On approximation of Stepanov's almost periodic functions by means of Marcinkiewicz

We study some questions of approximation of Stepanovs almost-periodic functions of partial Fourier sums and means of Marcinkiewicz, when the Fourier exponents of functions under consideration have a limit point in infinity.
Let $S_p$ ($p\geq1$) denote the class of Stepanovs almost-periodic functions, whose Fourier exponents take the following form:
$$ \lambda_0=0,\,\,\,\lambda_{-n}=-\lambda_n,\,\,\,\lim_{n\rightarrow\infty}\lambda_n=\infty,\,\,\, \lambda_n<\lambda_{n+1}\,\,\,(n=1,2,\ldots). $$

Consider the Fourier series for a functions $f(x)\in S_p$
$$ f(x)\sim\sum\limits_{n = -\infty}^\infty A_n e^{i\lambda_nx}, $$
where
$$ A_n=\lim_{T\rightarrow\infty}\frac{1}{2T}\int_{-T}^Tf(x)e^{-i\lambda_nx}dx $$
are Fourier coefficients of the function $f(x)\in S_p$ and
$$ S_\sigma(f;x)=\sum_{|\lambda_n|\leq\sigma}A_n e^{i\lambda_nx}\,\,\,(\sigma>0) $$
is a partial sum of Fourier series.
Let $\Phi_\sigma(t)$ is an arbitrary real continuous even function such that
$$ 1) \Phi_{\sigma}(0)=1; \,\,\,2) \Phi_{\sigma}(t)=0\,\,\,(|t|\leq\sigma). $$

We set
$$ U_{\sigma}(f;\varphi;x)=\sum_{|\lambda_{m}|\leq\sigma}A_m\Phi_{\sigma}(\lambda_{m})e^{i\lambda_{m}x}. $$

Let $S_p(R)$ stand for the space of bounded functions $f(x)\in S_p \,\,\,(p\geq 1)$ with the norm
$$ \|f(x)\|_{S_p}=\sup_{-\infty<x<\infty}\left\{\frac{1}{l}\int_x^{x+l}|f(x)|^p dx\right\}^{\frac{1}{p}}. $$

Consider the value
$$ R(f;x)=\left\|U_{\sigma}(f;\varphi;x)-f(x)\right\|_{S_p}, $$
where
$$ U_{\sigma}(f;\varphi;x)=\int_{-\infty}^{\infty} f(x+t) \Phi_{\sigma}(t)dt, $$

$$ \Phi_{\sigma}(t)=\frac{1}{2\pi}\int_{0}^{\infty}\varphi_{\sigma}(u)K_{u}(t)du,\,\,\, K_{u}(t)=2\frac{\sin(ut)}{t}, $$
$\varphi_{\sigma}(u)$ is some even function absolutely integrable on the interval $(0;\infty)$ with each fixed $\sigma>0$.
Theorem. If $f(x)\in S_p$, where Fourier exponents have no limit points at a finite distance, i.e. $\lambda_n\rightarrow\infty$, then the following bound is valid
$$ R(f;\varphi_{\sigma,a})\leq M\frac{\sigma+a}{\sigma-a}E_{\Lambda}(f)_{S_p}, $$
and
$$ \left\|f(x)-\frac{1}{n+1}\sum_{k=0}^{n}S_{k}(f;x)\right\|_{S_p}\leq\frac{M}{n+1}\sum_{k=0}^{n}E_{k}(f)_{S_p}, $$
where $M$—constant and
$$ E_{\Lambda}(f)_{S_p}=\inf_{A_m}\left\|f(x)-\sum_{|\lambda_{m}|\leq\Lambda}A_m e^{i\lambda_{m}x}\right\|_{S_p}. $$