n(t,x)\,d\theta(t)+\sum^m_{s=1}\sum^{N_s}_{i=0} M_{s,i}f^{(i)}(a_s)\frac{\partial^i F_n}{\partial t^i}(a_s, x), \\ n\in\mathbb{Z}_+, \qquad t,x\in[-1,1]. \end{gathered}
\end{equation}
\tag{4.3}
$$
A representation for the Fejér kernel follows from recurrence relation (3.3) and the Christoffel-Darboux formula. Lemma 4.1. For $t,x\in[-1,1]$ and $n\in\mathbb{Z}_+$, Proof. For the left-hand side of (4.4), using the Christoffel-Darboux formula we obtain
$$
\begin{equation*}
\begin{aligned} \, &[\pi_{N+1}(t)-\pi_{N+1}(x)]^2\sum^n_{k=0}D_k(t,x) \\ &\qquad =\sum^{N+1}_{j=1}\sum^n_{k=0}\sum^{k}_{s=k-j+1} d_{s+j,j}[\pi_{N+1}(t)- \pi_{N+1}(x)][\widehat{q}_{s+j}(t)\widehat{q}_s(x)-\widehat{q}_s(t)\widehat{q}_{s+j}(x)] \\ &\qquad =\sum^{(N+1)}_n (t,x)+\sum^{(N+1)}_n (x,t), \end{aligned}
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\begin{aligned} \, \sum^{(N+1)}_n (t,x) &=\sum^{N+1}_{j=1}\sum^n_{k=0}\sum^k_{s=k-j+1} d_{s+j,j}\pi_{N+1}(t)\widehat{q}_{s+j}(t)\widehat{q}_s(x) \\ &\qquad - \sum^{N+1}_{j=1}\sum^n_{k=0}\sum^k_{s=k-j+1} d_{s+j,j}\widehat{q}_{s+j}(t)\pi_{N+1}(x)\widehat{q}_{s}(x). \end{aligned}
\end{equation*}
\notag
$$
Next we use recurrence relation (3.3) and the formula
$$
\begin{equation*}
\sum^{N+1}_{j=1}\sum^n_{k=0}\sum^k_{s=k-j+1}\omega_s(j) =\sum^{N+1}_{j=1}j\sum^{n-j+1}_{s=0}\omega_s(j) +\sum^{N+1}_{j=1}\sum^n_{s=n-j+2}(n-s+1)\omega_s(j).
\end{equation*}
\notag
$$
which holds for any sequence $\{\omega_s(j)\}$, $s\in\mathbb{Z}_+$, $j\in\mathbb{N}$, where we set $\omega_s(j)=0$ for ${s=-1,-2,\dots}$ . We have
$$
\begin{equation*}
\begin{aligned} \, \sum^{(N+1)}_n (t,x) &=\sum^{N+1}_{j=1}j\sum^{n-j+1}_{s=0} d_{s+j,j}\pi_{n+1}(t)\widehat{q}_{s+j}(t)\widehat{q}_s(x) \\ &\qquad +\sum^{N+1}_{j=1}\sum^n_{s=n-j+2}(n-s+1)d_{s+j,j}\pi_{N+1}(t)\widehat{q}_{s+j}(t)\widehat{q}_s(x) \\ &\qquad -\sum^{N+1}_{j=1}j\sum^{n-j+1}_{s=0} d_{s+j,j}\widehat{q}_s(t)\pi_{n+1}(x)\widehat{q}_{s+j}(x) \\ &\qquad -\sum^{N+1}_{j=1}\sum^n_{s=n-j+2}(n-s+1)d_{s+j,j}\widehat{q}_s(t)\pi_{N+1}(x)\widehat{q}_{s+j}(x) \\ &=\sum^{N+1}_{j=1}j\sum^{N+1}_{l=0}\sum^{n-j+1}_{s=0}d_{s+j,j}d_{s+j+l,l} \widehat{q}_{s+j+l}(t)\widehat{q}_s(x) \\ &\qquad +\sum^{N+1}_{j=1}j\sum^{N+1}_{l=1}\sum^{n-j+1}_{s=0}d_{s+j,j}d_{s+j,l} \widehat{q}_{s+j-l}(t)\widehat{q}_s(x) \\ &\qquad +\sum^{N+1}_{j=1}\sum^{N+1}_{l=0}\sum^{n}_{s=n-j+2}(n-s+1) d_{s+j,j}d_{s+j+l,l}\widehat{q}_{s+j+l}(t)\widehat{q}_s(x) \\ &\qquad +\sum^{N+1}_{j=1}\sum^{N+1}_{l=1}\sum^{n}_{s=n-j+2}(n-s+1) d_{s+j,j}d_{s+j,l}\widehat{q}_{s+j-l}(t)\widehat{q}_s(x) \\ &\qquad +\sum^{N+1}_{j=1}j\sum^{N+1}_{l=0}\sum^{n-j+1}_{s=0} d_{s+j,j}d_{s+l,l}\widehat{q}_{s+j}(t)\widehat{q}_{s+l}(x) \\ &\qquad +\sum^{N+1}_{j=1}j\sum^{N+1}_{l=1}\sum^{n-j+1}_{s=0} d_{s+j,j}d_{s,l}\widehat{q}_{s+j}(t)\widehat{q}_{s-l}(x) \\ &\qquad +\sum^{N+1}_{j=1}\sum^{N+1}_{l=0}\sum^{n}_{s=n-j+2}(n-s+1) d_{s+j,j}d_{s+l,l}\widehat{q}_{s+j}(t)\widehat{q}_{s+l}(x) \\ &\qquad +\sum^{N+1}_{j=1}\sum^{N+1}_{l=1}\sum^{n}_{s=n-j+2}(n-s+1) d_{s+j,j}d_{s,l}\widehat{q}_{s+j}(t)\widehat{q}_{s-l}(x). \end{aligned}
\end{equation*}
\notag
$$
As a result,
$$
\begin{equation*}
\begin{aligned} \, \sum^{(N+1)}_n (t,x) &=\sum^{N+1}_{j=1}j\sum^{n}_{s=0}d_{s+j,j}[d_{s,0}-d_{s+j,0}] \widehat{q}_{s+j}(t)\widehat{q}_s(x) \\ &\qquad +\sum^{N+1}_{j=1}j\sum^{N+1}_{l=0}\sum^n_{s=0}(d_{s+j,j}d_{s+j+l,l}-d_{s+j+l,j}d_{s+l,l} ) \widehat{q}_{s+j+l}(t)\widehat{q}_s(x) \\ &\qquad +\sum^{N+1}_{j=1}j\sum^{N+1}_{l=1}\sum^n_{s=0}(d_{s+j,j}d_{s+j,l}-d_{s+j-l,j}d_{s,l} ) \widehat{q}_{s+j-l}(t)\widehat{q}_s(x) \\ &\qquad +\sum^{N+1}_{j=1}j\sum^{N+1}_{l=1}\sum^n_{s=n-j+l+2} d_{s+j-l,j}d_{s,l}\widehat{q}_{s+j-l}(t)\widehat{q}_s(x) \\ &\qquad -\sum^{N+1}_{j=1}j\sum^{N+1}_{l=1}\sum^n_{s=n-j+2}d_{s+j,j} d_{s+j+l,l}\widehat{q}_{s+j+l}(t)\widehat{q}_s(x) \\ &\qquad -\sum^{N+1}_{j=1}j\sum^{N+1}_{l=1}\sum^n_{s=n-j+2} d_{s+j,j}d_{s+j,l}\widehat{q}_{s+j-l}(t)\widehat{q}_s(x) \\ &\qquad - \sum^{N+1}_{j=1}j\sum^{N+1}_{l=1}\sum^n_{s=n-j-l+2} d_{s+j+l,j}d_{s+l,l}\widehat{q}_{s+j+l}(t)\widehat{q}_s(x) \\ &\qquad +\sum^{N+1}_{j=1}\sum^{N+1}_{l=0}\sum^n_{s=n-j+2}(n-s+1) d_{s+j,j}d_{s+j+l,l}\widehat{q}_{s+j+l}(t)\widehat{q}_s(x) \\ &\qquad - \sum^{N+1}_{j=1}\sum^{N+1}_{l=1}\sum^{n-l}_{s=n-j-l+2}(n-s-l+1) d_{s+j+l,j}d_{s+l,l}\widehat{q}_{s+j+l}(t)\widehat{q}_s(x) \\ &\qquad - \sum^{N+1}_{j=1}\sum^{N+1}_{l=1}\sum^n_{s=n-j+2}(n-s+1)d_{s+j,j} d_{s+j,l}\widehat{q}_{s+j-l}(t)\widehat{q}_s(x) \\ &\qquad - \sum^{N+1}_{j=1}\sum^{N+1}_{l=0}\sum^{n+l}_{s=n-j+l+2}(n-s+l+1) d_{s+j-l,j}d_{s,l}\widehat{q}_{s+j-l}(t)\widehat{q}_s(x). \end{aligned}
\end{equation*}
\notag
$$
Now formulae (4.4)–(4.7) easily follow. Lemma 4.1 is proved. Lemma 4.2. Let $\{\widehat{q}_n(x)\}^\infty_{n=0}$ be a polynomial system satisfying condition (3.7) and assume that the recurrence coefficients (see (3.3)) satisfy Them the Fejér kernel satisfies
$$
\begin{equation}
|F_n(t,x)|\leqslant (n+1)h(t)h(x), \qquad t,x\in\varepsilon_m, \quad |t-x|\geqslant 0,
\end{equation}
\tag{4.9}
$$
and
$$
\begin{equation}
|F_n(t,x)|\leqslant C_x\frac{h(t)h(x)}{(n+1)[\pi_{N+1}(t)-\pi_{N+1}(x)]^2}, \qquad t,x\in\varepsilon_m, \quad |t-x|> 0.
\end{equation}
\tag{4.10}
$$
Here the positive constant $C_x$ is independent of $t\in\varepsilon_m$ and $n\in\mathbb{Z}_+$. Moreover, for any compact subset $K$ of $\varepsilon_m$ and where the positive constant $C$ is independent of $t\in\varepsilon_m$, $x\in K$ and $n\in\mathbb{Z}_+$. Proof. Indeed, estimate (4.9) follows from (3.11), and relation (4.10) is a direct consequence of representation (4.4)–(4.7) and conditions (3.4), (3.5), (3.7) and (4.8). The uniform estimates (4.11) and (4.12) are secured by (4.9) and (3.4), (3.6), (3.7), (4.8) and the representations (4.4)–(4.7), respectively.
Lemma 4.2 is proved. Lemma 4.3. Let the polynomial system $\{\widehat{q}_n(x)\}^\infty_{n=0}$ satisfy conditions (3.7), (3.8) and (4.8). Then the function has an integrable humpbacked majorant $\widetilde{F}^*_n(t, x)$ satisfying the uniform estimate where $K$ is a compact subset of $\varepsilon_m$. Proof. Using estimates (4.11) and (4.12) we have where the constants $C>1$ do not depend on $t\in\varepsilon_m$, $x\in K$ and $n\in\mathbb{Z}_+$.
Therefore,
$$
\begin{equation}
|\widetilde{F}_n(t,x)|\leqslant \begin{cases} C(n+1), &t\in\varepsilon_m,\ x\in K,\ 0\leqslant|t-x|\leqslant\dfrac{1}{n+1}, \\ \dfrac{C}{(n+1)(t-x)^2}, & t\in\varepsilon_m,\ x\in K,\ |t-x|> \dfrac{1}{n+1}, \end{cases}
\end{equation}
\tag{4.15}
$$
where the constants $C>1$ do not depend on $t\in\varepsilon_m$, $x\in K$ and $n\in\mathbb{Z}_+$. We set
$$
\begin{equation*}
\widetilde{F}^*_n(t, x)=\frac{2C(n+1)}{1+(n+1)^2(t-x)^2}, \qquad t\in\varepsilon_m, \quad x\in K, \quad n\in\mathbb{Z}_+.
\end{equation*}
\notag
$$
Let us now verify that $\widetilde{F}^*_n(t, x)$ is an integrable humpbacked majorant of the kernel $\widetilde{F}_n(t, x)$. To do this we use (4.15). 1. Regarding the inequality $|\widetilde{F}_n(t,x)|\leqslant \widetilde{F}^*_n(t,x)$, we have: for $|t-x|\leqslant 1/(n+1)$, $(n+1)|t-x|\leqslant 1$ the estimate is equivalent to
$$
\begin{equation*}
C(n+1)\leqslant \frac{2C(n+1)}{1+(n+1)^2(t-x)^2}, \qquad 1+(n+1)^2(t-x)^2\leqslant 2;
\end{equation*}
\notag
$$
for $|t-x|>1/(n+1)$, $(n+1)|t-x|> 1$ the estimate is equivalent to the relation
$$
\begin{equation*}
\frac{C}{(n+1)(t-x)^2}\leqslant \frac{2C(n+1)}{1+(n+1)^2(t-x)^2}, \qquad 1+(n+1)^2(t-x)^2 \leqslant 2(n+1)^2(t-x)^2.
\end{equation*}
\notag
$$
2. That the humpbacked majorant $\widetilde{F}^*_n(t, x)$ is monotone is clear. 3. We show that the humpbacked majorant is integrable and verify (4.14). We have
$$
\begin{equation*}
\begin{aligned} \, J_n(x) &=\int^1_{-1} \widetilde{F}^*_n(t, x)h(t)\,d\theta(t)=\int^{a_{k_0+1}}_{a_{k_0}} \widetilde{F}^*_n(t, x)h(t)\,d\theta(t) \\ &\qquad +\int_{\substack{t\notin(a_{k_0}, a_{k_0+1})\\ t\in\varepsilon_m }} \widetilde{F}^*_n(t, x)h(t)\,d\theta(t)=\widetilde{J}_n^{(1)}(x)+\widetilde{J}_n^{(2)}(x). \end{aligned}
\end{equation*}
\notag
$$
As in the proof of Lemma 3.3, we assume that $t\geqslant x$ and consider a number $h>0$ and the intervals $\delta_1=[x, x+1/(n+1)]$, $\delta_2=[x+1/(n+1), a_{k_0+1}-h/2]$ and $\delta_3=[a_{k_0+1}-h/2,a_{k_0+1}]$, where $x\in[a_{k_0}+h, a_{k_0+1}-h]$. Then
$$
\begin{equation*}
\widetilde{J}^{(1)}_n(x)=\int_{\delta_1}+\int_{\delta_2}+\int_{\delta_3} =J^{(1)}_n(x)+J^{(2)}_n(x)+J^{(3)}_n(x).
\end{equation*}
\notag
$$
To estimate the first two terms we use the fact that the weight function $h(t)\omega(t)$ is bounded (see (3.7) and (3.8)). We have
$$
\begin{equation*}
\begin{aligned} \, J^{(1)}_n(x) &=\int_{\delta_1} \widetilde{F}^*_n(t, x)h(t)\,d\theta(t) =\int^{x+1/(n+1)}_{x} \frac{2C(n+1)}{1+(n+1)^2(t-x)^2}h(t)\omega(t)\,dt \\ &\leqslant \int^{x+1/(n+1)}_{x} \frac{2C(n+1)}{1+(n+1)^2(t-x)^2}\,dt=O(1) \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, J^{(2)}_n(x) &=\int_{\delta_2} \widetilde{F}^*_n(t, x)h(t)\,d\theta(t) =\int^{a_{k_0+1}-h/2}_{x+1/(n+1)} \frac{2C(n+1)}{1+(n+1)^2(t-x)^2}h(t)\omega(t)\,dt \\ &\leqslant 2C(n+1)\int^{a_{k_0+1}-h/2}_{x+1/(n+1)} \frac{dt}{1+(n+1)^2(t-x)^2} \leqslant C\int^\infty_{-\infty}\frac{dz}{1+z^2}=C\pi. \end{aligned}
\end{equation*}
\notag
$$
For $t\in\delta_3$ and $x\in[a_{k_0}+h, a_{k_0+1}-h]$ we have $t-x\geqslant h/2$, $1+(n+ 1)^2(t-x)^2\geqslant 1+(n+1)^2(h/2)^2$, hence
$$
\begin{equation*}
\begin{aligned} \, J^{(3)}_n(x) &=\int_{\delta_3} \widetilde{F}^*_n(t, x)h(t)\,d\theta(t) =\int^{a_{k_0+1}}_{a_{k_0+1}-h/2} \frac{2C(n+1)}{1+(n+1)^2(t-x)^2}h(t)\omega(t)\,dt \\ &\leqslant C(h)\int^1_{-1}h(t)\omega(t)\,dt\leqslant C(h). \end{aligned}
\end{equation*}
\notag
$$
As a result, where the positive constant $C(h)$ is independent of $n\in\mathbb{Z}_+$ and $x\in[{a_{k_0}+h}, {a_{k_0+1}-h}]$.To estimate the last term $\widetilde{J}^{(2)}_n(x)$ we use the fact that $|t-x|\geqslant h$ for $t\notin(a_{k_0}, a_{k_0+1})$ and $x\in[a_{k_0}+h, a_{k_0+1}-h]$, so that
$$
\begin{equation*}
\begin{aligned} \, &\widetilde{J}^{(2)}_n(x) =\int_{\substack{t\notin(a_{k_0}, a_{k_0+1})\\ t\in\varepsilon_m }} \widetilde{F}^*_n(t,x)h(t)\,d\theta(t) \\ &\quad=\int_{\substack{t\in(a_{k_0}, a_{k_0+1})\\ t\notin\varepsilon_m}} \frac{2C(n+1)}{1+(n+1)^2(t-x)^2}h(t)\omega(t)\,dt \leqslant C(h)\int^1_{-1}h(t)\omega(t)\,dt\leqslant C(h), \end{aligned}
\end{equation*}
\notag
$$
which completes the proof of Lemma 4.3. Corollary 4.1. Under the hypotheses of Lemma 4.3, the Lebesgue function satisfies uniformly on all compact subsets $K$ of $\varepsilon_m$. Lemma 4.4. Let conditions (3.7) and (4.8) be met and let $f\in\mathfrak{R}_p$, $1\leqslant p<\infty$, satisfy conditions (3.18). Then (i) at each Lebesgue point $x\in\varepsilon_m$ of the function $f$,
$$
\begin{equation}
\lim_{n\to\infty} \int^1_{-1} [f(t)-f(x)]F_n(t,x)\,d\theta(t)=0;
\end{equation}
\tag{4.17}
$$
(ii) if, in addition, the measure $d\theta(x)$ obeys (3.8) and $f$ is continuous on $[-1,1]$, then (4.17) holds uniformly on all compact subsets $K$ of $\varepsilon_m$. Proof. Proceeding as in the proof of Theorem 3.1, we verify assertion (i) for $p=1$.
Let $x\in\varepsilon_m$ be a Lebesgue point, let $x\in(a_{k_0}, a_{k_0+1})$, and let $n$ be a natural number such that $[x-1/(n+1), x+1/(n+1)]\subset(a_{k_0}, a_{k_0+1})$. Then
$$
\begin{equation}
\begin{aligned} \, \notag &\int^1_{-1} [f(t)-f(x)]F_n(t,x)\,d\theta(t) =\int_{\substack{|t-x|\leqslant1/(n+1)\\ t\in(a_{k_0}, a_{k_0+1})}} [f(t)-f(x)]F_n(t,x)\,d\theta(t) \\ \notag &\qquad+\int_{\substack{|t-x|>1/(n+1)\\ t\in(a_{k_0}, a_{k_0+1})}} [f(t)-f(x)]F_n(t,x)\,d\theta(t) \\ &\qquad+\int_{\substack{|t-x|>1/(n+1)\\ t\notin(a_{k_0}, a_{k_0+1})}} [f(t)-f(x)]F_n(t,x)\,d\theta(t)=J^{(1)}_n(x)+J^{(2)}_n(x)+J^{(3)}_n(x). \end{aligned}
\end{equation}
\tag{4.18}
$$
An appeal to (4.9), conditions (3.18) and relation (3.17) shows that
$$
\begin{equation}
|J^{(1)}_n(x)|\leqslant C_x(n+1)\int_{\substack{|t-x|\leqslant1/(n+1)\\ t\in(a_{k_0}, a_{k_0+1}) }} |f(t)-f(x)|h(t)\,d\theta(t)=o_x(1), \qquad n\to\infty.
\end{equation}
\tag{4.19}
$$
Using inequality (4.10) and (3.5) this gives
$$
\begin{equation*}
\begin{aligned} \, |J^{(2)}_n(x)| &\leqslant C_x\frac{1}{n+1} \int_{\substack{|t-x|>1/(n+1)\\ t\in(a_{k_0}, a_{k_0+1}) }} \frac{|f(t)-f(x)|}{[\pi_{N+1}(t)- \pi_{N+1}(x)]^2}h(t)\,d\theta(t) \\ &\leqslant C_x\frac{1}{n+1}\int_{\substack{|t-x|>1/(n+1)\\ t\in(a_{k_0}, a_{k_0+1})}} \frac{|f(t)-f(x)|}{(t-x)^2}h(t)\,d\theta(t). \end{aligned}
\end{equation*}
\notag
$$
We consider the integral
$$
\begin{equation*}
\widetilde J^{(2)}_n(x)\leqslant C_x\frac{1}{n+1} \int^{a_{k_0+1}}_{x+1/(n+1)} \frac{|f(t)-f(x)|}{(t-x)^2}h(t)\,d\theta(t),
\end{equation*}
\notag
$$
because
$$
\begin{equation*}
\frac{1}{n+1} \int^{x-1/(n+1)}_{a_{k_0}} \frac{|f(t)-f(x)|}{(t-x)^2}h(t)\,d\theta(t)
\end{equation*}
\notag
$$
is estimated similarly.
Let $M>0$, $M=M(n, x, k_0)$, be such that
$$
\begin{equation*}
x+\frac{2^{M-1}}{n+1}\leqslant a_{k_0+1}<x+\frac{2^M}{n+1}.
\end{equation*}
\notag
$$
Then
$$
\begin{equation*}
\begin{aligned} \, |\widetilde J^{(2)}_n(x)| &\leqslant C_x\frac{1}{n+1}\sum^M_{j=1} \biggl(\frac{n+1}{2^{j-1}}\biggr)^2 \int_{x+2^{j-1}/(n+1)}^{x+2^j/(n+1)} |f(t)-f(x)|h(t)\,d\theta(t) \\ &=C_x\sum^M_{j=1}\frac{n+1}{2^{2j}}\int_{|t-x|\leqslant 2^j/(n+1)} |f(t)-f(x)|h(t)\,d\theta(t) \\ &=o_x\biggl(\sum^{M}_{j=1}\frac{1}{2^j}\biggr)=o_x(1), \qquad n\to\infty. \end{aligned}
\end{equation*}
\notag
$$
Thus, we have shown that Using estimates (4.10) and (3.5) we obtain
$$
\begin{equation*}
\begin{aligned} \, |J^{(3)}_n(x)| &\leqslant C_x\frac{1}{n+1} \int_{\substack{t\notin(a_{k_0}, a_{k_0+1})\\ x\in K_0 }} \frac{|f(t)-f(x)|}{[\pi_{N+1}(t)-\pi_{N+1}(x)]^2}h(t)\,d\theta(t) \\ &\leqslant C_x\frac{1}{n+1}\int_{\substack{t\notin(a_{k_0}, a_{k_0+1})\\ x\in K_0 }} \frac{|f(t)-f(x)|}{(t-x)^2}h(t)\,d\theta(t) \\ &=O_x(1)\frac{1}{n+1}\int_{\substack{t\notin(a_{k_0}, a_{k_0+1})\\ x\in (a_{k_0}, a_{k_0+1}) }} |f(t)-f(x)|h(t)\,d\theta(t) \end{aligned}
\end{equation*}
\notag
$$
and now, by (3.18)
Substituting (4.19)–(4.21) into (4.18) we arrive at (4.17). (ii) That (4.17) holds uniformly for $x\in K\subset\varepsilon_m$ can be derived in a standard way from Corollary 4.1 (see the proof of Theorem 3.1, (ii)). This completes the proof of Lemma 4.4. Theorem 4.1. Let the orthonormal polynomial system $\{\widehat{q}_n(x)\}^\infty_{n=0}$ satisfy conditions (3.7) and (4.8). (i) Let $f\in\mathfrak{R}_p$, $1\leqslant p<\infty$, satisfy (3.18). Then at each Lebesgue point $x\in\varepsilon_m$ of $f$ the Fejér means $\sigma_nf(x)$ of the Fourier-Sobolev series (1.4), (1.5) converge to $f(x)$, that is, (ii) If, in addition, the measure $d\theta(x)$ satisfies (3.8) and $f$ is continuous on $[-1,1]$, then (4.22) holds uniformly on all compact subsets $K$ of $\varepsilon_m$. Proof. From Lemma 3.2 and since the Fejér means are Toeplitz regular, for $x\in\varepsilon_m$ we have where the convergence is uniform on compact subsets $K$ of $\varepsilon_m$.
The rest of the proof is secured by Lemma 4.4. Theorem 4.1 is proved. Now we estimate the norm of a majorant for the Fejér means of the Fourier-Sobolev series(see (4.3)), namely, of
$$
\begin{equation*}
\begin{aligned} \, \sigma_*f(x) &=\sup_{n\in\mathbb{Z}_+}|\sigma_nf(x)| =\sup_{n\in\mathbb{Z}_+}\biggl|\int^1_{-1}f(t) F_n(t,x)\,d\theta(t) \\ &\qquad +\sum^m_{s=1}\sum^{N_s}_{i=0} M_{s,i}f^{(i)}(a_s)\frac{\partial^i F_n}{\partial t^i}(a_s,x)\biggr|, \qquad n\in\mathbb{Z}_+, \quad x\in\varepsilon_m. \end{aligned}
\end{equation*}
\notag
$$
We have
$$
\begin{equation}
\begin{aligned} \, \notag \sigma_*f(x) &\leqslant \sup_{n\in\mathbb{Z}_+}\biggl|\int^1_{-1}f(t) F_n(t,x)\,d\theta(t)\biggr| \\ &\qquad +\sup_{n\in\mathbb{Z}_+}\biggl|\sum^m_{s=1}\sum^{N_s}_{i=0} M_{s,i}f^{(i)}(a_s)\frac{\partial^i F_n}{\partial t^i}(a_s,x)\biggr|, \qquad n\in\mathbb{Z}_+, \quad x\in\varepsilon_m. \end{aligned}
\end{equation}
\tag{4.23}
$$
Consider
$$
\begin{equation}
\|\sigma_*f(x)\|^p_{L^p_\theta(K)},\qquad \sum^m_{s=1}\sum^{N_s}_{i=0} M_{s,i} \Bigl[\sup_{n\in\mathbb{Z}_+}|(\sigma_nf)^{(i)}(a_s)|\Bigr]^p,
\end{equation}
\tag{4.24}
$$
where $K$ is a compact subset of $\varepsilon_m$. Let us estimate the first expression in (4.24). Lemma 4.5. Let the polynomial system $\{\widehat{q}_n(x)\}^\infty_{n=0}$ satisfy conditions (3.7), (3.8), (3.36), (4.8) and let $f\in\mathfrak{R}_p$ satisfy (3.18) (for some $1<p<\infty$). Then where the positive constant $C_p$ is independent of the function $f\in W^p_\theta(K)$. Proof. From (4.23) and by Minkowski’s inequality we have
$$
\begin{equation}
\begin{aligned} \, \notag \|\sigma_*f\|_{L^p_\theta(K)}^{p} &\leqslant C_p \biggl\|\sup_{n\in\mathbb{Z}_+}\biggl|\int^1_{-1} f(t)F_n(t,x)\,d\theta(t)\biggr|\biggr\|^p_{L_\theta^p(K)} \\ &\qquad +C_p \biggl\|\sum^m_{s=1}\sum^{N_s}_{i=0} M_{s,i} |f^{(i)}(a_s)|\sup_{n\in\mathbb{Z}_+}\biggl|\frac{\partial^i F_n}{\partial t^i}(a_s,x)\biggr|\biggr\|^p_{L^p_\theta(K)}, \end{aligned}
\end{equation}
\tag{4.26}
$$
where $C_p>0$ is independent of $f$.
From (2.8), using Lemma 4.3 and condition (3.18) we obtain
$$
\begin{equation*}
\sup_{n\in\mathbb{Z}_+}\biggl|\int^1_{-1} f(t)F_n(t,x)\,d\theta(t)\biggr| \leqslant C_p M_\theta[fh](x)h(x), \qquad x\in K.
\end{equation*}
\notag
$$
Hence, since the function $h(x)$ is bounded on $K$, from (2.9) we deduce
$$
\begin{equation*}
\begin{aligned} \, &\biggl\|\sup_{n\in\mathbb{Z}_+}\biggl|\int^1_{-1} f(t)F_n(t,x)\,d\theta(t)\biggr|\biggr\|^p_{L_\theta^p(K)} \leqslant C_p \|M_\theta[fh](x)h(x)\|^p_{L_\theta^p(K)} \\ &\qquad\leqslant C_p \|M_\theta[fh](x)\|^p_{L_\theta^p(K)} \leqslant C_p \|fh\|^p_{L_\theta^p(K)}. \end{aligned}
\end{equation*}
\notag
$$
Using again the boundedness of $h(x)$ on $K$, we have the estimate
$$
\begin{equation}
\biggl\|\sup_{n\in\mathbb{Z}_+}\biggl|\int^1_{-1} f(t)F_n(t,x)\,d\theta(t)\biggr|\biggr\|^p_{L_\theta^p(K)} \leqslant C_p \|f\|^p_{L_\theta^p(K)},
\end{equation}
\tag{4.27}
$$
where the positive constant $C_p$ is independent of the function $f$.
Let us estimate the second term in the sum (4.26). From (3.7) and (3.36) we obtain
$$
\begin{equation}
\begin{aligned} \, \notag &\sup_{n\in\mathbb{Z}_+}\biggl|\frac{\partial^i F_n}{\partial t^i}(a_s,x)\biggr| =\sup_{n\in\mathbb{Z}_+}\biggl|\sum^n_{l=0} \biggl(1-\frac{l}{n+1}\biggr)\widehat{q}^{\,(i)}_l(a_s)\widehat{q}_l(x)\biggr| \\ &\qquad\leqslant Ch(x), \qquad x\in\varepsilon_m. \end{aligned}
\end{equation}
\tag{4.28}
$$
Hence by (3.40) and (4.28),
$$
\begin{equation}
\begin{aligned} \, \notag &\biggl\|\sum^m_{s=1}\sum^{N_s}_{i=0} M_{s,i}|f^{(i)}(a_s)|\sup_{n\in\mathbb{Z}_+} \biggl|\frac{\partial^i F_n}{\partial t^i}(a_s,x)\biggr|\biggr\|^p_{L^p_\theta(K)} \\ &\qquad \leqslant C_p\biggl\{\sum^m_{s=1}\sum^{N_s}_{i=0}M_{s,i}|f^{(i)}(a_s)|\biggr\}^p \int_K h^p(x)\,d\theta(x) \leqslant C_p\sum^m_{s=1}\sum^{N_s}_{i=0} M_{s,i} |f^{(i)}(a_s) |^p. \end{aligned}
\end{equation}
\tag{4.29}
$$
So, using (4.26), (4.27), (4.29) and (3.32) we find that
$$
\begin{equation*}
\|\sigma_*f\|_{L^p_\theta(K)}^p\leqslant C_p\biggl\{ \|f\|^p_{L_\theta^p(K)}+\sum^m_{s=1}\sum^{N_s}_{i=0} M_{s,i} |f^{(i)}(a_s)|^p\biggr\} =C_p\|f\|^p_{W_\theta^p(K)},
\end{equation*}
\notag
$$
proving Lemma 4.5. Lemma 4.6. Let conditions (3.7), (3.8), (3.36), (4.8), (3.41) be met for an orthonormal polynomial system $\{\widehat{q}_n(x)\}^\infty_{n=0}$ for some $p$, $1<p<\infty$, and let $f$ be a function in $W^p_\theta([-1,1])$. Then where the positive constant $C$ is independent of the function $f$. Proof. We have
$$
\begin{equation*}
(\sigma_nf)^{(j)}(a_k)=\int^1_{-1}f(t)\frac{\partial^j F_n}{\partial x^j}(t, a_k)\,d\theta(t) +\sum^m_{s=1}\sum^{N_s}_{i=0} M_{s,i}f^{(i)}(a_s)\, \frac{\partial^{j+i}F_n}{\partial x^j \, \partial t^i}(a_s, a_k),
\end{equation*}
\notag
$$
hence
$$
\begin{equation*}
\begin{aligned} \, &\sup_{n\in\mathbb{Z}_+}|(\sigma_nf)^{(j)}(a_k)|\leqslant \sup_{n\in\mathbb{Z}_+}\biggl|\int^1_{-1}f(t)\, \frac{\partial^j F_n}{\partial x^j}(t, a_k)\,d\theta(t)\biggr| \\ &\qquad +\sup_{n\in\mathbb{Z}_+}\biggl|\sum^m_{s=1}\sum^{N_s}_{i=0} M_{s,i}f^{(i)}(a_s)\, \frac{\partial^{j+i}F_n}{\partial x^j\, \partial t^j}(a_s, a_k)\biggr|. \end{aligned}
\end{equation*}
\notag
$$
Therefore,
$$
\begin{equation}
\begin{aligned} \, \notag &\sum^m_{k=1}\sum^{N_k}_{j=0} M_{k,j}\Bigl[\sup_{n\in\mathbb{Z}_+}|(\sigma_nf)^{(j)}(a_k)|\Bigr]^p \\ \notag &\leqslant C_p \sum^m_{k=1}\sum^{N_k}_{j=0} M_{k,j}\biggl\{\sup_{n\in\mathbb{Z}_+} \biggl|\int^1_{-1} f(t)\, \frac{\partial^j F_n}{\partial x^j}(t, a_k)\,d\theta(t)\biggr|\biggr\}^p \\ &\ +C_p \sum^m_{k=1}\sum^{N_k}_{j=0} M_{k,j}\biggl\{\sum^m_{s=1}\sum^{N_s}_{i=0} M_{s,i}|f^{(i)}(a_s)|\sup_{n\in\mathbb{Z}_+} \biggl|\sum^n_{l=0}\biggl(1\,{-}\frac{l}{n+1}\biggr)\widehat{q}^{\,(i)}_l(a_s) \widehat{q}^{\,(j)}_l(a_k)\biggr|\biggr\}^p\!. \end{aligned}
\end{equation}
\tag{4.31}
$$
From estimates (3.7), (3.36) and (4.28) we obtain
$$
\begin{equation}
\begin{gathered} \, \biggl|\frac{\partial^{j+i}F_n}{\partial x^j\, \partial t^i}(a_s, a_k)\biggr| \leqslant C \quad\text{and}\quad \biggl|\frac{\partial^j F_n}{\partial x^j}(t, a_k)\biggr|\leqslant C h(t), \\ n\in\mathbb{Z}_+, \qquad t\in\varepsilon_m, \quad j=0,1,\dots,N_k, \qquad i=0,1,\dots, N_s, \qquad s,k=1,\dots,m. \end{gathered}
\end{equation}
\tag{4.32}
$$
Applying (3.41) and (4.32) and using Hölder’s inequality, this establishes that
$$
\begin{equation}
\begin{aligned} \, \notag &\sum^m_{k=1}\sum^{N_k}_{j=0} M_{k,j} \biggl\{\sup_{n\in\mathbb{Z}_+}\biggl|\int^1_{-1} f(t)\, \frac{\partial^j F_n}{\partial x^j}(t, a_k)\,d\theta(t)\biggr|\biggr\}^p \\ &\qquad \leqslant C_p (\|f\|_{L^p_\theta([-1,1])}\|h\|_{L^q_\theta([-1,1])})^p \leqslant C_p(\|f\|_{L^p_\theta([-1,1])})^p, \end{aligned}
\end{equation}
\tag{4.33}
$$
where the positive constant $C_p$ is independent of the function $f\in L^p_\theta([-1,1])$. Using the first estimate in (4.32) and relation (3.40) we find that
$$
\begin{equation}
\begin{aligned} \, \notag &\sum^m_{k=1}\sum^{N_k}_{j=0} M_{k,j}\biggl\{\sum^m_{s=1}\sum^{N_s}_{i=0} M_{s,i}|f^{(i)}(a_s)|\sup_{n\in\mathbb{Z}_+}\biggl|\frac{\partial^{j+i}F_n}{\partial x^j\, \partial t^i}(a_s, a_k)\biggr|\biggr\}^p \\ &\qquad \leqslant C_p \sum^m_{k=1}\sum^{N_k}_{j=0} M_{k,j}\biggl\{\sum^m_{s=1}\sum^{N_s}_{i=0} M_{s,i}|f^{(i)}(a_s)|\biggr\}^p \leqslant C_p\sum^m_{s=1}\sum^{N_s}_{i=0} M_{s,i}|f^{(i)}(a_s)|^p. \end{aligned}
\end{equation}
\tag{4.34}
$$
Substituting (4.33) and (4.34) into (4.31) we arrive at (4.30). Lemma 4.6 is proved. The next result follows from the definition (3.32) and Lemmas 4.5 and 4.6. Theorem 4.2. Let the polynomial system $\{\widehat{q}_n(x)\}^\infty_{n=0}$ satisfy conditions (3.7), (3.8), (3.36), (3.41), (4.8). Then for any function $f\in W^p_\theta([-1,1])$ satisfying (3.18), for some $p$, $1<p<\infty$, on an arbitrary compact subset $K$ of $\varepsilon_m$, where the positive constant $C_p $ is independent of $n\in\mathbb{Z}_+$ and $f$. § 5. Multipliers for Fourier-Sobolev seriesConsider the partial sums
$$
\begin{equation}
T_nf(x;\Phi)=\sum^n_{k=0}\phi_k c_k(f)\widehat{q}_k(x), \qquad x\in[-1,1], \quad n\in\mathbb{Z}_+,
\end{equation}
\tag{5.1}
$$
of the multiplier series (1.7) generated by the sequence (1.6):
$$
\begin{equation*}
\Phi=\bigl\{\phi_n,\ n\in\mathbb{Z}_+;\ \phi_0=1,\ \{\phi_n\}^\infty_{n=0}\in l^\infty\bigr\}.
\end{equation*}
\notag
$$
We set
$$
\begin{equation*}
\Delta\phi_n=\phi_n-\phi_{n+1}, \quad \Delta^2\phi_n=\Delta(\Delta\phi_n)=\phi_n- 2\phi_{n+1}+\phi_{n+2}, \qquad n\in\mathbb{Z}_+.
\end{equation*}
\notag
$$
A sequence $\Phi=\{\phi_n\}^\infty_{n=0}$ is called quasi-convex if Lemma 5.1 (see [30], Ch. 7, 7.1.3). (i) If the sequence $\Phi=\{\phi_n\}^\infty_{n=0}$ is quasi-convex and bounded, then it has bounded variation and the sequence $n\Delta\phi_n$ is bounded (ii) If the sequence $\Phi=\{\phi_n\}^\infty_{n=0}$ is quasi-convex and has a finite limit, then it has a bounded variation and Remark 5.1. If the sequence $\Phi=\{\phi_n\}^\infty_{n=0}$ is quasi-convex and bounded, then by (5.3) Lemma 5.2. Let $s_n$ and $\sigma_n$ be the partial sums and arithmetic means of a series $\sum^\infty_{k=0}u_k$, respectively. If the $\sigma_n$ converge and if $s_n=o(\mu_n)$, where $\{{1}/{\mu_n}\}^\infty_{n=0}$ is quasi-convex and tends to zero as $n\to\infty$, then the series $\sum^\infty_{k=0}{u_k}/{\mu_k}$ is convergent. Proof. Indeed, applying the Abel transform twice we obtain
$$
\begin{equation*}
\sum^n_{k=0}\frac{u_k}{\mu_k} =\sum^{n-2}_{k=0}(k+1)\sigma_k\Delta^2\biggl(\frac{1}{\mu_k}\biggr) +n\Delta\biggl(\frac{1}{\mu_{n-1}}\biggr)\sigma_{n-1}+s_n\frac{1}{\mu_n}, \qquad n\in\mathbb{Z}_+.
\end{equation*}
\notag
$$
Using assertion (ii) of Lemma 5.1 (see (5.4)), under the hypotheses of Lemma 5.2 we obtain
$$
\begin{equation*}
\sum^n_{k=0}\frac{u_k}{\mu_k}\to \sum^\infty_{k=0}(k+1)\sigma_k\Delta^2\biggl(\frac{1}{\mu_k}\biggr)
\end{equation*}
\notag
$$
as $n\to\infty$. Now it is clear that the series $\sum^\infty_{k=0}(k+1)\Delta^2(1/\mu_k)\sigma_k$ is absolutely convergent. Lemma 5.2 is proved. Theorem 5.1. Let the orthonormal polynomial system $\{\widehat{q}_n(x)\}^\infty_{n=0}$ satisfy conditions (3.7) and (4.8) and let the quasi-convex sequence $\Phi=\{\phi_n\}^\infty_{n=0}$ satisfy Also let $f\in\mathfrak{R}_p$, $1\leqslant p <\infty$, satisfy conditions (3.18), Then: 1) the multiplier series (1.7) is convergent at each Lebesgue point $x\in\varepsilon_m$, so that
$$
\begin{equation*}
Tf(x;\Phi)=\sum^\infty_{k=0} \phi_k c_k(f)\widehat{q}_k(x) \quad\textit{for $\theta$-almost all }x\in[-1,1];
\end{equation*}
\notag
$$
2) if, in addition, the function $f$ is continuous on $[-1,1]$ and (3.8) is met, then the multiplier series (1.7) converges uniformly on each compact subset $K\subset\varepsilon_m$, that is, with respect to the topology of unform convergence on compact sets. Corollary 5.1. Let the polynomial system $\{\widehat{q}_n(x\}^\infty_{n=0}$ satisfy conditions (3.7), (4.8) and let $f \in \mathfrak{R}_p$, $1 \leqslant p < \infty$, satisfy (3.18). Then both the series Indeed, it is easily checked that condition (5.5) of Theorem 5.1 is satisfied for the sequences $\phi_k={1}/{\ln(k+2)}$ and $\phi_k={1}/{(k+1)^\gamma}$, $\gamma>0$. Consider now the problem of estimating the norm of a majorant of the partial sums of series (1.7). Theorem 5.2. Let the orthonormal polynomial system $\{\widehat{q}_n(x)\}^\infty_{n=0}$ satisfy the hypotheses of Theorem 4.2. Also let the quasi-convex sequence (1.6) satisfy (5.5) and $f\in W^p_\theta([-1,1])$, $1<p<\infty$, satisfy (3.18). Then Proof. We have (see (3.1))
$$
\begin{equation*}
S_nf(x)=\int^1_{-1}f(t)D_n(t,x)\,d\theta(t)+\sum^m_{s=1}\sum^{N_s}_{i=0} M_{s,i}f^{(i)}(a_s)\, \frac{\partial^i D_n}{\partial t^i}(a_s, x), \qquad n\in\mathbb{Z}_+.
\end{equation*}
\notag
$$
Hence, by the definition of the function $G_nf(x)$ and from (5.5) we obtain
$$
\begin{equation*}
\begin{aligned} \, |\phi_nS_nf(x)| &\leqslant |\phi_n\ln(n+2)|\,\biggl|\frac{1}{\ln(n+2)}\int^1_{-1}f(t)D_n(t,x)\,d\theta(t)\biggr| \\ &\qquad +|\phi_n|\,\biggl|\sum^m_{s=1}\sum^{N_s}_{i=0}M_{s,i}f^{(i)}(a_s)\, \frac{\partial^i D_n}{\partial t^i}(a_s, x)\biggr| \\ &\leqslant C|G_nf(x)|+C\biggl|\sum^m_{s=1}\sum^{N_s}_{i=0}M_{s,i}f^{(i)}(a_s)\, \frac{\partial^i D_n}{\partial t^i} (a_s,x)\biggr|. \end{aligned}
\end{equation*}
\notag
$$
Applying Lemmas 3.4, 3.5 and taking (3.35), (3.37) and (3.42) into account we see that
$$
\begin{equation}
\|\phi_nS_nf\|_{W^p_\theta(K)}\leqslant C_p\|f\|_{W^p_\theta([-1,1])},
\end{equation}
\tag{5.8}
$$
where the positive constant $C_p$ is independent of $n\in\mathbb{Z}_+$, $f\in W^p_\theta([-1,1])$, and the sequence $\{\phi_n\}^\infty_{n=0}$.
Applying the Abel transform twice we obtain
$$
\begin{equation*}
T_nf(x;\Phi)=\phi_nS_nf(x)+n(\Delta\phi_{n-1})\sigma_{n-1}f(x) +\sum^{n-2}_{k=0}(k+1)(\Delta^2\phi_k)\sigma_kf(x).
\end{equation*}
\notag
$$
As a result,
$$
\begin{equation*}
|T_nf(x;\Phi)| \leqslant |\phi_nS_nf(x)|+(n|\Delta\phi_{n-1}|)|\sigma_{n-1}f(x)| +\sup_{n\in\mathbb{Z}_+} \sum^{n-2}_{k=0}(k+1)|\Delta^2\phi_k|\,|\sigma_kf(x)|.
\end{equation*}
\notag
$$
Now Theorem 5.2 follows from estimates (4.35), (5.3) and (5.8). Theorem 5.2 is proved. The following theorem is the main result of our paper. Theorem 5.3. Let the hypotheses of Theorem 5.2 on a function $f$, a system of orthonormal polynomials $\{\widehat{q}_n(x)\}^\infty_{n=0}$, and a multiplier sequence (1.6) be met for some $p$, $1<p<\infty$. Then the multiplier operator $T$ satisfies the estimate Theorem 5.3 is a direct consequence of Fatou’s lemma (Lemma 2.4), Theorem 5.1 and estimate (5.7). Consider the multiplier series (5.6) and define the multiplier operator
$$
\begin{equation}
Pf(x)\sim\sum^\infty_{k=0}\frac{c_k(f)\widehat{q}_k(x)}{\ln(k+2)}, \qquad x\in[-1,1].
\end{equation}
\tag{5.9}
$$
Setting $\phi_k={1}/{\ln(k+2)}$ in Theorem 5.3 we arrive at the following result. Corollary 5.2. Let the polynomial system $\{\widehat{q}_n(x)\}^\infty_{n=0}$ and the function $f$ satisfy the hypotheses of Theorem 5.2. Then the series (5.9) converges almost everywhere on $[-1,1]$, and its sum $Pf(x)$ satisfies on any compact subset $K$ of $\varepsilon_m$, where the positive constant $C_p$ is independent of the function $f\in W^p_\theta([-1,1])$. A similar estimate also holds for the multiplier operator § 6. Symmetric Gegenbauer-Sobolev polynomialsConsider the inner product
$$
\begin{equation*}
\begin{aligned} \, \langle f,g\rangle_\alpha &=\int^1_{-1} f(x)g(x)\omega_\alpha(x)\,dx+M[f(1)g(1)+f(-1)g(-1)] \\ &\qquad +N[f'(1)g'(1)+f'(-1)g'(-1)], \qquad M\geqslant0, \quad N\geqslant0, \end{aligned}
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\omega_\alpha(x)=\frac{\Gamma(2\alpha)}{2^{2\alpha+1}\Gamma^2(\alpha+1)} (1-x^2)^\alpha, \qquad \alpha>-\frac{1}{2}.
\end{equation*}
\notag
$$
Let $\{\widehat{B}^{(\alpha)}_n(x)\equiv \widehat{B}^{(\alpha)}_n(x;M,N)\}$ be the system of symmetric orthonormal Gegenbauer-Sobolev polynomials:
$$
\begin{equation*}
\begin{aligned} \, &\int^1_{-1} \widehat{B}^{(\alpha)}_n(x) \widehat{B}^{(\alpha)}_m(x)\omega_\alpha(x)\,dx +M[\widehat{B}^{(\alpha)}_n(1)\widehat{B}^{(\alpha)}_m(1) +\widehat{B}^{(\alpha)}_n(-1)\widehat{B}^{(\alpha)}_m(-1)] \\ &\qquad+N[\{\widehat{B}^{(\alpha)}_n(1)\}'\{\widehat{B}^{(\alpha)}_m(1)\}' +\{\widehat{B}^{(\alpha)}_n(-1)\}'\{\widehat{B}^{(\alpha)}_m(-1)\}']=\delta_{n,m}, \qquad n,m\,{\in}\,\mathbb{Z}_+. \end{aligned}
\end{equation*}
\notag
$$
For $\alpha=0$ we have the system of symmetric orthonormal Legendre-Sobolev polynomials. For $M>0$ and $N>0$, the polynomials $\widehat{B}_n^{(\alpha)}(x;M,N)$ have some properties not shared by the classical Gegenbauer (ultraspherical) polynomials $\widehat{P}_n^{(\alpha)}(x)$ which are orthonormal with respect to the weight $\omega_\alpha(x)$ (the case $M=0$, $N=0$). Recall some properties of the polynomials $\widehat{B}_n^{(\alpha)}(x;M,N)$ (see [34]–[43] and the references given there). 1. For sufficiently large $n$ there exists a pair of real zeros lying outside $[-1,1]$ (all the zeros of $\widehat{P}^{(\alpha)}_n(x)$ lie in the interval $(-1,1)$). 2. The polynomials $\widehat{B}^{(\alpha)}_n(x)$ are eigenfunctions of a linear differential operator (usually, of infinite order). Only for $\alpha=0,1,2,\dots$ does this class contain an operator of finite order, namely,
$$
\begin{equation*}
\begin{aligned} \, 2 &\quad\text{for }\ M=0 \quad\text{and}\quad N=0, \\ 2\alpha+4 &\quad\text{for }\ M>0 \quad\text{and}\quad N=0, \\ 2\alpha+8 &\quad\text{for }\ M=0 \quad\text{and}\quad N>0, \\ 4\alpha+10 &\quad\text{if }\ M>0,\ N>0. \end{aligned}
\end{equation*}
\notag
$$
3. The values at the endpoints of the orthogonality interval are
$$
\begin{equation}
|\widehat{B}^{(\alpha)}_n(\pm 1)|\approx n^{-\alpha-3/2}\quad\text{and}\quad |\{\widehat{B}^{(\alpha)}_n\}'(\pm 1)|\approx n^{-\alpha-7/2}, \qquad n\to\infty,
\end{equation}
\tag{6.1}
$$
here, $A_n\approx B_n(n\to\infty)$ means that $\lim_{n\to\infty}{A_n}/{B_n}=1$ (it is known that $|\widehat{P}^{(\alpha)}_n(\pm 1)|\approx n^{\alpha+1/2}$ and $|\{\widehat{P}^{(\alpha)}_n\}'(\pm 1)|\approx n^{\alpha+5/2}$). 4. The following recurrence relations hold:
$$
\begin{equation}
(x^3-3x)\widehat{B}^{(\alpha)}_n(x) =a_{n+3}\widehat{B}^{(\alpha)}_{n+3}(x)+b_{n+1}\widehat{B}^{(\alpha)}_{n+1}(x) +b_n\widehat{B}^{(\alpha)}_{n-1}(x)+a_n\widehat{B}^{(\alpha)}_{n-3}(x),
\end{equation}
\tag{6.2}
$$
where
$$
\begin{equation}
\sum^\infty_{k=0}(|\Delta a_k|+|\Delta b_k|)<\infty,
\end{equation}
\tag{6.3}
$$
and
$$
\begin{equation}
\begin{aligned} \, \notag (x^2-1)^{2}\widehat{B}^{(\alpha)}_n(x) &={\epsilon}_{n+4}\widehat{B}^{(\alpha)}_{n+4}(x) +{\beta}_{n+2}\widehat{B}^{(\alpha)}_{n+2}+{\gamma}_n\widehat {B}^{(\alpha)}_{n}(x) \\ &\qquad+{\beta}_{n}\widehat{B}^{(\alpha)}_{n-2}(x)+{\epsilon}_{n}\widehat{B}^{(\alpha)}_{n-4}(x), \end{aligned}
\end{equation}
\tag{6.4}
$$
where
$$
\begin{equation*}
\sum^\infty_{k=0}(|\Delta {\epsilon}_k|+|\Delta {\beta}_k|+|\Delta{\gamma}_k|)<\infty.
\end{equation*}
\notag
$$
That the recurrence coefficients in (6.4) have bounded variation was shown in [36]. The proof of (6.3) for recurrence relation (6.2) is similar. Recall that the classical Gegenbauer polynomials satisfy a three-term recurrence relation. Condition (4.8) for the polynomials $\widehat {B}^{(\alpha)}_n(x)$ is satisfied if estimate (6.3) holds. Note that for the Gegenbauer-Sobolev polynomials one can get by with recurrence relation (6.2). 5. The weight estimate for the polynomials $\widehat{B}^{(\alpha)}_n(x)$, as well as for $\widehat{P}^{(\alpha)}_n(x)$, has the form
$$
\begin{equation*}
|\widehat{B}^{(\alpha)}_n(x)|\leqslant C(1-x^2)^{-(\alpha/2+1/4)}, \qquad -1<x<1, \quad\alpha>-\frac{1}{2},
\end{equation*}
\notag
$$
where the positive constant $C$ is independent of $n=1,2,\dots$ and $x\in (-1,1)$. We set
$$
\begin{equation*}
\begin{aligned} \, W^p_{\omega_\alpha}(F) &=\biggl\{f\colon \|f\|_{W^p_{\omega_\alpha}(F)}<\infty,\text{ where }\|f\|^p_{W^p_{\omega_\alpha}(F)}=\|f\|^p_{L^p_{\omega_\alpha}(F)} \\ &\qquad+\sum^m_{k=1}\sum^{N_k}_{i=0} M_{k,i} |f^{(i)}(a_k)|^p\biggr\}, \quad\text{for } F\subseteq [-1,1]\quad\text{and} \quad 1\leqslant p<\infty. \end{aligned}
\end{equation*}
\notag
$$
With each function $f\in W^p_{\omega_\alpha}([-1,1])$, $\alpha>-1/2$, we associate a Fourier-Gegenbauer-Sobolev series:
$$
\begin{equation*}
\begin{aligned} \, &f(x)\sim\sum^\infty_{k=0} c_k^{(\alpha)}(f)\widehat{B}^{(\alpha)}_k(x), \\ &c_k^{(\alpha)}(f)=\int^1_{-1} f(x)\widehat{B}^{(\alpha)}_k(x)\omega_\alpha(x)\,dx +M\bigl[f(1)\widehat{B}^{(\alpha)}_n(1)+f(-1)\widehat{B}^{(\alpha)}_n(-1)\bigr] \\ &\qquad\qquad +N\bigl[f'(1)\{\widehat{B}^{(\alpha)}_n\}'(1)+f'(-1)\{\widehat{B}^{(\alpha)}_n\}'(-1)\bigr] \end{aligned}
\end{equation*}
\notag
$$
(we consider the case $M>0$, $N>0$). Consider the series
$$
\begin{equation}
\sum^\infty_{k=0} \frac{c_k^{(\alpha)}\widehat{B}^{(\alpha)}_k(x)}{\ln(k+2)}, \qquad x\in[-1,1],
\end{equation}
\tag{6.5}
$$
and
$$
\begin{equation}
\sum^\infty_{k=0} \frac{c_k^{(\alpha)}\widehat{B}^{(\alpha)}_k(x)}{(k+1)^\gamma}, \qquad x\in[-1,1].
\end{equation}
\tag{6.6}
$$
Let $[a,b]$ be an arbitrary closed interval in $(-1,1)$. An analysis similar to that used in Corollaries 5.1 and 5.2 gives the following result. Theorem 6.1. Let $\alpha>-1/2$ and $f\in W^p_{\omega_\alpha}([-1,1])$, and let Then the series (6.5) converges almost everywhere on $(-1,1)$. Moreover, for
$$
\begin{equation}
\frac{4(\alpha+1)}{2\alpha+3}<p<\frac{4(\alpha+1)}{2\alpha+1}
\end{equation}
\tag{6.8}
$$
the sum
$$
\begin{equation*}
P^{(\alpha)}f(x)=\sum^\infty_{k=0}\frac{c_k^{(\alpha)}(f)\widehat{B}^{(\alpha)}_k(x)}{\ln(k+2)}, \qquad x\in[-1,1],
\end{equation*}
\notag
$$
of this series satisfies the estimate
$$
\begin{equation*}
\|P^{(\alpha)}f\|_{W^p_{\omega_\alpha}([a,b])}\leqslant C_p\|f\|_{W^p_{\omega_\alpha([-1,1])}}
\end{equation*}
\notag
$$
on each compact interval $[a,b]\subset(-1,1)$, where $C_p>0$ is independent of the function $f$. A similar result also holds for the series (6.6). Proof. Let us check that conditions (3.18) and (3.41) are satisfied for the majorant $h(x)$ of the system $\{\widehat{B}^{(\alpha)}_n(x)\equiv\widehat{B}^{(\alpha)}_n(x;M,N)\}$
We have $\omega_\alpha(x)=(1-x^2)^{-(\alpha/2+1/4)}$, $-1<x<1$, $\alpha>-1/2$, hence from (3.18) and (3.41) we obtain
$$
\begin{equation*}
\int^1_{-1} (1-x^2)^{\alpha-p(\alpha/2+1/4)}\,dx < \infty, \quad \int^1_{-1} (1-x^2)^{\alpha-q(\alpha/2+1/4)}\,dx < \infty,\qquad \frac{1}{p}+\frac{1}{q}=1.
\end{equation*}
\notag
$$
Moreover, the convergence of the first integral implies inequality (6.7), and the convergence of both integrals secures (6.8). Conditions (3.36) are direct consequences of (6.1), and (4.8) follows from (6.3). Theorem 6.1 is proved. The convergence of Fourier-Gegenbauer-Sobolev series (and their generalizations) was studied in [44] and [45].
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