Approximation of classes of Poisson integrals by rectangular Fejér means

The article is devoted to the problem of approximation of classes of periodic functions by rectangular linear means of Fourier series. Asymptotic equalities are found for upper bounds of deviations in the uniform metric of rectangular Fejér means on classes of periodic functions of several variables generated by sequences that tend to zero at the rate of geometric progression. In one-dimensional cases, these classes consist of Poisson integrals, namely functions that can be regularly extended in the fixed strip of a complex plane.

1 Introduction

Let ℝ^d be the Euclidean space of vectors $\bar{x}=(x_1;x_2;\dots ;x_d)$. Let $f(\bar{x})$ be a function 2π-periodic in each variable x_i, $i\in \overline{\left\{1,d\right\}}$ and summable on the set 𝕋^d = [−π; π]^d, i.e., f ∈ L(𝕋^d), let

$S\left[f\right](\bar{x})={\displaystyle \sum _{\bar{k}\in {\mathbb{Z}}_{+}^d}}{2}^{-\gamma (\bar{k})}{\displaystyle \sum _{\bar{s}\in {\left\{0;1\right\}}^d}}{a}_{\bar{k}}^{\bar{s}}\left[f\right]{\displaystyle \prod _{i=1}^d}cos(k_i{x}_i-\frac{s_i\pi }{2})$

be the complete Fourier series of function f, where

$a_{\bar{k}}^{\bar{s}}\left[f\right]={\pi }^{-d}{\displaystyle \underset{{𝕋}^d}{\int }}f(\bar{x}){\displaystyle \prod _{i=1}^d}cos\left(k_i{x}_i-\frac{s_i\pi }{2}\right)d{x}_i,$

are the Fourier coefficients of the function f, corresponding to the vectors $\bar{k}\in {\mathbb{Z}}_{+}^d$, $\bar{s}\in {\left\{0;1\right\}}^d,$ and $\gamma (\bar{k})$ is the number of zero coordinates of the vector $\bar{k}$.

Let $\overline{\text{Λ}}=({\text{Λ}}_1;{\text{Λ}}_2;\dots ;{\text{Λ}}_d)$ be the fixed set of infinite triangular matrices of numbers ${\text{Λ}}_i=\left\{{\lambda }_{k_i}^{(n_i)}\right\},$ $i\in \overline{\left\{1,d\right\}}$ such that ${\lambda }_0^{(n_i)}=1$, ${\lambda }_{k_i}^{(n_i)}=0$, k_i ≥ n_i. Denote ${\lambda }_{\bar{k}}^{(\bar{n})}={\displaystyle \prod _{i=1}^d}{\lambda }_{k_i}^{(n_i)}$, and ${\text{𝔾}}_{\bar{n}}={\displaystyle \prod _{i=1}^d}\left[0;n_i-1\right]$. If $\bar{k}\notin {\text{𝔾}}_{\bar{n}}$, then ${\lambda }_{\bar{k}}^{(\bar{n})}=0$. For function f ∈ L(𝕋^d) the set $\overline{\text{Λ}}$ defines a family of trigonometric polynomials

$U_{\bar{n}}\left[f;\overline{\text{Λ}}\right](\bar{x})={\displaystyle \sum _{\bar{k}\in {\text{𝔾}}_{\bar{n}}}}{2}^{-\gamma (\bar{k})}{\lambda }_{\bar{k}}^{(\bar{n})}{\displaystyle \sum _{\bar{s}\in {\left\{0;1\right\}}^d}}{a}_{\bar{k}}^{\bar{s}}\left[f\right]{\displaystyle \prod _{i=1}^d}cos(k_i{x}_i-\frac{s_i\pi }{2}).$

The polynomials $U_{\bar{n}}\left[f;\overline{\text{Λ}}\right](\bar{x})$ are called rectangular linear means for $S\left[f\right](\bar{x}).$ In particular, if ${\lambda }_{k_i}^{(n_i)}=1,$ $\bar{k}\in {\text{𝔾}}_{\bar{n}},$ then $U_{\bar{n}}\left[f;\overline{\text{Λ}}\right](\bar{x})=S_{\bar{n}-1}\left[f\right](\bar{x})$ are the rectangular partial sums of $S\left[f\right](\bar{x})$, and if ${\lambda }_{k_i}^{(n_i)}=1-\frac{k_i}{n_i}$, $\bar{k}\in {\text{𝔾}}_{\bar{n}},$ then

$U_{\bar{n}}\left[f;\overline{\text{Λ}}\right](\bar{x})={\sigma }_{\bar{n}}\left[f\right](\bar{x})={\displaystyle \prod _{i=1}^d}{n}_i^{-1}{\displaystyle \sum _{\bar{k}\in {\text{𝔾}}_{\bar{n}}}}{S}_{\bar{n}}\left[f\right](\bar{x})$

are the rectangular Fejér means of $S\left[f\right](\bar{x})$.

Basic results relating to the approximation of functional classes by linear methods of summation of Fourier series can be found in books Timan [1], Lorentz [2], and Dyachenko [3]. Linear summation methods are widely used both for the solution of practical problems and for development of more advanced approximation methods. This chapter of approximation theory has been intensively developed over the past decades [4–9]. Here it is difficult to mention all the relevant published research papers in this area. Recently, we have seen the publication of several important works [10–15].

Let C(𝕋^d) be the space of continuous 2π-periodic in each variable's functions $f(\bar{x})$ with the norm

$||f||:=||f{||}_C={\displaystyle \underset{\bar{x}\in {\text{𝕋}}^d}{max}}\left|f(\bar{x})\right|.$

Let $J(r)$ be the arbitrary subset of the set $\overline{\left\{1;d\right\}},$ where r is the number of elements of the set $J(r)$. Denote by $C^{\bar{q}}({\text{𝕋}}^d)$, $\bar{q}\in {(0;1)}^d$ the set of functions f ∈ C(𝕋^d) such that $\forall J:=J(r)\subseteq \overline{\left\{1;d\right\}}$, the series

$\begin{array}{ll}{\displaystyle \sum _{\genfrac{}{}{0ex}{}{\bar{k}\in {\mathbb{Z}}_{+}^d,}{k_j\ne 0,j\in J}}}{2}^{-\gamma (\bar{k})}{\displaystyle \prod _{j\in J}}{q}_j^{-k_j}{\displaystyle \sum _{\bar{s}\in {\left\{0;1\right\}}^d}}{a}_{\bar{k}}^{\bar{s}}\left[f\right]{\displaystyle \prod _{i=1}^d}cos(k_i{x}_i-\frac{s_i\pi }{2})& (1)\end{array}$

are the Fourier series of certain functions ${\phi }_{\bar{q}}^{(J)}(\bar{x})\in L({\text{𝕋}}^d)$, which are almost everywhere bounded by a unity, and the Fourier series of functions ${\phi }_{\bar{q}}^{(J)}(\bar{x})$ do not contain terms independent of the variables x_i, $i\in J(r)$.

For example, in the case d = 2, the series (Equation 1) is as follows:

$\begin{array}{c}S\left[{\phi }_{\bar{q}}^{(1)}\right](\bar{x})={\displaystyle \sum _{\bar{k}\in \mathbb{N}\times {\mathbb{Z}}_{+}}}{2}^{-\gamma (\bar{k})}{q}_1^{-k_1}{\displaystyle \sum _{\bar{s}\in {\left\{0;1\right\}}^2}}{a}_{\bar{k}}^{\bar{s}}\left[f\right]\\ cos(k_1{x}_1-\frac{s_1\pi }{2})cos(k_2{x}_2-\frac{s_2\pi }{2}),\\ S\left[{\phi }_{\bar{q}}^{(2)}\right](\bar{x})={\displaystyle \sum _{\bar{k}\in {\mathbb{Z}}_{+}\times \mathbb{N}}}{2}^{-\gamma (\bar{k})}{q}_2^{-k_2}{\displaystyle \sum _{\bar{s}\in {\left\{0;1\right\}}^2}}{a}_{\bar{k}}^{\bar{s}}\left[f\right]\\ cos(k_1{x}_1-\frac{s_1\pi }{2})cos(k_2{x}_2-\frac{s_2\pi }{2}),\\ S\left[{\phi }_{\bar{q}}^{(J)}\right](\bar{x})={\displaystyle \sum _{\overrightarrow{k}\in {\mathbb{N}}^2}}{2}^{-\gamma (\bar{k})}{q}_1^{-k_1}{q}_2^{-k_2}{\displaystyle \sum _{\bar{s}\in {\left\{0;1\right\}}^2}}{a}_{\bar{k}}^{\bar{s}}\left[f\right]\\ cos(k_1{x}_1-\frac{s_1\pi }{2})cos(k_2{x}_2-\frac{s_2\pi }{2}).\end{array}$

In the one-dimensional case, the classes C^q(𝕋¹), q∈(0;1) consist of continuous 2π-periodic functions, given by the convolution

$f(x)=A_0+{\pi }^{-1}{\displaystyle \underset{{\text{𝕋}}^1}{\int }}{\phi }_q^{(1)}(x+t)P_q(t)dt,A_0-\text{const},$

where

$P(q;t)={\displaystyle \sum _{k=0}^{\infty }}{q}^{k}coskt=\frac{1-qcost}{1-2qcost+q^2},q\in (0;1)$

is the well-known Poisson kernel, the function ${\phi }_q^{(1)}\in L({\text{𝕋}}^1)$ $(J(1)=i,i=1)$ satisfies almost everywhere the conditions $\left|{\phi }_q^{(1)}(t)\right|\le 1$, ${\phi }_q^{(1)}\perp 1.$

In this work, we consider the problem of the exact upper bound for the approximation of periodic functions by linear means of the Fourier series. We employed methods for studying integral representations of deviations of polynomials, generated by linear summation methods of Fourier series of continuous periodic functions, developed in the works of Nikolskii [16], Telyakovskii [17], Stepanets [18], and others. This topic is currently being developed in the works of many authors [19–21].

Nikolskii [22] established the asymptotic equality as n → ∞

$\begin{array}{c}sup\left\{||f-S_n\left[f\right]||:f\in C^q({\text{𝕋}}^1)\right\}=\\ sup\left\{||\frac{1}{\pi }{\displaystyle \underset{{\text{𝕋}}^1}{\int }}{\displaystyle {\phi }_q^{(1)}}(x+t){\displaystyle \sum _{k=n+1}^{\infty }}{q}^{k}cosktdt||:\left|{\displaystyle {\phi }_q^{(1)}}(t)\right|\le 1,{\displaystyle {\phi }_q^{(1)}}\perp 1\right\}\\ =\frac{8q^{n+1}}{{\pi }^2}K(q)+O(1)\frac{q^n}{n},\end{array}$

where $K(q)=\underset{0}{\overset{\frac{\pi }{2}}{\int }}{(1-q^2{sin}^{2}u)}^{-\frac{1}{2}}du$ is the complete elliptic integral of the first kind and O(1) is a quantity uniformly bounded with respect to n. Regarding the summability of Fourier series by Fejér means σ_n[f], we proved the following two theorems [23–25].

Theorem 1. Let q₀ be the only root of the equation q⁴ − 2q³ − 2q² − 2q+1 = 0, that belongs to the interval (0;1), $q_0={(2+\sqrt{5}-2\sqrt{2+\sqrt{5}})}^{1/2}=0.346\dots .$ If q ∈ (0;q₀], then the equality hold as n → ∞

$sup\left\{||f-{\sigma }_n\left[f\right]||:f\in C^q({\text{𝕋}}^1)\right\}=\frac{4q}{\pi n(1+q^2)}+O(1)\frac{q^n}{n},$

where O(1) is a quantity uniformly bounded with respect to n.

Theorem 2. If q ∈ [q₀; 1), then the equality hold as n → ∞

$\begin{array}{c}sup\left\{||f-{\sigma }_n\left[f\right]||:f\in C^q({\text{𝕋}}^1)\right\}\\ =\frac{2}{\pi n}\frac{{(1+q^2)}^2}{(1-q^2)(1-q^2+\sqrt{2(1+q^4)})}+O(1)\frac{q^n}{n{(1-q)}^3},\end{array}$

where O(1) is uniformly bounded with respect to n, q.

The purpose of this paper is to present the asymptotic equalities for upper bounds of deviations of rectangular Fejér means taken over multidimensional analogs of classes C^q(𝕋¹). Similar asymptotic expansions for other rectangular linear methods can be found in Rukasov et al. [26] and Rovenska [27].

2 Result

The main result is the following.

Theorem 3. Let $\bar{q}\in {(0;1)}^d$. Then

$\begin{array}{c}sup\left\{||f-{\sigma }_{\bar{n}}\left[f\right]||:f\in C^{\bar{q}}({\text{𝕋}}^d)\right\}=\frac{4}{\pi }{\displaystyle \sum _{i=1}^d}\frac{A(q_i)}{n_i}\\ +O(1)\left({\displaystyle \sum _{i=1}^d}\frac{q_i^{n_i}}{n_i{(1-q_i)}^3}+{\displaystyle \sum _{r=2}^d}{\displaystyle \sum _{J(r)\subset \overline{\left\{1,d\right\}}}}{\displaystyle \prod _{j\in J(r)}}\frac{1}{n_j{(1-q_j)}^3}\right),& (2)\end{array}$

where

$A(q)=\{\begin{array}{ll}\frac{q}{1+q^2},\hfill & q\in (0;q_0]\hfill \\ \frac{{(1+q^2)}^2}{2(1-q^2)(1-q^2+\sqrt{2(1+q^4)})},\hfill & q\in [q_0;1),\hfill \end{array}$

q₀ is the only root of the equation q⁴−2q³ − 2q² − 2q+1 = 0, that belongs to the interval (0;1), q₀ = 0.346…, O(1) is a quantity, uniformly bounded with respect to q_i, n_i, $i\in \overline{\left\{1,d\right\}}$.

Proof

First we find the upper estimate for the quantity

$\begin{array}{ll}sup\left\{||f-{\sigma }_{\bar{n}}\left[f\right]||:f\in C^{\bar{q}}({\text{𝕋}}^d)\right\}.& (3)\end{array}$

Based on Theorem 1 in Rukasov et al. [26], $\forall f\in C^{\bar{q}}({\text{𝕋}}^d)$, the equality holds

$\begin{array}{c}f(\bar{x})-{\sigma }_{\bar{n}}\left[f\right](\bar{x})={\displaystyle \sum _{\bar{k}\in {\mathbb{Z}}_{+}^d}}{2}^{-\gamma (\bar{k})}{\displaystyle \sum _{\bar{s}\in {\left\{0;1\right\}}^d}}{a}_{\bar{k}}^{\bar{s}}\left[f\right]{\displaystyle \prod _{i=1}^d}cos(k_i{x}_i-\frac{s_i\pi }{2})-\\ {\displaystyle \sum _{\bar{k}\in {\text{𝔾}}_{\bar{n}}}}{2}^{-\gamma (\bar{k})}{\displaystyle \prod _{i=1}^d}(1-\frac{k_i}{n_i}){\displaystyle \sum _{\bar{s}\in {\left\{0;1\right\}}^d}}{a}_{\bar{k}}^{\bar{s}}\left[f\right]{\displaystyle \prod _{i=1}^d}cos(k_i{x}_i-\frac{s_i\pi }{2})=\\ \frac{1}{\pi }{\displaystyle \sum _{i=1}^d}\frac{1}{n_i}{\displaystyle \underset{{\text{𝕋}}^1}{\int }}{\phi }_{q_i}^{(i)}(\bar{x}+t_i{ē}_i){\displaystyle \sum _{k_i=0}^{n_i-1}}{\displaystyle \sum _{{\nu }_i=k_i+1}^{\infty }}{q}_i^{{\nu }_i}cos{\nu }_i{t}_{i}d{t}_i+\\ {\displaystyle \sum _{r=2}^d}{(-1)}^{r+1}\frac{1}{{\pi }^r}{\displaystyle \sum _{J(r)\subset \overline{\left\{1,d\right\}}}}{\displaystyle \underset{{\text{𝕋}}^r}{\int }}{\phi }_{\bar{q}}^{(J)}\left(\bar{x}+{\displaystyle \sum _{j\in J(r)}}{t}_j{ē}_j\right)\\ {\displaystyle \prod _{j\in J(r)}}\frac{1}{n_j}{\displaystyle \sum _{k_j=0}^{n_j-1}}{\displaystyle \sum _{{\nu }_j=k_j+1}^{\infty }}{q}_j^{{\nu }_j}cos{\nu }_j{t}_{j}d{t}_j.& (4)\end{array}$

In Novikov et al. [24] and Rovenska [25] it was shown that

$\begin{array}{c}\sup \{\Vert \frac{1}{n}{\displaystyle \underset{T^1}{\int }{\phi }_q^{(1)}}(x+t)\sum _{k=0}^{n-1}\sum _{\nu =k+1}^{\infty }{q}^{\nu }\cos \nu tdt\Vert \\ :|{\phi }_q^{(1)}(t)|\le 1,{\phi }_q^{(1)}\perp 1\}=\\ \frac{1}{n}{\displaystyle \underset{T^1}{\int }{\phi }_q^{*(1)}}(t)\sum _{k=0}^{n-1}\sum _{\nu =k+1}^{\infty }{q}^{\nu }\cos \nu tdt\\ =\frac{A(q)}{n}+O(1)\frac{q^n}{n{(1-q)}^3},& (5)\end{array}$

where

$\begin{array}{ll}{\phi }_q^{*(1)}(t)=\{\begin{array}{c}\text{sign}\left(\frac{\partial P(q;t)}{\partial q}-\frac{\partial P(q;t)}{\partial q}{|}_{t=\frac{\pi }{2}}\right),q\in (0;q_0],\\ \text{sign}\left(\frac{\partial P(q;t)}{\partial q}-\frac{\partial P(q;t)}{\partial q}{|}_{t=t_q}\right),q\in [q_0;1),\end{array}& (6)\end{array}$

and t_q is determined by the condition

$\frac{\partial P(q;t)}{\partial q}{|}_{t=t_q}=\frac{\partial P(q;t)}{\partial q}{|}_{t=t_q+\frac{\pi }{2}},0\le t_q\le \frac{\pi }{2}.$

Combining Equations 4, 5, and 6, we obtain

$\begin{array}{c}sup\left\{||f-{\sigma }_{\bar{n}}\left[f\right]||:f\in C^{\bar{q}}({\text{𝕋}}^d)\right\}\le \frac{4}{\pi }{\displaystyle \sum _{i=1}^d}\frac{A(q_i)}{n_i}\\ +O(1)\left({\displaystyle \sum _{i=1}^d}\frac{q_i^{n_i}}{n_i{(1-q_i)}^3}+{\displaystyle \sum _{r=2}^d}{\displaystyle \sum _{J(r)\subset \overline{\left\{1,d\right\}}}}{\displaystyle \prod _{j\in J(r)}}\frac{1}{n_j{(1-q_j)}^3}\right).& (7)\end{array}$

Next, we find the lower estimate of Equation 3. We construct the function $f^{*}(\bar{x})\in C^{\bar{q}}({\text{𝕋}}^d)$ for which estimate Equation 7 cannot be improved. Based on equality Equation 3 we have

$\begin{array}{c}f(\bar{0})-{\sigma }_{\bar{n}}\left[f\right](\bar{0})\\ =\frac{1}{\pi }\sum _{i=1}^d\frac{1}{n_i}{\displaystyle \underset{{\text{𝕋}}^1}{\int }}{\phi }_{q_i}^{(i)}(\bar{0}+t_i{ē}_i)\sum _{k_i=0}^{n_i-1}\sum _{{\nu }_i=k_i+1}^{\infty }{q}_i^{{\nu }_i}cos{\nu }_i{t}_{i}d{t}_i+\\ \sum _{r=2}^d{(-1)}^{r+1}\frac{1}{{\pi }^r}{\displaystyle \sum _{J(r)\subset \overline{\left\{1,d\right\}}}}{\displaystyle \underset{{\text{𝕋}}^r}{\int }}{\phi }_{\bar{q}}^{(J)}\left(\bar{0}+{\displaystyle \sum _{j\in J(r)}}{t}_j{ē}_j\right)\\ {\displaystyle {\prod }_{j\in J(r)}}\frac{1}{n_j}{\displaystyle \sum _{k_j=0}^{n_j-1}}{\displaystyle \sum _{{\nu }_j=k_j+1}^{\infty }}{q}_j^{{\nu }_j}cos{\nu }_j{t}_{j}d{t}_j.\end{array}$

Since the functions ${\phi }_{\bar{q}}^{(J)}$ satisfy the condition $\left|{\phi }_{\bar{q}}^{(J)}(\bar{x})\right|\le 1$ almost everywhere, and

$\begin{array}{c}{\displaystyle \underset{{\text{𝕋}}^1}{\int }}|\sum _{k_j=0}^{n_j-1}\sum _{{\nu }_j=k_j+1}^{\infty }{q}_j^{{\nu }_j}cos{\nu }_j{t}_j|d{t}_j\\ ={\displaystyle \underset{{\text{𝕋}}^1}{\int }}\left|\frac{\partial P(q_j;t_j)}{\partial q_j}\right|d{t}_j=O(1)\frac{1}{{(1-q_j)}^3},i\in \overline{\left\{1,d\right\}},\end{array}$

then

$\begin{array}{c}f(\bar{0})-{\sigma }_{\bar{n}}\left[f\right](\bar{0})=\frac{1}{\pi }\sum _{i=1}^d\frac{1}{n_i}{\displaystyle \underset{{\text{𝕋}}^1}{\int }}{\phi }_{q_i}^{(i)}(\bar{0}+t_i{ē}_i)\sum _{k_i=0}^{n_i-1}\\ \sum _{{\nu }_i=k_i+1}^{\infty }{q}_i^{{\nu }_i}cos{\nu }_i{t}_{i}d{t}_i\\ +O(1)\left(\sum _{r=2}^d{\displaystyle \sum _{J(r)\subset \overline{\left\{1,d\right\}}}}{\displaystyle \prod _{j\in J(r)}}\frac{1}{n_j{(1-q_j)}^3}\right).\end{array}$

Denote by ${\phi }_{q_i}^{*(i)}(\bar{x}),$ $\bar{x}\in {\text{𝕋}}^d$ an arbitrary continuation on the set 𝕋^d of the function ${\phi }_{q_i}^{*(i)}(x_i),$ $x_i\in {\text{𝕋}}^1$, and denote by $f_i^{*}(\bar{x})$, $\bar{x}\in {\text{𝕋}}^d$ the function, such that

$S\left[{\phi }_{q_i}^{*(i)}\right](\bar{x})={\displaystyle \sum _{\genfrac{}{}{0ex}{}{\bar{k}\in {\mathbb{Z}}_{+}^d,}{k_i\ne 0}}}{2}^{-\gamma (\bar{k})}{q}_i^{-k_i}{\displaystyle \sum _{\bar{s}\in {\left\{0;1\right\}}^d}}{a}_{\bar{k}}^{\bar{s}}\left[f_i^{*}\right]{\displaystyle \prod _{i=1}^d}cos\left(k_i{x}_i-\frac{s_i\pi }{2}\right).$

Let $f^{*}(\bar{x}):={\displaystyle \sum _{i=1}^d}{f}_i^{*}(\bar{x}).$ It's clear that $f^{*}(\bar{x})\in C^{\bar{q}}({\text{𝕋}}^d)$. Therefore, we have

$\begin{array}{c}{f}^{*}(\bar{0})-{\sigma }_{\bar{n}}\left[f^{*}\right](\bar{0})=\frac{1}{\pi }{\displaystyle \sum _{i=1}^d}\frac{1}{n_i}{\displaystyle \underset{{\text{𝕋}}^1}{\int }}{\phi }_{q_i}^{*(i)}(t_i){\displaystyle \sum _{k_i=0}^{n_i-1}}{\displaystyle \sum _{{\nu }_i=k_i+1}^{\infty }}{q}_i^{{\nu }_i}cos{\nu }_i{t}_{i}d{t}_i\\ +O(1)\left({\displaystyle \sum _{r=2}^d}{\displaystyle \sum _{J(r)\subset \overline{\left\{1,d\right\}}}}{\displaystyle \prod _{j\in J(r)}}\frac{1}{n_j{(1-q_j)}^3}\right).& (8)\end{array}$

Combining Equations 5, 7, and 8, we obtain equality (Equation 2). The proof is complete.

Remark 1. Formula Equation 2 is asymptotically exact for any $\bar{q}\in {(0;1)}^d.$

Remark 2. In the case d = 2, formula Equation 2 is simplified as follows:

$\begin{array}{c}sup\left\{||f-{\sigma }_{\bar{n}}\left[f\right]||:f\in C^{\bar{q}}({\text{𝕋}}^2)\right\}\\ =\frac{4}{\pi }{\displaystyle \sum _{i=1,2}}\frac{A(q_i)}{n_i}\\ +O(1)\left({\displaystyle \sum _{i=1,2}}\frac{q_i^{n_i}}{n_i{(1-q_i)}^3}+\prod _{j=1,2}\frac{1}{n_j{(1-q_j)}^3}\right).\end{array}$

3 Conclusion

In this study, we propose an approach to define the multidimensional analogs of classes of Poisson integrals, which allows us to take into account the rate of decrease of each sequence that determine the class. The problem connected with the search for upper bounds of approximation errors with respect to a fixed class of functions and with the choice of an approximation tool is considered.In the certain case, our approach turned out to be effective for obtaining exact asymptotic. The key point in this approach is to construct the function $f^{*}(\bar{x})\in C^{\bar{q}}({\text{𝕋}}^d)$ that implements the upper bound.

Our study may be useful for solving the upper bound problem in other particular cases. In particular, our ideas can be used to obtain the corresponding asymptotic equalities on classes, which in one-dimensional cases are determined by the Poisson kernels ${\tilde{P}}_q(t)={\displaystyle \sum _{k=1}^{\infty }}sinkt$, $P_q^{\beta }(t)={\displaystyle \sum _{k=0}^{\infty }}cos\left(kt+\frac{\beta \pi }{2}\right),$ β ∈ ℝ, etc.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material; further inquiries can be directed to the corresponding author.

Author contributions

OR: Writing – review & editing, Writing – original draft.

Funding

The author(s) declare financial support was received for the research, authorship, and/or publication of this article. This research during 2020–2023 was supported by the Volkswagen Foundation project “From Modeling and Analysis to Approximation.”

Conflict of interest

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher's note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

References

1. Timan AF. Theory of Approximation of Functions of a Real Variable. New York: Macmillan. (1963). doi: 10.1016/B978-0-08-009929-3.50008-7

Crossref Full Text | Google Scholar

2. Lorentz GG. Approximation of Functions. Holt: Rinehart and Winston. (1966).

Google Scholar

3. Dyachenko MI. Convergence of multiple Fourier series: main results and unsolved problems. In: Fourier analysis and related topics. Warsaw: Banach Center Publication (2002). doi: 10.4064/bc56-0-3

PubMed Abstract | Crossref Full Text | Google Scholar

4. Singh LB. On absolute summability of Fourier-Jacobi series. Univ Timisoara Seria St Matematica. (1984) 22:89–99.

Google Scholar

5. Szabowski PJ. A few remarks on Riesz summability of orthogonal series. Proc Am Math Soc. (1991) 113:65–75. doi: 10.1090/S0002-9939-1991-1072349-8

PubMed Abstract | Crossref Full Text | Google Scholar

6. Liflyand E, Nakhman A. On linear means of multiple fourier integrals defined by special domains. Rocky Mountain J Math. (2002) 32:969–80. doi: 10.1216/rmjm/1034968426

Crossref Full Text | Google Scholar

7. Rhoades BE, Savacs E. On absolute Norlund summability of Fourier series. Tamkang J Math. (2002) 33:359–64. doi: 10.5556/j.tkjm.33.2002.284

Crossref Full Text | Google Scholar

8. Sonker S, Singh U. Degree of approximation of conjugate of signals (functions) belonging to Lip(α, r)-class by (C, 1)(E, q) means of conjugate trigonometric Fourier series. J Inequal Appl. (2012) 2012:1–7. doi: 10.1186/1029-242X-2012-278

Crossref Full Text | Google Scholar

9. Krasniqi XhZ. On absolute almost generalized Nörlund summability of orthogonal series. Kyungpook Math J. (2012) 52:279–290. doi: 10.5666/KMJ.2012.52.3.279

Crossref Full Text | Google Scholar

10. Trigub R. Asymptotics of approximation of continuous periodic functions by linear means of their Fourier series. Izvestiya: Mathem. (2020) (3):608–24. doi: 10.1070/IM8905

Crossref Full Text | Google Scholar

11. Duman O. Generalized Cesáro summability of Fourier series and its applications. Constr Math Anal. (2021) 4:135–44. doi: 10.33205/cma.838606

Crossref Full Text | Google Scholar

12. Meleteu AD, Păltănea R. On a method for uniform summation of the Fourier-Jacobi series. Results Math. (2022) 77:153. doi: 10.1007/s00025-022-01703-7

Crossref Full Text | Google Scholar

13. Munjal A. Absolute linear method of summation for orthogonal series. In:Singh S, Sarigl MA, Munjal A., , editors. Algebra, Analysis, and Associated Topics Trends in Mathematics. Cham: Birkhäuser. (2022). doi: 10.1007/978-3-031-19082-7_4

Crossref Full Text | Google Scholar

14. Aral A. On a new approach in the space of measurable functions. Constr Math Anal. (2023) 6:237–48. doi: 10.33205/cma.1381787

Crossref Full Text | Google Scholar

15. Anastassiou G. Trigonometric derived rate of convergence of various smooth singular integral operators. Modern Math Methods. (2024) 2:27–40.

Google Scholar

16. Nikolskii SM. Approximations of periodic functions by trigonometrical polynomials. Tr Mat Inst Steklova. (1945) 15:1–76.

Google Scholar

17. Telyakovskii SA. Approximation of differentiable functions by partial sums of their Fourier series. Math Notes. (1968) 4:668–73. doi: 10.1007/BF01116445

Crossref Full Text | Google Scholar

18. Stepanets AI. On a problem of A N Kolmogorov in the case of functions of two variables. Ukr Math J. (1972) 24:526–36. doi: 10.1007/BF01090536

Crossref Full Text | Google Scholar

19. Chaichenko S, Savchuk V, Shidlich A. Approximation of functions by linear summation methods in the Orlicz-type spaces. J Math Sci. (2020) 249:705–19. doi: 10.1007/s10958-020-04967-y

Crossref Full Text | Google Scholar

20. Serdyuk AS, Sokolenko IV. Approximation by Fourier sums in the classes of Weyl-Nagy differentiable functions with high exponent of smoothness. Ukr Math J. (2022) 74:783–800. doi: 10.1007/s11253-022-02101-6

Crossref Full Text | Google Scholar

21. Stasyuk SA. Yanchenko SY. Approximation of functions from Nikolskii-Besov type classes of generalized mixed smoothness. Anal Math. (2015) 41:311–34. doi: 10.1007/s10476-015-0305-0

Crossref Full Text | Google Scholar

22. Nikolskii SM. Approximation of the functions by trigonometric polynomials in the mean. Izv Akad Nauk SSSR Ser Mat. (1946) 10:207–56.

PubMed Abstract | Google Scholar

23. Novikov OO, Rovenska OG. Approximation of periodic analytic functions by Fejér sums. Matematchni Studii. (2017) 47:196–201. doi: 10.15330/ms.47.2.196-201

Crossref Full Text | Google Scholar

24. Novikov OO, Rovenska OG, Kozachenko YuA. Approximation of classes of Poisson integrals by Fejér sums. Visn V N Karazin Kharkiv Nat Univer, Ser Math, Appl Math, Mech. (2018) 87:4–12. doi: 10.26565/2221-5646-2023-97-01

Crossref Full Text | Google Scholar

25. Rovenska O. Approximation of classes of Poisson integrals by Fejé r means. Matematychni Studii. (2023) 59:201–4. doi: 10.30970/ms.59.2.201-204

Crossref Full Text | Google Scholar

26. Rukasov VI, Novikov OA, Bodraya VI. Approximation of classes of $\overline{\psi }$-integrals of periodic functions of many variables by rectangular linear means of their Fourier series. Ukr Math J. (2005) 57:678–86. doi: 10.1007/s11253-005-0219-2

Crossref Full Text | Google Scholar

27. Rovenska O. Approximation of differentiable functions by rectangular repeated de la Vallée Poussin means. J Anal. (2023) 31:691–703. doi: 10.1007/s41478-022-00479-x

Crossref Full Text | Google Scholar