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This article is cited in 2 scientific papers (total in 2 papers) Sharp Bernstein-type inequalities for Fourier-Dunkl multipliers O. L. Vinogradov Saint Petersburg State University, St. Petersburg, Russia
Russian version:
Matematicheskii Sbornik, 2023, Volume 214, Number 1, DOI: https://doi.org/10.4213/sm9724 
Bibliographic databases:
   § 1. Introduction1.1. A short surveyThe classical sharp inequalities
$$
\begin{equation}
\|f^{(r)}\|\leqslant\biggl(\frac{\sigma}{2\sin(\sigma h/2)}\biggr)^{r}\|\delta_h^rf\|, \qquad 0<h<\frac{2\pi}{\sigma},
\end{equation}
\tag{1.2}
$$
and
$$
\begin{equation}
\frac{\|\delta_u^rf\|}{\sin^r(\sigma u/2)} \leqslant\frac{\|\delta_h^rf\|}{\sin^r(\sigma h/2)}, \qquad 0<u<h<\frac{2\pi}{\sigma},
\end{equation}
\tag{1.3}
$$
for entire functions of exponential type, and, in particular, for trigonometric polynomials (see, for example, [1], § 84 and Appendix 2.53 in the Russian edition, and [2], §§ 4.8.2 and 4.8.6) play an important role in approximation theory. Here $r\in\mathbb N$, $\sigma>0$, $f$ is a function of type at most $\sigma$ which is bounded on $\mathbb R$,
$$
\begin{equation*}
\delta_h^rf(x)=\sum_{k=0}^{r}(-1)^kC_r^k f\biggl(x+\frac{(r-2k)h}2\biggr)
\end{equation*}
\notag
$$
is the $r$th central difference of $f$, and $\|f\|=\sup_{x\in\mathbb R}|f(x)|$. An inequality of the form (1.1) was first established by Bernstein, who proved it for trigonometric polynomials first (see [3] and the comments at the end of that book) and then for entire functions of exponential type (see [4]). An inequality of type (1.2) (for trigonometric polynomials) was established by M. Riesz [5] (also see [6]–[8]), an inequality of type (1.3) was obtained by Boas [9]. Similar inequalities also hold for $L_p$-spaces on the line or period interval. Differentiation and difference operators are multipliers in terms of the Fourier transform. A method for the derivation of inequalities for operators of this type which is based on interpolation formulae was proposed by Civin [10] (also see [2], § 4.8.6, and [11], § 3.2). Subsequently, this method was developed further in [12], for trigonometric polynomials, and in [13], for entire functions of exponential type. In the present paper we consider analogous questions for operators defined by means of multipliers in terms of the Fourier-Dunkl transform. In particular, we consider powers of the Dunkl operator $\Lambda$ and generalized differences. By analogy with Fourier multipliers, such operators can naturally be called Fourier-Dunkl multipliers. For such operators we construct Civin-type interpolation formulae, from which we derive inequalities of the form (1.1)–(1.3) in $L_p$-spaces with power weights. The operator $\Lambda^2$ coincides with the Bessel operator on the set of even functions, hence inequalities for powers of the Bessel operator are particular cases of our inequalities. For the uniform norm, analogues of inequalities (1.1)–(1.3) are shown to be sharp. In particular, we prove sharp inequalities of the form (1.1) for the operators $\Lambda^r$, $r\in\mathbb N$. For even $r$ these sharp inequalities were previously obtained in [14], order inequalities of the form (1.1)–(1.3) were presented in [15] and [16], and order inequalities of the form (1.1) with arbitrary $r\in\mathbb{N}$ were established in [17]. For earlier results on Bessel operators, see [18] and [19]. The paper is organized as follows. Some preliminary facts on Dunkl operators and the related transmutation operators are collected in § 2. In § 3 we present a general scheme of construction of interpolation formulae, show how inequalities can be derived from such formulae, and then discuss the sharpness of these inequalities. Applications of the general scheme to concrete operators are presented in § 4, where we also give detailed comments on some inequalities. 1.2. NotationIn what follows, $\mathbb C$, $\mathbb R$, $\mathbb R_+$, $\mathbb Z$, ${\mathbb Z}_+$ and $\mathbb N$ are the sets of complex, real, nonnegative real numbers, integers, nonnegative integers and positive integers, respectively. Throughout, unless otherwise mentioned, it will be assumed that $\alpha>-1/2$. In the case when $\alpha=-1/2$, a shift, a difference, a convolution and other objects will be called ordinary or classical; this case will be mentioned for the comparison of results. The index $\alpha$ will frequently be dropped. The function spaces considered below are real or complex, unless otherwise stated. We consider the following function spaces: $C(E)$, $\mathrm{CB}(E)$, $C^{(r)}(E)$ and $L_1(E)$ (the classes of continuous, bounded continuous, $r$ times continuously differentiable and absolutely integrable functions on $E$, respectively). Next, $L_{p,\alpha}$, $p\in[1,+\infty)$, is the space of measurable functions $f$ on $\mathbb R$ such that
$$
\begin{equation*}
\|f\|_{p,\alpha}=\biggl(\int_{\mathbb R} |f(x)|^p|x|^{2\alpha+1}\,dx\biggr)^{1/p}<+\infty;
\end{equation*}
\notag
$$
$L_{\infty,\alpha}=L_{\infty}$ is the space of essentially bounded functions on $\mathbb R$ equipped with the $\mathrm{ess\,sup}$-norm $\|\cdot\|_{\infty,\alpha}=\|\cdot\|_{\infty}$; the uniform norm of a continuous function is denoted by the same symbol; $\mathbf E_{\sigma}$ is the class of entire functions of exponential type at most $\sigma$, $\mathbf B_{\sigma}=\mathrm{CB}(\mathbb R)\cap \mathbf E_{\sigma}$ and $L_{p,\alpha,\sigma}=L_{p,\alpha}\cap \mathbf E_{\sigma}$. The domain and range of an operator $A$ are denoted by $\mathcal D(A)$ and $\mathcal R(A)$, respectively; $\lfloor x\rfloor$ and $\{x\}$ are the integral and fractional parts of the number $x$, and $x^{\langle r\rangle }=|x|^r\operatorname{sign}x$;
$$
\begin{equation*}
f_c(x)=\frac{f(x)+f(-x)}{2}\quad\text{and} \quad f_s(x)=\frac{f(x)-f(-x)}{2}
\end{equation*}
\notag
$$
are the even and odd parts of a function $f$, respectively. At points of removable discontinuity functions are defined by continuity; in other cases, $0/0$ is set to be $0$. The empty sum is defined to be 0, the empty product is defined to be 1; $\sum_{k\in\mathbb Z}c_k=\lim_{N\to\infty} \sum_{k=-N}^{N}c_k$. A sequence $\{c_k\}_{k\in\mathbb Z}$ is said to alternate in sign if there exists $\varepsilon\in\mathbb C\setminus\{0\}$ such that $\varepsilon(-1)^kc_k\geqslant0$ for all $k\in\mathbb Z$.
§ 2. Preliminaries2.1. Harmonic analysis generated by Dunkl operatorsThe differential-difference operators $\Lambda$ defined by are known as Dunkl operators. These operators were introduced (in the multivariate case right away) by Dunkl in a series of papers, of which we mention [20].If a function $f$ is even and twice differentiable, then $\Lambda^2f=Lf$, where $L$ is a Bessel operator, The harmonic analysis generated by the Dunkl operator on functions defined on $\mathbb R$ is an extension of the harmonic analysis generated by the Bessel operator on the set of even functions (or, equivalently, on functions defined on ${\mathbb R}_+$). Recall some facts about the harmonic analysis generated by the Dunkl operator.The normalized Bessel function $j_{\alpha}$ is defined by the power series
$$
\begin{equation*}
j_{\alpha}(z)= \Gamma(\alpha+1)\sum_{k=0}^{\infty}\frac{(-1)^k}{k!\,\Gamma(k+\alpha+1)} \biggl(\frac{z}{2}\biggr)^{2k}, \qquad z\in\mathbb C.
\end{equation*}
\notag
$$
This is an even entire function of exponential type $1$. Consider the functions
$$
\begin{equation*}
\varphi_{y}(x)=j_{\alpha}(yx)\quad\text{and} \quad e_{0}=1, \qquad e_{y}=\varphi_{y}+\frac{1}{iy}\varphi'_{y}, \quad y\ne0.
\end{equation*}
\notag
$$
The function $e_{y}$ is expressed in terms of the Bessel functions by
$$
\begin{equation*}
e_{y}(x)= j_{\alpha}(yx)+ \frac{i}{2(\alpha+1)}yx\,j_{\alpha+1}(yx).
\end{equation*}
\notag
$$
For any $y\in\mathbb C$ the function $\varphi_{y}$ is a unique solution of the Cauchy problem (see, for example, § 1.2.2 in [21]), and the function $e_{y}$ is the unique solution of the problem (see, for example, Lemma 1 in [22]). In the classical case $\alpha=-1/2$ the functions $\varphi_y$ and $e_y$ are cosines and exponentials, respectively:
$$
\begin{equation*}
\varphi_y(x)=\cos{yx}\quad\text{and}\quad e_y(x)=e^{iyx}.
\end{equation*}
\notag
$$
The direct and inverse Fourier-Dunkl transforms are defined on $L_{1,\alpha}$ by
$$
\begin{equation*}
\mathcal Ff(y) =\int_{\mathbb R}f(t)e_{-y}(t) |t|^{2\alpha+1}\,dt
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\mathcal F^{-1}g(x) = \frac{1}{(2^{\alpha+1}\Gamma(\alpha+1))^2}\int_{\mathbb R}g(y)e_{y}(x)|y|^{2\alpha+1}\,dy.
\end{equation*}
\notag
$$
If $f,\mathcal F\in L_{1,\alpha}$ and $f$ is continuous, then the inversion formula ${\mathcal F}^{-1}{\mathcal F}f=f$ holds (see Theorem 4.20 in [23]). Let ${\mathbf V}_{\alpha,\sigma}$ be the set of functions $f$ representable in the form where $\rho$ is a function of bounded variation, or, equivalently, a signed Borel measure on $[-\sigma,\sigma]$. Interpreting a signed measure $\rho$ as a distribution we deduce that $\rho$ is defined uniquely by $f$, which follows from the uniqueness theorem for the inverse Fourier transform (see [24], Lemma 3.3, and [25], Theorem 4.5). We denote this signed measure by $\rho(f)$. Since $e_y\in \mathbf E_y$ and $|e_y|\leqslant1$ for all $y$, we have ${\mathbf V}_{\alpha,\sigma}\subset{\mathbf B}_{\sigma}$. If $f\in L_{1,\alpha,\sigma}$, then by the Paley-Wiener theorem (see [25], Theorem 4.5) $\mathcal Ff=0$ outside $[-\sigma,\sigma]$. Moreover, the inversion formula (2.1) holds, in which
$$
\begin{equation*}
d\rho(f,y)=\mathcal Ff(y) \frac{|y|^{2\alpha+1}\,dy}{(2^{\alpha+1}\Gamma(\alpha+1))^2}.
\end{equation*}
\notag
$$
As a result, $L_{1,\alpha,\sigma}\subset{\mathbf V}_{\alpha,\sigma}$. Given a function $f\in C^{(\infty)}(\mathbb R)$, let $Tf$ be the bivariate function defined as the unique solution of the Cauchy problem for the differential-difference equation (see, for example, Theorem 5 in [22]). Here $\Lambda_x$ and $\Lambda_y$ denote, respectively, the operator $\Lambda$ applied to the first and second argument, that is, $\Lambda_xv(x,y)=\Lambda v(\cdot,y)(x)$ and $\Lambda_yv(x,y)=\Lambda v(x,\cdot)(y)$. Below $Tf(x,y)$ and $T^yf(x)$ are synonyms. The operator $T^y$ is known as the generalized translation operator generated by the operator $\Lambda$.In particular, $T^0f=f$ and $T^ye_u=e_ye_u$. It is known (see Theorem 2.4 in [24] and also [26]) that for $x,y\ne0$ the operator $T$ can be represented as
$$
\begin{equation}
Tf(x,y)=\int_{\mathbb R} f(z)W(x,y,z)|z|^{2\alpha+1}\,dz,
\end{equation}
\tag{2.2}
$$
where the kernel $W$ has the following properties: $W(x,y,\cdot)\!\in\! L_{1,\alpha}$ and ${W(x,y,z)\!=\!0}$ for $|z|\notin\bigl[\bigl||x|-|y|\bigr|,\,|x|+|y|\bigr]$. The explicit form of the kernel $W$ is not required in what follows. If $y\in\mathbb R\setminus\{0\}$ and a function $f$ is locally absolutely integrable with weight $|\cdot|^{2\alpha+1}$, then the right-hand side of (2.2) exists for almost all $x$ (see Corollary 5 in [26]). Hence formula (2.2) extends the operator $T$ to the class of such functions. We mention some properties of the operator $T$. T1. If $y\in\mathbb R$, $p\in[1,+\infty]$ and $f\in L_{p,\alpha}$, then $T^yf\in L_{p,\alpha}$. If $f\in \mathrm{CB}(\mathbb R)$, then $Tf\in \mathrm{CB}(\mathbb R^2)$ and, in particular, $T^yf\in \mathrm{CB}(\mathbb R)$. Property T1 for $L_{p,\alpha}$ is contained in Lemma 1 in [27] and for $\mathrm{CB}(\mathbb R)$ in Theorem 3.1 in [24]; for the result in both cases, see [26], Theorem 4 and Corollary 6. T2. If $f\in\mathbf V_{\alpha,\sigma}$, then $T^hf\in\mathbf V_{\alpha,\sigma}$ and $d\rho(T^hf,y)=e_y(h)\,d\rho(f,y)$. For a proof it suffices to substitute (2.1) into (2.2) and change the order of integration. For functions $f$ in $L_{1,\alpha}$ (in particular, in the Schwartz class $\mathcal S$) or in $L_{2,\alpha}$ it is known that $\mathcal FT^hf=e_h\cdot\mathcal Ff$ (see, for example, formulae (4.2) in [28] and (2.25) in [29]). However, the operators $\mathcal F$ and $T^y$ can be defined in the usual way on the space $\mathcal S'$ of tempered distributions (for the Fourier-Dunkl transform this was done, for example, in [25], [29] and [17], and for the generalized translation, in [17]); of course, the above equality remains valid here. Property T2 is a particular case of this equality (recall that $e_y(h)=e_h(y)$). T3. If $y\in\mathbb R$, $p\in[1,+\infty]$ and $f\in L_{p,\alpha}$, then
$$
\begin{equation}
\|T^yf\|_{p,\alpha}\leqslant2^{|1-2/p|}\|f\|_{p,\alpha},
\end{equation}
\tag{2.3}
$$
$$
\begin{equation}
\|T^yf\pm T^{-y}f\|_{p,\alpha}\leqslant2\|f\|_{p,\alpha}.
\end{equation}
\tag{2.4}
$$
Property T3 (in full form) was proved in Theorems 3, 4 and Corollary 4 in [26]. For further purposes (at least, for $p=+\infty$) it is important that inequalities (2.4) hold with constant $2$. Unlike the ordinary shift, the operator $T^y$ is not positive for $y\ne0$. Hence its norm (as an operator from $L_{\infty}$ to $L_{\infty}$) is greater than $1$. This makes the situation more involved in comparison with the classical case. However, as in the classical case, the mean $\square_{y}=(T^y+T^{-y})/2$ is known to be a norm-one positive operator, and the norm of the generalized difference $T^y-T^{-y}$ is at most 2 (in Remark 7 in [26] the last inequality was shown to be sharp for $p=1,+\infty$). The operator $\square_{y}$ was introduced in a different way in [30], formula (14). In [30], an integral representation for this operator (different from the arithmetic mean of the expressions in (2.2) by a change of the variable) was also obtained, the operator $\square_{y}$ was shown to be positive, and its norm was estimated. In Theorem 3.1 in [15] an analogous estimate was obtained in the multivariate case. In [26] inequalities (2.3) and (2.4) were proved (in the case of one variable) for the generalized shift constructed on the basis of more general differential-difference operators. 2.2. Absolutely monotone functionsIn what follows we consider absolutely monotone and logarithmically absolutely monotone functions on intervals of the form $[0,b\rangle \subset\mathbb R$, where $b\in(0,+\infty]$. (Throughout, an angular bracket in notation like $[0,b\rangle$ indicates that the corresponding endpoint lies or does not lie in the interval). Definition. A function $f$ is said to be absolutely monotone on an interval $[0,b\rangle $ if $f$ admits a power series expansion on $[0,b\rangle $ with nonnegative coefficients: A function $f$ is said to be logarithmically absolutely monotone on $[0,b\rangle $ if $f$ is continuous on $[0,b\rangle $, differentiable on $[0,b)$, $f>0$, and $(\ln f)'$ is absolutely monotone on $[0,b)$. For equivalent definitions of absolute monotonicity, see [31]. Absolute monotone and logarithmically absolutely monotone functions have great value in probability; for details, see [32]. In [12], [13] and [33] absolute monotonicity was used to strengthen sharp inequalities for derivatives and differences of trigonometric polynomials and entire functions of exponential type. We require the following elementary properties of absolutely monotone functions (cf. [33]). M1. Each logarithmically absolutely monotone function is absolutely monotone. M2. If a function $f$ is even and absolutely monotone on $[0,b\rangle $, then it is nonnegative and convex on $\langle -b,b\rangle $. M3. If $f$ is logarithmically absolutely monotone on $[0,b\rangle $, then for any $r>0$ the $r$th power $f^r$ of $f$ is logarithmically absolutely monotone on $[0,b\rangle $. M4. If $f$ is logarithmically absolutely monotone on $[0,b\rangle $ and $\beta> \alpha>0$, then the function $z\mapsto f(\beta z)/f(\alpha z)$ is logarithmically absolutely monotone on $[0,b/\beta\rangle $. Lemma 1. If $\alpha\geqslant-1/2$, then the function $1/j_{\alpha}$ is logarithmically absolutely monotone on $[0,\tau_1)$, where $\tau_{1}$ is the first positive zero of $j_{\alpha}$. Proof. The infinite product expansion of the Bessel function $j_{\alpha}$ is as follows:
$$
\begin{equation*}
j_{\alpha}(x)=\prod_{s=1}^{\infty}\biggl(1-\frac{x^2}{\tau_s^2}\biggr);
\end{equation*}
\notag
$$
here the $\tau_s$ are the positive zeros of $j_{\alpha}$ arranged in increasing order (see, for example, [34], § 15.41). It is clear that $j_{\alpha}>0$ on $(-\tau_1,\tau_1)$. It remains to insert the product formula for $j_{\alpha}$ into the denominator and use the logarithmic absolute monotonicity of each factor,
Lemma 1 is proved. Remark 1. Another method of the proof of Lemma 1 is to find a recurrent relation for the Taylor coefficients of the function $\ln(1/j_{\alpha})$ from the Bessel differential equation. If $\alpha=-1/2$, then $j_{\alpha}(x)=\cos{x}$; in this case the result of the lemma is known (see [33]). 2.3. The transmutation operator (intertwining operator)For further purposes we need to expand a function $g$ defined on $[-\sigma,\sigma]$ in a Schlömilch-type series
$$
\begin{equation}
g(y)=\sum_{l\in\mathbb Z}c_l e_{y}\biggl((l+\theta)\frac{\pi}{\sigma}\biggr)
\end{equation}
\tag{2.5}
$$
satisfying the additional requirement This can conveniently be done using the transmutation operator. For an account of Schlömilch series, see, for example, [34], Ch. 19. A Schlömilch series expansion (2.5) is not unique in general. However, as we will see below from property P2, for fixed $\theta$ and under the additional condition (2.6) a Schlömilch series expansion is unique if exists. Schlömilch series expansions were used in the proof of Bernstein-type inequalities in [19], and some expansions of the form (2.5) were employed in [17]. The Poisson operator
$$
\begin{equation}
Rf(y)= \frac{2\Gamma(\alpha+1)}{\Gamma(\alpha+1/2)\sqrt{\pi}} \int_{0}^{1}f(yt)(1-t^2)^{\alpha-1/2}\,dt
\end{equation}
\tag{2.7}
$$
has proved to be widely useful in the theory of Fourier-Bessel expansions (see, for example, [35] and [21], Ch. 2). The index $\alpha$ will be dropped as usual. The operator $P$ defined by
$$
\begin{equation}
Pf(y)=Rf_c(y)+\frac{1}{y}R\bigl(x\mapsto x f_s(x),y\bigr)
\end{equation}
\tag{2.8}
$$
plays an equally important role in the theory of Fourier-Dunkl expansions. Note that $P=R$ for even functions. The operators $P$ were introduced by Dunkl ([36], Theorem 5.1). The domains of the operators $R$ and $P$ and those of the functions to which they are applied can be chosen in various ways. It will be important for us to be able to apply such operators not only to functions on $\mathbb R_+$ or $\mathbb R$, but also to ones defined on a bounded interval. The operator $R$ applies both to functions $f$ defined on an interval of the form $[0,b\rangle $, where $b\in(0,+\infty]$, and to even functions $f$ defined on a symmetric interval $\langle -b,b\rangle $. In the first case we will consider the even extensions of functions. The operator $P$ applies to functions $f$ defined on a symmetric interval $\langle -b,b\rangle $. The definition (2.7) makes sense for locally absolutely integrable functions $f$. In order that the definition (2.8) be meaningful it suffices to assume that the functions $f_c$ and $x\mapsto xf_s(x)$ are locally absolutely integrable. Note that the domains of the functions $Rf$ and $Pf$ are equal to that of $f$. If a function $f$ is locally absolutely integrable (hence so are the functions $f_c$ and $x\mapsto xf_s(x)$), then
$$
\begin{equation*}
Pf(y)=\frac{\Gamma(\alpha+1)} {\Gamma(\alpha+1/2)\sqrt{\pi}} \int_{-1}^{1}f(yt)(1-t^2)^{\alpha-1/2}(1+t)\,dt.
\end{equation*}
\notag
$$
Recall several known facts about the inversion of the operators $R$ and $P$ on classes of absolutely integrable functions. We can assume without loss of generality that $b\in(0,+\infty)$, $\mathcal D(R)=L_1[0,b]$ and $\mathcal D(P)=\bigl\{f\colon f_c\in L_1[-b,b], \,x\mapsto xf_s(x)\in L_1[-b,b]\bigr\}$. The parameter $b$ being arbitrary, the corresponding results also hold for locally absolutely integrable functions. The Poisson operator is expressed in terms of the Riemann-Liouville fractional integration operator
$$
\begin{equation*}
I^{\gamma}_{0+}f(x)=\frac{1}{\Gamma(\gamma)} \int_{0}^{x}(x-t)^{\gamma-1}f(t)\,dt, \qquad \gamma>0, \quad x>0
\end{equation*}
\notag
$$
(here we follow the notation in [37]), by the formula
$$
\begin{equation}
Rf(y)= \frac{\Gamma(\alpha+1)}{\sqrt{\pi}}y^{-2\alpha} I^{\alpha+1/2}_{0+} \biggl(\frac{f(\sqrt{\cdot}\,)}{\sqrt{\cdot}},y^2\biggr), \qquad y>0.
\end{equation}
\tag{2.9}
$$
This can easily be verified by changing the variable. The operator $I^{\gamma}_{0+}$ is injective on its domain $L_1[0,b] $ (see Lemma 2.5 in [37]). A function $f$ is absolutely integrable if and only if $f(\sqrt{\cdot}\,)/\sqrt{\cdot}$ is too. Hence by (2.9) the operator $R$ is injective. Applying these results to the even and odd parts of the function, we find that the operator $P$ is injective. The range of the Riemann-Liouville operator was described in Theorem 2.3 of [37] in terms of fractional integrals. The range of the Poisson operator $\mathcal R(R)$ can be described by combining this theorem with (2.9). By (2.8), the inclusion $g\in\mathcal R(P)$ is equivalent to the pair of inclusions $g_c\in\mathcal R(R)$, $y\mapsto yg_s(y)\in\mathcal R(R)$. No explicit description of the ranges of these operators will be required in what follows. The inverse operator $R^{-1}$ can be found using fractional differentiation (see Theorem 2.4 in [37]) and (2.9). Its explicit expression is given by formula (18.17) in [37] for $a=0$, $\sigma=2$, $\eta=-1/2$ and $\alpha$ replaced by $\alpha+1/2$. If $g\in\mathcal R(R)$, then
$$
\begin{equation}
\begin{aligned} \, R^{-1}g(x) &=\frac{2\sqrt{\pi}x}{\Gamma(\alpha+1)\Gamma(1-\{\alpha+1/2\})} \nonumber \\ &\qquad\times \biggl(\frac{d}{dx^2}\biggr)^{\lfloor\alpha+1/2\rfloor+1} \int_{0}^{x} (x^2-y^2)^{-\{\alpha+1/2\}}y^{2\alpha+1}g(y)\,dy. \end{aligned}
\end{equation}
\tag{2.10}
$$
For $\alpha=k+1/2$, $k\in\mathbb Z_+$, formula (2.10) assumes a simpler form:
$$
\begin{equation}
R^{-1}g(x)= \frac{\sqrt{\pi}x}{\Gamma(\alpha+1)} \biggl(\frac{d}{dx^2}\biggr)^{k+1}(x^{2k+1}g(x)).
\end{equation}
\tag{2.11}
$$
The inverse operator $P^{-1}$ is obtained by considering the even and odd parts of the function separately: if $g\in\mathcal R(P)$, then
$$
\begin{equation}
P^{-1}g(x)=R^{-1}g_c(x)+\frac{1}{x}R^{-1}\bigl(y\mapsto yg_s(y),x\bigr).
\end{equation}
\tag{2.12}
$$
Equalities (2.10) and (2.11) are well known. For $\alpha\in(-1/2,1/2)$ the right-hand side of (2.10) defines the Sonine operator (for example, in the terminology of [38], which is slightly different from that used in [35]). In § 9 of [35], formulae (2.10) and (2.11) were derived for even functions $g\in C^{(k+1)}(\mathbb R)$ (in an equivalent form); the corresponding expressions for even functions $g\in C^{(\infty)}(\mathbb R)$ are given in Theorem 2.1.2 of [21]. Formula (2.12) was obtained in Remark 2 of [22] and Remark 3.4 of [25] for functions $g\in C^{(\infty)}(\mathbb R)$. Note that in [21], formula (2.1.32), and in [22], Remark 1.1.b, the coefficient in (2.11) is wrong. In our paper the only nonsmooth functions to which the operator $P^{-1}$ is applied are linear combinations of functions of the form $y\mapsto|y|^r\psi_1(y)$ and $y\mapsto y^{\langle r\rangle }\psi_2(y)$, where $r\geqslant0$, and $\psi_1$ and $\psi_2$ are infinitely differentiable even functions. That these functions lie in $\mathcal R(P)$ is clear, since by Taylor’s theorem they can be represented as sums of several monomials (with generally noninteger exponents) and a sufficiently smooth residual term. Remark 2. The functions $R^{-1}g$ and $P^{-1}g$ are defined up to equivalence. If an equivalence class contains a continuous representative, then we assume that $R^{-1}g$ (or $P^{-1}g$) is set to be this representative. Let us mention some other properties of the operators $P$. P1. If $u,y\in\mathbb R$, then $P(\exp(iu\cdot),y)=e_{y}(u)$. This is the core property of the transmutation operator; it follows from Poisson’s formula for Bessel functions (see, for example, [21], formula (2.1.15), and [24], Lemma 2.1). P2. If
$$
\begin{equation}
\sum_{l\in\mathbb Z}|c_{l}|<+\infty\quad\textit{and} \quad f(x)=\sum_{l\in\mathbb Z}c_{l}e^{i((l+\theta)\pi/\sigma)x},
\end{equation}
\tag{2.13}
$$
then
$$
\begin{equation*}
Pf(y)= \sum_{l\in\mathbb Z}c_{l}e_{y}\biggl((l+\theta)\frac{\pi}{\sigma}\biggr).
\end{equation*}
\notag
$$
Property P2 is secured by P1. By property P2 a function $g$ can be written as (2.5) under condition (2.6) if and only if $g\in P(A_{\theta})$, where $A_{\theta}$ is the set of functions $f$ of the form (2.13). Substituting $x=\sigma$ into (2.13) we find that
$$
\begin{equation}
e^{-i\theta\pi}P^{-1}g(\sigma)=e^{i\theta\pi}P^{-1}g(-\sigma).
\end{equation}
\tag{2.14}
$$
As a result, a necessary condition for expansibility is $|P^{-1}g(\sigma)|=|P^{-1}g(-\sigma)|$. If this condition is met and, in addition $P^{-1}g(\sigma)\ne0$, then the parameter $\theta$ can be found from (2.14) uniquely modulo $1$ (the change of $\theta$ by $\theta+1$ corresponds to a shift in the summation index). As the system of exponentials is orthogonal, the coefficients $c_l$ are uniquely defined, for fixed $\theta$, by the equality
$$
\begin{equation*}
c_l=\frac{1}{2\sigma}\int_{-\sigma}^{\sigma} P^{-1}g(t)e^{-i((l+\theta){\pi}/{\sigma})t}\,dt.
\end{equation*}
\notag
$$
We denote these coefficients by $c_l^{(\theta)}(g)$. P3. The following equalities hold:
$$
\begin{equation}
P\bigl(t\mapsto |t|^r,y\bigr) =\frac{\Gamma((r+1)/2)\Gamma(\alpha+1)} {\sqrt{\pi}\, \Gamma(r/2+\alpha+1)}|y|^r, \qquad r>-1,
\end{equation}
\tag{2.15}
$$
and
$$
\begin{equation}
P\bigl(t\mapsto t^{\langle r\rangle },y\bigr) =\frac{\Gamma(r/2+1)\Gamma(\alpha+1)} {\sqrt{\pi}\, \Gamma((r+3)/2+\alpha)}y^{\langle r\rangle }, \qquad r>-2.
\end{equation}
\tag{2.16}
$$
These formulae can easily be obtained by writing integrals in terms of the beta function. For $r\in\mathbb Z_+$ such formulae can be found in Theorem 5.1 in [36]. P4. If $R\in(0,+\infty]$ and then
$$
\begin{equation}
Pf(y)=\sum_{k=0}^{\infty}b_kc_{k}y^k, \qquad |y|<R,
\end{equation}
\tag{2.18}
$$
where
$$
\begin{equation}
b_k=\begin{cases} \dfrac{\Gamma((k+1)/2)\Gamma(\alpha+1)}{\sqrt{\pi}\, \Gamma(k/2+\alpha+1)}, &k\textit{ is even}, \\ \dfrac{\Gamma(k/2+1)\Gamma(\alpha+1)}{\sqrt{\pi}\,\Gamma((k+3)/2+\alpha)}, &k\textit{ is odd}. \end{cases}
\end{equation}
\tag{2.19}
$$
The series (2.17) and (2.18) have the same radius of convergence. Indeed, for each $b\in(0,R)$ the series (2.17) converges uniformly on the interval $[-b,b]$. Hence the operator $P$ can be applied termwise to the sum of the series, after which property P3 should be used. The sequence $\{b_k\}$ decays with a power rate, and so the radii of convergence are equal. Remark 3. Formula (2.19) and the Abel test show that if $ R < + \infty$ and if the series (2.17) converges at $R$ or $-R$, then the same is also true for the series (2.18). However, the converse result fails. From property P4, one can express the inverse operator $P^{-1}$ in terms of power series; in parallel with formulae (2.10)–(2.12) property P4 can be used in approximate calculations. The following result is immediate from property P4. P5. A function $f$ is analytic on $(-R,R)$ if and only if $Pf$ is too. The operator $P$ preserves the sign of the Maclaurin coefficients. In particular, the operators $P$ and $P^{-1}$ send absolutely monotone functions on $[0,R)$ to absolutely monotone ones. § 3. General sharp inequalities for entire functions of exponential typeLet $\lambda,\mu\in C[-\sigma,\sigma]$. Consider the operators
$$
\begin{equation}
Uf(x)=\int_{-\sigma}^{\sigma} \lambda(y)e_{y}(x)\,d\rho(f,y)
\end{equation}
\tag{3.1}
$$
and
$$
\begin{equation}
Vf(x)=\int_{-\sigma}^{\sigma} \mu(y)e_{y}(x)\,d\rho(f,y)
\end{equation}
\tag{3.2}
$$
defined on the set ${\mathbf V}_{\alpha,\sigma}$ of functions of the form (2.1). It is clear that $U,V$: ${{\mathbf V}_{\alpha,\sigma}\to{\mathbf V}_{\alpha,\sigma}}$. We are interested in an inequality of the form
$$
\begin{equation}
\|Uf\|_{p,\alpha}\leqslant M\|Vf\|_{p,\alpha}, \qquad f\in {\mathbf V}_{\alpha,\sigma},
\end{equation}
\tag{3.3}
$$
and in the best constant $M$ in this inequality. 3.1. Upper estimates Lemma 2. 1. Let $\sigma>0$, $\theta\in\mathbb R$, $\mu\in C[-\sigma,\sigma]$, 2. If, in addition, $p\in[1,+\infty]$ and $Vf\in L_{p,\alpha}$, then $Uf\in L_{p,\alpha}$ and If, under the above assumptions, or then Proof. 1. Substituting the series expansion of $g$ into (3.1) and integrating this series termwise (integration is justified by uniform convergence) we arrive at (3.4):
$$
\begin{equation*}
\begin{aligned} \, Uf(x)&=\int_{-\sigma}^{\sigma} \sum_{l\in\mathbb Z}c_l^{(\theta)}(g)e_{y} \biggl((l+\theta)\frac{\pi}{\sigma}\biggr)e_{y}(x)\,d\rho(Vf,y) \\ &=\sum_{l\in\mathbb Z}c_l^{(\theta)}(g)\int_{-\sigma}^{\sigma} e_{y}(x)\,d\rho(T^{(l+\theta)\pi/\sigma}Vf,y) =\sum_{l\in\mathbb Z}c_l^{(\theta)}(g)T^{(l+\theta)\pi/\sigma}Vf(x). \end{aligned}
\end{equation*}
\notag
$$
2. By (2.3) the series on the right in (3.4) converges in the $L_{p,\alpha}$-norm; this implies the inclusion $Uf\in L_{p,\alpha}$ and estimate (3.6). We verify (3.9). In the case (3.7) we have $c_{-l}^{(0)}(g)=c_{l}^{(0)}(g)$, so that
$$
\begin{equation*}
Uf(x)=c_0^{(0)}(g)Vf(x)+\sum_{l=1}^{\infty}c_l^{(0)}(g) (T^{l\pi/\sigma}+T^{-l\pi/\sigma})Vf(x).
\end{equation*}
\notag
$$
In the case (3.8) we have $c_{-l-1}^{(1/2)}(g)=-c_{l}^{(1/2)}(g)$, so that
$$
\begin{equation*}
Uf(x)= \sum_{l=0}^{\infty}c_l^{(1/2)}(g) (T^{(l+1/2)\pi/\sigma}-T^{-(l+1/2)\pi/\sigma}) Vf(x).
\end{equation*}
\notag
$$
It remains to use inequalities (2.4).
Lemma 2 is proved. Corollary 1. 1. Let $\sigma>0$, $\theta\in\mathbb R$, 2. If, in addition, $p\in[1,+\infty]$ and $f\in L_{p,\alpha}$, then $Uf\in L_{p,\alpha}$ and
$$
\begin{equation*}
\|Uf\|_{p,\alpha}\leqslant 2^{|1-2/p|} \biggl(\sum_{l\in\mathbb Z}|c_l^{(\theta)}(\lambda)|\biggr) \|f\|_{p,\alpha}.
\end{equation*}
\notag
$$
If, under the above assumptions, either condition (3.7) or (3.8) is met, then For a proof it suffices to put $\mu\equiv1$ in Lemma 2. In many cases the conclusions of Lemma 2 and Corollary 1 hold on the class ${\mathbf B}_{\sigma}$, which is wider than ${\mathbf V}_{\alpha,\sigma}$. We say that a sequence of functions $f_n$ defined on $\mathbb R$ converges boundedly locally uniformly to a function $f$ if $f_n\to f$ uniformly on each finite closed interval and the functions $f_n$ are uniformly bounded. An operator $V\colon {\mathbf B}_{\sigma}\to \mathrm{CB}({\mathbb R})$ is said to be closed under bounded locally uniform convergence if, for all functions $f_n$ and $f$ in ${\mathbf B}_{\sigma}$ and $h\in \mathrm{CB}({\mathbb R})$, the equality $h=Vf$ holds whenever $f_n$ and $Vf_n$ converge boundedly locally uniformly to $f$ and $h$, respectively. In addition to the continuous operators from $\mathrm{CB}({\mathbb R})$ into $\mathrm{CB}({\mathbb R})$, examples of operators which are closed under bounded locally uniform convergence include the differentiation operators $D^r$ and thus the operators $\Lambda^r$ and $L^r$, $r\in\mathbb N$. Lemma 3. For each function $f\in{\mathbf B}_{\sigma}$ there exists a sequence of functions $f_n$ in $L_{1,\alpha,\sigma}$ (hence in ${\mathbf V}_{\alpha,\sigma}$) such that $f_n\to f$ uniformly on any finite closed interval and $\|f_n\|_{\infty}\leqslant\|f\|_{\infty}$. To construct such a sequence it suffices to consider the functions
$$
\begin{equation*}
h_{\varepsilon}(t)=f\biggl(\biggl(1-\frac{\varepsilon}{\sigma}\biggr)t\biggr)\biggl( \frac{N}{\varepsilon t}\sin\frac{\varepsilon t}{N}\biggr)^N,\qquad \varepsilon\in(0,\sigma),
\end{equation*}
\notag
$$
which lie in $L_{1,\alpha,\sigma}$ for $N>2\alpha+2$, and to set $f_n=h_{\varepsilon_n}$, where $\varepsilon_n\to0+$. Lemma 4. Let an operator $U\colon {\mathbf B}_{\sigma}\to \mathrm{CB}({\mathbb R})$ be defined for $f\in {\mathbf V}_{\alpha,\sigma}$ by (3.1) and be closed under bounded locally uniform convergence. Then, under the hypotheses of Corollary 1, the condition $f\in{\mathbf V}_{\alpha,\sigma}$ can be replaced by $f\in{\mathbf B}_{\sigma}$. Proof. Let $f\in{\mathbf B}_{\sigma}$ and let $h(x)$ be the sum of the series on the right-hand side of (3.10). Using Lemma 3 we find a sequence of functions $f_n$ from ${\mathbf V}_{\alpha,\sigma}$ such that $f_n\to f$ uniformly on any finite closed interval and $\|f_n\|_{\infty}\leqslant\|f\|_\infty$. Hence
We claim that $Uf_n\to h$ uniformly on any closed interval $[-a,a]$; Once this is shown, the closedness of $U$ will imply that $Uf=h$. Given $\varepsilon>0$, we find $N \in \mathbb N$ such that $\sum_{|l|>N}|c_l^{(\theta)}(\lambda)|\,\|f\|_{\infty} < {\varepsilon}/{8}$. We also set $a_N\,{=}\,a+(|\theta|+N)\pi/\sigma$. The kernel of the operator $T^y$ vanishes for $|z|>|x|+|y|$; hence for all $x\in[-a,a]$, we have
$$
\begin{equation*}
\begin{aligned} \, &|Uf_n(x)-h(x)| \\ &\qquad \leqslant 2\sum_{|l|>N}|c_l^{(\theta)}(\lambda)|\,\|f_n-f\|_{\infty} +\sum_{l=-N}^{N}|c_l^{(\theta)}(\lambda)|\,\bigl|T^{(l+\theta)\pi/\sigma}(f_n-f)(x)\bigr| \\ &\qquad\leqslant 4\sum_{|l|>N}|c_l^{(\theta)}(\lambda)|\,\|f\|_{\infty}+ 2\sum_{l\in\mathbb Z}|c_l^{(\theta)}(\lambda)| \max_{[-a_N,a_N]}|f_n-f|. \end{aligned}
\end{equation*}
\notag
$$
It remains to apply the uniform convergence of $f_n$ to $f$ on $[-a_N,a_N]$ and find $n_0$ such that, for all $n > n_0$, the second term is smaller than ${\varepsilon}/{2}$.
This proves Lemma 4. Remark 4. Under the hypotheses of Corollary 1 equality (3.10) provides a natural way to extending the operator $U$ to the set ${\mathbf B}_{\sigma}$. Lemma 5. Let $\sigma>0$, $\theta,\beta\in{\mathbb R}$, Proof. Assume that a sequence $\{f_n\}_{n=1}^{\infty}\subset{\mathbf V}_{\alpha,\sigma}$ converges boundedly locally uniformly to $f\in{\mathbf B}_{\sigma}$. By Lemma 2,
$$
\begin{equation*}
Uf_n(x)=\sum_{l\in\mathbb Z}c_l^{(\theta)}(g) T^{(l+\theta)\pi/\sigma}Vf_n(x),
\end{equation*}
\notag
$$
while by Lemma 4 the sequence $Vf_n$ converges boundedly locally uniformly to $Vf$ and $Vf\in{\mathbf B}_{\sigma}$. It now remains to repeat the argument from the proof of Lemma 4 for $\lambda$, $f_n$ and $f$ replaced by $g$, $Vf_n$, and $Vf$, respectively.
This proves Lemma 5. Li, Su and Ivanov (see Theorem 5.5 in [17]) proved that if $\lambda\in C^{(\infty)}(\mathbb{R})$ and $\lambda$ grows polynomially, then for any $\sigma>0$, $p\in[1,+\infty]$ and $f\in L_{p,\alpha,\sigma}$ we have where the coefficients $d_k$ depend on $\sigma$. In the general case this expansion differs from (3.10), but these expansions are equal for even and odd functions $f$, since in this case $d_{2k-1}=0$ or $d_{2k}=0$. As a particular case (see [17], Theorem 5.6),
$$
\begin{equation*}
\Lambda f(x)=\frac{(2\alpha+2)\sigma}{\pi^2}\sum_{l\in\mathbb Z}\frac{(-1)^l}{(l+1/2)^2}T^{(l+1/2)\pi/\sigma}f(x);
\end{equation*}
\notag
$$
for $\alpha=-1/2,$ and this formula becomes M. Riesz’s well-known identity for the derivative. 3.2. Lower estimatesIn this section, for $p=+\infty$ we, in some cases, estimate sharp constants from below and show that our inequalities are sharp. Lemma 6. Under the hypotheses of Lemma 2 or Lemma 5 let $p=+\infty$, let the sequence $\{c_l^{(\theta)}(g)\}$ alternate in sign, and let there exist a function $f^{*}_{\sigma}\in \mathbf B_{\sigma}$ such that Proof. We have Hence completing the proof of the lemma. Equation (3.11) is trivially solvable if $V$ is the identity operator. However, a function $\mu$ can vanish in many important examples, which is an obstacle to the invertibility of the operator $V$. The next lemma shows that if $\mu\ne0$ almost everywhere on $[-\sigma,\sigma]$, then (3.11) is solvable in a certain approximate sense, which is sufficient for a lower estimate. Lemma 7. Under the hypotheses of Lemma 2 or Lemma 5 let $p=+\infty$, let the sequence $\{c_l^{(\theta)}(g)\}$ alternate in sign, and let $\mu\ne0$ almost everywhere on $[-\sigma,\sigma]$. Then the sharp constant in inequality (3.6) is no smaller than $\sum_{l\in\mathbb Z}|c_l^{(\theta)}(g)|$. As a result, inequality (3.9) is sharp. Proof. We set $h(x)=\cos(\sigma x-\theta\pi)$. Using Lemma 3 we construct functions ${h_{\varepsilon}\in L_{1,\alpha,\sigma}}$ such that $\|h_{\varepsilon}\|_{\infty}\leqslant\|h\|_{\infty}$ and $h_{\varepsilon}\to h$ as $\varepsilon\to0+$ uniformly on any finite closed interval.
Let $E$ be the zero set of the function $\mu$ on $[-\sigma,\sigma]$. For each $n\in\mathbb N$ let $G_n\supset E$ be an open set of measure <$1/n$. We also set $H_{n,\varepsilon}(y)={\mathcal F}h_{\varepsilon}(y)$ for $y\in \mathbb R\setminus G_n$ and $H_{n,\varepsilon}(y)=0$ for $y\in G_n$. We have $H_{n,\varepsilon}\to {\mathcal F}h_{\varepsilon}$ in $L_{1,\alpha}$, and therefore ${\mathcal F}^{-1}H_{n,\varepsilon}\to h_{\varepsilon}$ uniformly on $\mathbb R$ as $n\to\infty$. Now set
$$
\begin{equation*}
f_{n,\varepsilon}(x) =\frac{1}{(2^{\alpha+1}\Gamma(\alpha+1))^2}\int_{-\sigma}^{\sigma} \frac{H_{n,\varepsilon}(y)}{\mu(y)}e_y(x)|y|^{2\alpha+1}\,dy.
\end{equation*}
\notag
$$
This definition is consistent because the function ${H_{n,\varepsilon}}/{\mu}$ is bounded. Hence $f_{n,\varepsilon}\in \mathbf V_{\alpha,\sigma}$ and $Vf_{n,\varepsilon}={\mathcal F}^{-1}H_{n,\varepsilon}$.
Using (3.5) we obtain
$$
\begin{equation*}
\begin{aligned} \, \lim_{\varepsilon\to0+}\lim_{n\to\infty}|Uf_{n,\varepsilon}(0)| &= \lim_{\varepsilon\to0+}\lim_{n\to\infty}\biggl| \sum_{l\in\mathbb Z}c_l^{(\theta)}(g) Vf_{n,\varepsilon}\biggl(\frac{(l+\theta)\pi}{\sigma}\biggr)\biggr| \\ &=\lim_{\varepsilon\to0+}\biggl|\sum_{l\in\mathbb Z}c_l^{(\theta)}(g) h_{\varepsilon}\biggl(\frac{(l+\theta)\pi}{\sigma}\biggr)\biggr| \\ &=\biggl|\sum_{l\in\mathbb Z}c_l^{(\theta)}(g) h\biggl(\frac{(l+\theta)\pi}{\sigma}\biggr)\biggr| =\sum_{l\in\mathbb Z}|c_l^{(\theta)}(g)|. \end{aligned}
\end{equation*}
\notag
$$
Taking the limit is justified here in both cases by the dominated convergence theorem. Next, which proves the lemma. The next theorem combines the above results. Theorem 1. Let $\sigma>0$, $g\in C[-\sigma,\sigma]\cap\mathcal R(P)$, let either condition (3.7) or (3.8) be met, let $P^{-1}g$ be a continuous function, let $\{c_l^{(\theta)}(g)\}$ alternate in sign, let $\mu\in C[-\sigma,\sigma]$ and $\lambda(y)=g(y)\mu(y)$ for $|y|\leqslant\sigma$, and let the operators $U$ and $V$ be defined by (3.1) and (3.2). 1. If $p\in[1,+\infty]$, $f\in{\mathbf V}_{\alpha,\sigma}$ and $Vf\in L_{p,\alpha}$, then $Uf\in L_{p,\alpha}$ and
$$
\begin{equation}
\|Uf\|_{p,\alpha}\leqslant |P^{-1}g(\sigma)| \, \|Vf\|_{p,\alpha}.
\end{equation}
\tag{3.12}
$$
2. If $\sum_{l\in\mathbb Z}|c_l^{(\beta)}(\mu)|<+\infty$ for some $\beta\in{\mathbb R}$, and the operators $U,V\colon {\mathbf B}_{\sigma}\to \mathrm{CB}({\mathbb R})$ are defined by (3.1) and (3.2) at the functions in ${\mathbf V}_{\alpha,\sigma}$ and are closed under bounded locally uniform convergence, then in part 1 the condition $f\in{\mathbf V}_{\alpha,\sigma}$ can be replaced by $f\in{\mathbf B}_{\sigma}$. 3. If $\mu\ne0$ almost everywhere on $[-\sigma,\sigma]$, then for $p=+\infty$ inequality (3.12) is sharp. Proof. 1. Since the function $P^{-1}g$ is continuous and its Fourier coefficients alternate in sign, its Fourier series converges to it everywhere (see, for example, Remark 4 in [12] and Lemma 6 in [13]):
$$
\begin{equation*}
\sum_{l\in\mathbb Z} c_l^{(\theta)}(g)e^{i((l+\theta)\pi/\sigma) x}= P^{-1}g(x), \qquad |x|\leqslant\sigma.
\end{equation*}
\notag
$$
For $x=\sigma$ we have
$$
\begin{equation*}
\sum_{l\in\mathbb Z}|c_l^{(\theta)}(g)| =\biggl|\sum_{l\in\mathbb Z}(-1)^lc_l^{(\theta)}(g)\biggr|=|P^{-1}g(\sigma)|.
\end{equation*}
\notag
$$
It remains to use Lemma 2.
2, 3. Assertion 2 of the theorem follows from Lemma 5, and assertion 3 follows from Lemma 7. This proves Theorem 1. Remark 5. If $\theta=0$, $g$ is even, and the function $P^{-1}g$ is continuous, nonnegative and convex on $[-\sigma,\sigma]$, then $(-1)^lc_l^{(0)}(g)\geqslant0$ for all $l\in{\mathbb Z}_+$ (see, for example, [2], § 4.8.61). Remark 6. The proof of Lemma 7 shows that, under condition (3.7), inequality (3.12) is sharp on the set of even functions, while under condition (3.8) it is sharp on the set of odd functions. Remark 7. If $p \!=\! 2$, the operators $U$ and $V$ are given by (3.1) and (3.2), ${g\!\in\! L_{\infty}[-\sigma,\sigma]}$ and $\lambda=g\mu$, then the sharp constant in inequality (3.3) is equal to § 4. Applications4.1. Bernstein-type inequalitiesFor $r>0$ and $\beta\in\mathbb R$ we define analogues of the Weyl-Nagy differentiation operators (see, for example, § 3.6 in [40]). These analogues, denoted by $\Lambda^{r,\beta}$, are defined on the classes $\mathbf B_{\sigma}$. Given $\sigma>0$, let $f\in\mathbf V_{\alpha,\sigma}$ be as in (2.1). Set
$$
\begin{equation*}
\Lambda^{r,\beta}f(x)=\int_{-\sigma}^{\sigma}e^{(i\beta\pi/2)\operatorname{sign}y}|y|^r e_{y}(x)\,d\rho(f,y).
\end{equation*}
\notag
$$
Let us show that this definition can be carried over to the classes $\mathbf B_{\sigma}$ by (3.10). To this end we set
$$
\begin{equation*}
\lambda(y)=e^{(i\beta\pi/2)\operatorname{sign}y}|y|^r= \cos\frac{\beta\pi}{2}|y|^r+i\sin\frac{\beta\pi}{2}y^{\langle r\rangle }.
\end{equation*}
\notag
$$
In view of (2.15) and (2.16) the function $P^{-1}\lambda$ has a similar form: Let the parameter $\theta$ be defined by (2.14), where $g$ is replaced by $\lambda$. The $2\sigma$-periodic extension of the function $x\mapsto e^{-(i\theta\pi/\sigma)x}P^{-1}\lambda(x)$ has bounded variation and is Lipschitzian. Hence (see Theorem 6.3.6 in [41]) it can be expanded in an absolutely convergent Fourier series in the system $\{e^{(il\pi)/\sigma)x}\}_{l\in\mathbb Z}$. This is equivalent to saying that the function $P^{-1}\lambda$ can be expanded in an absolutely convergent Fourier series (2.13) on the interval $[-\sigma,\sigma]$. So the assumptions of Corollary 1 are met. It is easily seen that the definition is correct in the following sense: if $f\in\mathbf B_{\sigma}$ and $\sigma_1>\sigma$, then the definitions of $\Lambda^{r,\beta}f$ with parameters $\sigma$ and $\sigma_1$ are the same. Considering even and odd $r$ separately, one can easily check that for $\beta=r\in\mathbb N$,
$$
\begin{equation*}
e^{(i\beta\pi/2)\operatorname{sign}y}|y|^r=(iy)^r, \qquad \Lambda^{r,\beta}f=\Lambda^{r}f.
\end{equation*}
\notag
$$
This shows that, for noninteger $r$, the operator $\Lambda^{r,r}$ can also be looked upon as the $r$th power of $\Lambda$. This is analogous to the definition of Weyl-Nagy fractional differentiation (and coincides with this definition in the nonweighted case ${\alpha=-1/2}$). Similarly, for $r\in\mathbb N$ and $\beta=r-1$ we have
$$
\begin{equation*}
e^{(i\beta\pi/2)\operatorname{sign}y}|y|^r=(-i\operatorname{sign}y)(iy)^r;
\end{equation*}
\notag
$$
hence $\Lambda^{r,r-1}f$ is an analogue of the $r$th derivative of the trigonometrically conjugate function. On the class of even functions the operator $D^{r,0}$ coincides with the operator $(-L)^{{r}/{2}}$, which was constructed in [19] also with the use of interpolation formulae. Fractional powers of the operator $-\Lambda^2$ on Lizorkin spaces of generalized functions were constructed in § 3 of [16]. If functions in $\mathbf B_{\sigma}$ are viewed as distributions, then the operator $D^{r,0}$ acts on these functions precisely as the operator $(-\Lambda^2)^{{r}/{2}}$ in [16]. Theorem 2. Let $r\geqslant1$, $\sigma>0$, $p\in[1,+\infty]$ and $f\in L_{p,\alpha,\sigma}$. Then For $p=+\infty$ this inequality is sharp. Proof. Setting
$$
\begin{equation*}
\lambda(y)=|y|^r, \qquad\mu(y)=1\quad\text{and} \quad\theta=0
\end{equation*}
\notag
$$
in Theorem 1 we obtain $g(y)=|y|^r$. In view of (2.15) the function $P^{-1}g$ differs only by a constant factor from $g$; moreover, this function is even, nonnegative and convex on $[-\sigma,\sigma]$. By Remark 5 its Fourier coefficients alternate in sign. It follows from (2.15) that the sharp constant is
This proves Theorem 2. By Remark 6 Theorem 2 gives a sharp inequality for powers of the Bessel operator on the set of even functions. Theorem 3. Let $r\geqslant1$, $\sigma>0$, $p\in[1,+\infty]$ and $f\in L_{p,\alpha,\sigma}$. Then For $p=+\infty$ this inequality is sharp. Proof. Setting
$$
\begin{equation*}
\lambda(y)=i|y|^r\operatorname{sign}y, \qquad\mu(y)=1\quad\text{and} \quad\theta=\frac12
\end{equation*}
\notag
$$
in Theorem 1 we obtain $g(y)=i|y|^r\operatorname{sign}y$. In view of (2.16) the function $P^{-1}g$ differs only by a constant factor from $g$. Its Fourier coefficients are well known to alternate in sign (see [42]). By (2.16) the sharp constant is
This proves Theorem 3. We formulate separately a direct analogue of Bernstein’s inequality, that is, the inequality for integer powers of the operator $\Lambda$. Corollary 2. Let $r\in\mathbb N$, $\sigma>0$, $p\in[1,+\infty]$ and $f\in L_{p,\alpha,\sigma}$. Then For even $r$ Corollary 2 follows from Theorem 2, and for odd $r$, from Theorem 3. Platonov proved (4.3) for even functions and verified its sharpness for $p=+\infty$ (see Theorem 1.3 in [19]). Previously, for integer and half-integer $\alpha$ this result (in other terms, namely, in the form of Bernstein’s inequality for the Laplace operator in $\mathbb R^n$, where $\alpha=n/2-1$) was established by Kamzolov [18]. Inequality (4.2) was proved in [14], where it was also shown that it is sharp for $p=+\infty$ and even $r$. (In [14] the corresponding constants were given in the form $\binom{2s}{s}^{-1}4^s\binom{s+\alpha}{s}$, where $s=r/2$.) An account of the available results on constants in Bernstein-type inequalities can also be found in [14]. Inequality (4.2) with a not sharp constant was presented in Theorem 2.3 in [17]. Order estimates of the form (4.1), and, in particular, for noninteger $r$, were established in Corollary 5.3 of [16]. Previously, for even functions, that is, for (not necessarily integer) powers of the operator $-L$ such estimates were obtained in Theorem 3.4 of [19]; for even $r$ (that is, for integer powers of the operator $-\Lambda^2$) such estimates were proved in [15], Theorem 7.3. 4.2. Riesz-type and Boas-type inequalitiesThere are different nonidentical natural analogues of classical difference operators. The main cause for their noncoincidence is the lack of the group property of generalized shift operators; in general, $T^{u}T^{v}\ne T^{u+v}$. As a rule, in the classical case, central differences turn out to be more convenient that forward and backward ones. This is why we use analogies with central differences in our generalizations. The first-order central difference with step size $h$ is defined by $\delta_h=T^{h/2}-T^{-h/2}$. Let $\delta_h^r$ for $r\in\mathbb Z_+$ be the $r$th power of the operator $\delta_h$. If a function $f\in \mathbf V_{\alpha,\sigma}$ is given by (2.1), then by property T2
$$
\begin{equation*}
\begin{aligned} \, \delta_h^rf(x) &=\int_{-\sigma}^{\sigma}\biggl(e_{y}\biggl(\frac{h}{2}\biggr) -e_{y}\biggl(-\frac{h}{2}\biggr)\biggr)^re_{y}(x)\,d\rho(f,y) \\ &=\int_{-\sigma}^{\sigma} \biggl(\frac{ihy}{2(\alpha+1)}j_{\alpha+1}\biggl(\frac{hy}{2}\biggr)\biggr)^r e_{y}(x)\,d\rho(f,y). \end{aligned}
\end{equation*}
\notag
$$
We extend this definition to fractional $r$ similarly to what we did for derivatives. For $r>0$ and $h,\beta\in\mathbb R$ we define difference analogues $\delta^{r,\beta}_h$ of the Weyl-Nagy differentiation operators. We define them only on the classes $\mathbf B_{\sigma}$, and only for $|h|<2\tau_1/\sigma$, where $\tau_1$ is the first positive zero of the function $j_{\alpha+1}$. Let $\sigma>0$ and let the function $f\in\mathbf V_{\alpha,\sigma}$ be given by (2.1). Then we set
$$
\begin{equation*}
\delta^{r,\beta}_hf(x)= \int_{-\sigma}^{\sigma} e^{(i\beta\pi/2)\operatorname{sign}hy} \biggl|e_{y}\biggl(\frac{h}{2}\biggr)-e_{y}\biggl(-\frac{h}{2}\biggr)\biggr|^re_{y}(x)\,d\rho(f,y).
\end{equation*}
\notag
$$
This definition extends to the classes $\mathbf B_{\sigma}$ via (3.10) (this is justified in the same way as for differentiation operators). Considering even and odd $r$ separately one can easily check that for $\beta=r\in\mathbb N$, $|y|\leqslant\sigma$ and $|h|<2\tau_1/\sigma$ we have
$$
\begin{equation*}
e^{(i\beta\pi/2)\operatorname{sign}hy} \biggl|e_{y}\biggl(\frac{h}{2}\biggr)-e_{y}\biggl(-\frac{h}{2}\biggr)\biggr|^r =\biggl(e_{y}\biggl(\frac{h}{2}\biggr)-e_{y}\biggl(-\frac{h}{2}\biggr)\biggr)^r.
\end{equation*}
\notag
$$
Hence $\delta^{r,r}_h=\delta^r_h$ for $r\in\mathbb N$. We also take this equality for the definition of the difference $\delta^r_h$ for fractional $r$. Another analogy with the classical differences leads to the following definition. Given $r\in\mathbb Z_+$ and $h\in\mathbb R$, we set
$$
\begin{equation*}
\delta_{r,h}=\sum_{j=0}^{r}(-1)^{r-j}C_{r}^{j}T^{(j-r/2)h}.
\end{equation*}
\notag
$$
Then for $f\in\mathbf V_{\alpha,\sigma}$ of the form (2.1) we have
$$
\begin{equation*}
\delta_{r,h}f(x)=\int_{-\sigma}^{\sigma}\psi_r(hy)e_{y}(x)\,d\rho(f,y),
\end{equation*}
\notag
$$
where
$$
\begin{equation}
\psi_r(t)=\sum_{j=0}^{r}(-1)^{r-j}C_{r}^{j} e_t\biggl(j-\frac{r}{2}\biggr).
\end{equation}
\tag{4.4}
$$
For even $r=2m$ we have
$$
\begin{equation*}
\delta_{2m,h}=\sum_{j=0}^{2m}(-1)^{j}C_{2m}^{j}T^{(j-m)h}= \sum_{l=-m}^{m}(-1)^{m-l}C_{2m}^{m-l}T^{lh}.
\end{equation*}
\notag
$$
In particular, $\delta_{2,h}=T^h-2I+T^{-h}$, and if $f\in\mathbf V_{\alpha,\sigma}$ has the form (2.1), then
$$
\begin{equation*}
\begin{aligned} \, \delta_{2,h}f(x) &=\int_{-\sigma}^{\sigma}\bigl(e_y(h)-2+e_y(-h)\bigr)e_{y}(x)\,d\rho(f,y) \\ &=2\int_{-\sigma}^{\sigma}\bigl(j_{\alpha}(hy)-1\bigr)e_{y}(x)\,d\rho(f,y). \end{aligned}
\end{equation*}
\notag
$$
The difference $\delta_{2,h}$ coincides with the difference $\Delta^2_h$ up to a constant factor; the integer powers of the latter were used in [15] and before that, for even functions, in [19]: $\Delta^2_h=-\frac 12\delta_{2,h}$, and, therefore, $\Delta_h^{2m}=(-2)^{-m}\delta_{2,h}^{m}$ for all $m\in\mathbb Z_+$. The following notational distinction should be mentioned: in [15], formula (6.5), and in [19], formula (1.6), the difference $\Delta_h^{2m}$ was denoted by $\Delta_h^{m}$. In [16], formula (3.9), the authors defined (and denoted by $\Delta^r_h$) the powers of the operator $\Delta^2_h$ with (fractional) exponents $r/2>0$, that is,
$$
\begin{equation*}
\widetilde{\delta}_h^r=(\Delta^2_h)^{r/2}= \sum_{s=0}^{\infty}(-1)^sC_{r/2}^{s}(\square_h)^s, \qquad \square_{h}=\frac{T^h+T^{-h}}{2}.
\end{equation*}
\notag
$$
It is also worth noting that in the nonweighted case, for odd $r$ the operator $\widetilde{\delta}_h^r$ cannot be reduced to the classical $r$th difference. It is easily seen that if a function $f\in\mathbf V_{\alpha,\sigma}$ is given by (2.1), then
$$
\begin{equation*}
\widetilde{\delta}_h^rf(x)=\int_{-\sigma}^{\sigma}\bigl(1-j_{\alpha}(hy)\bigr)^{r/2} e_{y}(x)\,d\rho(f,y).
\end{equation*}
\notag
$$
Since $|j_{\alpha}(t)|<1$ for all $t\in\mathbb R\setminus\{0\}$, the power under the integral sign is real. As the operator $\widetilde{\delta}_h^r$ is continuous, we obtain (3.10) for $\theta=0$ for functions $f\in\mathbf B_{\sigma}$. Remark 8. The following equalities are clear: $\delta^r_{-h}=(-1)^r\delta^r_{h}$ ($r\in\mathbb Z_+$), $\delta_{r,-h}=(-1)^r\delta_{r,h}$ ($r\in\mathbb Z_+$) and $\widetilde{\delta}_{-h}^r=\widetilde{\delta}_{h}^r$ ($r>0$). Using these equalities, we can limit ourselves to positive values of $h$ in norm estimates. Other operators of difference type have also been considered (see, for example, formulae (6.6) and (6.7) in [15], formula (2.8) in [17] and formula (4.1) in [19]). From the point of view of M. Riesz-type (see (1.2)) and Boas-type (see (1.3)) estimates, it is most convenient to work with the differences $\delta^r_h$. Theorem 4. Let $r>0$, $\beta\in\mathbb R$, $\sigma>0$, $p\in[1,+\infty]$, $f\in L_{p,\alpha,\sigma}$, let $\tau_1$ be the first positive zero of the function $j_{\alpha+1}$, assume that $0<h<2\tau_1/{\sigma}$, and let Proof. Setting $\theta=0$,
$$
\begin{equation*}
\lambda(y)=e^{(i\beta\pi/2)\operatorname{sign}y}|y|^r\quad\text{and} \quad \mu(y)=e^{(i\beta\pi/2)\operatorname{sign}y} \biggl|e_{y}\biggl(\frac{h}{2}\biggr)- e_{y}\biggl(-\frac{h}{2}\biggr)\biggr|^r
\end{equation*}
\notag
$$
in Theorem 1 we obtain ${\lambda}/{\mu}=g$. The function $g$ is even. From Lemma 1 and property M3 it follows that this function is logarithmically absolutely monotone on the half-open interval $[0,{2\tau_1}/{h}) \supset [0,\sigma]$. By property P5 the function $P^{-1}g$ is even and absolutely monotone on this interval. By property M2 and Remark 5 the Fourier coefficients of $P^{-1}g$ alternate in sign. It remains to use Theorem 1.
This proves Theorem 4. Theorem 5. Let $r>0$, $\beta\in\mathbb R$, $\sigma>0$, $p\in[1,+\infty]$, $f\in L_{p,\alpha,\sigma}$, let $\tau_1$ be the first positive zero of the function $j_{\alpha+1}$, assume that $0<u<h<{2\tau_1}/{\sigma}$, and let Proof. Setting $\theta=0$,
$$
\begin{equation*}
\lambda(y) =e^{(i\beta\pi/2)\operatorname{sign}y} \biggl|e_{y}\biggl(\frac{u}{2}\biggr)-e_{y}\biggl(-\frac{u}{2}\biggr)\biggr|^r
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\mu(y) = e^{(i\beta\pi/2)\operatorname{sign}y} \biggl|e_{y}\biggl(\frac{h}{2}\biggr)-e_{y}\biggl(-\frac{h}{2}\biggr)\biggr|^r
\end{equation*}
\notag
$$
in Theorem 1 we obtain ${\lambda}/{\mu}=g$. The function $g$ is even. By Lemma 1 and properties M3 and M4 this function is logarithmically absolutely monotone on the half-open interval $[0,{2\tau_1}/{h})\supset [0,\sigma]$. By property P5 the function $P^{-1}g$ is even and absolutely monotone on the same interval. By property M2 and Remark 5 the Fourier coefficients of $P^{-1}g$ alternate in sign. It now remains to use Theorem 1.
Theorem 5 is proved. Recall that the function $\psi_r$ was defined in (4.4). Lemma 8. Let $r\in\mathbb N$, $q\in(0,1)$, $g_0(z)={(iz)^r}/{\psi_r(z)}$ and $g_1(z)={\psi_r(qz)}/{\psi_r(z)}$, $l=0,1$. Then $g_l(0)>0$ and $g_l''(0)>0$. Proof. We have $\psi_r(z)=\delta_1^re_z(0)$, where $\delta_1^r$ is a classical difference. Let
$$
\begin{equation*}
e_z(x)=\sum_{k=0}^{\infty}i^{k}a_{k}x^{k}z^{k}, \qquad a_k>0,
\end{equation*}
\notag
$$
be the power series expansion of $e_z$. Then
$$
\begin{equation*}
\begin{gathered} \, \psi_r(z)= \sum_{k=0}^{\infty}i^ka_kz^k\delta_1^r((\cdot)^k,0)=i^rb_rz^{r}+i^{r+2}b_{r+2}z^{r+2}+\dotsb, \qquad b_k>0, \\ g_0(z)=\frac{z^r}{b_rz^{r}-b_{r+2}z^{r+2}+\dotsb}= \frac{1}{b_r}\biggl(1+\frac{b_{r+2}}{b_r}z^2+\dotsb\biggr) \end{gathered}
\end{equation*}
\notag
$$
and
This proves Lemma 4. Theorem 6. For any $r\in\mathbb N$ there exists $\tau>0$ such that if $\sigma>0$, $p\in[1,+\infty]$, $f\in L_{p,\alpha,\sigma}$, $0<h<{\tau}/{\sigma}$ and $g(y)={(iy)^r}/{\psi_r(hy)}$, then For $p=+\infty$ this inequality is sharp. Proof. In Theorem 1 we set
$$
\begin{equation*}
\lambda(y)=(iy)^r, \qquad \mu(y)=\psi_r(hy), \qquad \theta=0.
\end{equation*}
\notag
$$
As a result, $g={\lambda}/{\mu}$. Set $z=hy$ and $g_0(z)=h^{-r}g(y)={(iz)^r}/{\psi_r(z)}$. Then it is clear that the function $g_0$ is independent of $h$, even and analytic in some neighbourhood of the origin. Hence so is the function $P^{-1}g_0$. Therefore, for sufficiently small $b>0$ the function $g_0$ expands as
$$
\begin{equation*}
g_0(z)=\sum_{l\in\mathbb Z}c_{l,b}e_{z}\biggl(\frac{l\pi}{b}\biggr), \qquad |z|\leqslant b.
\end{equation*}
\notag
$$
By Lemma 8 $g_0(0)>0$ and $g_0''(0)>0$. By property P5 the same is also true for $P^{-1}g_0$. Hence this function is positive and convex in a neighbourhood of the origin. By Remark 5, for sufficiently small $b$ we have $(-1)^lc_{l,b}\geqslant0$ for all $l\in\mathbb Z$. We set By the above $\tau>0$.
After reintroducing the variable $y$ we obtain
$$
\begin{equation*}
g(y)=\sum_{l\in\mathbb Z}c_{l,b}e_{y}\biggl(\frac{lh\pi}{b}\biggr).
\end{equation*}
\notag
$$
Setting $b=\sigma h$ we have $b<\tau$, and therefore $(-1)^lc_{l,b}\geqslant0$. It remains to use Theorem 1.
Theorem 6 is proved. Theorem 7. For all $r\in\mathbb N$ and $q\in(0,1)$ there exists $\tau>0$ such that if $\sigma>0$, $p\in[1,+\infty]$, $f\in L_{p,\alpha,\sigma}$, $0<h< {\tau}/{\sigma}$, $u=qh$ and $g(y)={\psi_r(uy)}/{\psi_r(hy)}$, then For $p=+\infty$ this inequality is sharp. Proof. Setting
$$
\begin{equation*}
\lambda(y)=\psi_r(uy), \qquad \mu(y)=\psi_r(hy)\quad\text{and} \qquad \theta=0
\end{equation*}
\notag
$$
in Theorem 1, we have $g={\lambda}/{\mu}$. We also define $z=hy$ and consider $g_1(z)=g(y)={\psi_r(qz)}/{\psi_r(z)}$. It is clear that the function $g_1$ is independent of $h$, even and analytic. By Lemma 8 $g_1(0)>0$ and $g_1''(0)>0$. The rest of the proof goes through as in Theorem 6. Theorem 8. For any $r>0$ there exists $\tau>0$ such that if $\sigma>0$, $p\in[1,+\infty]$, $f\in L_{p,\alpha,\sigma}$, $0<h<{\tau}/{\sigma}$ and Proof. Setting
$$
\begin{equation*}
\lambda(y)=|y|^r, \qquad \mu(y)=(1-j_{\alpha}(hy))^{r/2}\quad\text{and} \quad \theta=0
\end{equation*}
\notag
$$
in Theorem 1 we have ${\lambda}/{\mu}=g$. We set $z=hy$ and $g_0(z)=h^{-r}g(y)$. Then $g_0$ is independent of $h$, even and analytic. It is clear that $g_0(0)>0$, $g_0''(0)>0$. The rest of the proof goes through as in Theorem 6. Theorem 9. For all $r>0$ and $q\in(0,1)$ there exists $\tau>0$ such that if $\sigma>0$, $p\in[1,+\infty]$, $f\in L_{p,\alpha,\sigma}$, $0<h< {\tau}/{\sigma}$, $u=qh$ and then For $p=+\infty$ this inequality is sharp. Proof. Setting
$$
\begin{equation*}
\lambda(y)=(1-j_{\alpha}(uy))^{r/2}, \qquad \mu(y)=(1-j_{\alpha}(hy))^{r/2}\quad\text{and} \quad \theta=0
\end{equation*}
\notag
$$
in Theorem 1 we obtain $g=\lambda/\mu$. We set $z=hy$ and $g_1(z)={g(qz)}/{g(z)}$. The function $g_1$ is independent of $h$, is even and analytic. It is clear that $g_1(0)>0$ and $g_1''(0)>0$. The rest of the proof is as in Theorem 6. Order estimates of the form (4.5) and (4.6), and, in particular, for fractional $r$, were presented in [16], Corollaries 5.5 and 5.7. Previously, for even $r$ such estimates had been obtained in [15], Theorems 7.5 and 7.7, Remark 6.8, and, for even functions and even $r$, in [19], Corollaries 4.1 and 4.3. 4.3. Estimates for the norms of differencesThe following results show that the differences $\delta_{r,h}$ are the most simple ones from the point of view of sharp estimates in terms of the norm of the function itself. Theorem 10. Let $r\in\mathbb N$, $\sigma>0$, $p\in[1,+\infty]$, $f\in L_{p,\alpha,\sigma}$ and $0<h\leqslant\pi/\sigma$. Then For $p=+\infty$ this inequality is sharp. Proof. Setting $\mu(y)=1$, $\lambda(y)=g(y)=\psi_r(hy)$ (recall that $\psi_r$ is defined by (4.4)), $\theta=0$ for even $r$ and $\theta=1/2$ for odd $r$ in Theorem 1 we have
$$
\begin{equation*}
P^{-1}g(x)= \sum_{j=0}^{r}(-1)^{r-j}C_{r}^{j} e^{i(j-r/2)hx} =\bigl(e^{ihx/2}-e^{-ihx/2}\bigr)^{r}= \biggl(2i\sin\frac{hx}{2}\biggr)^{r}.
\end{equation*}
\notag
$$
It is well known that the Fourier coefficients of this function alternate in sign; this property underlies inequality (4.7) for the classical difference. For $r=1$, see, for example, [6], [12] and [13]; this case yields directly the corresponding result for $r>1$ because the property of alternation of the Fourier coefficients in sign is preserved under multiplication of functions.
This proves Theorem 10. The proof of the following two theorems is similar to the proof of Theorem 6. Theorem 11. For any $r>0$ there exists $\tau>0$ such that if $\sigma>0$, $p\in[1,+\infty]$, $f\in L_{p,\alpha,\sigma}$, $0<h<{\tau}/{\sigma}$ and then For $p=+\infty$ this inequality is sharp. Order estimates of the form (4.8) (in particular, for noninteger $r$) were obtained in [16], Corollary 5.4. Previously, for even $r$ similar estimates had been obtained in [15], Corollary 7.4, and for even functions and even $r$, in [19], Corollary 4.3. Theorem 12. For any $r>0$ there exists $\tau>0$ such that if $\sigma>0$, $p\in[1,+\infty]$, $f\in L_{p,\alpha,\sigma}$, $0<h<{\tau}/{\sigma}$ and then For $p=+\infty$ this inequality is sharp. Remark 9. The question about the greatest value of $\tau$ in Theorems 6–9, 11 and 12 requires a special investigation. 
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