Order estimates of the uniform approximations by Zygmund sums on the classes of convolutions of periodic functions

  • A.S. Serdyuk Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereschenkivska str., 01601, Kyiv, Ukraine
  • U.Z. Hrabova Lesya Ukrainka Volyn National University, 9 Potapova str., 43025, Lutsk, Ukraine
Keywords: best approximation, Zygmund sum, Fejér sum, subspace of trigonometric polynomials, order estimate
Published online: 2021-04-21


The Zygmund sums of a function $f\in L_{1}$ are trigonometric polynomials of the form $$Z^{s}_{n-1}(f;t):=\frac{a_{0}}{2}+\sum_{k=1}^{n-1}\Big(1-\big(\frac{k}{n}\big)^{s}\Big) \big(a_{k}(f)\cos kt+b_{k}(f)\sin kt\big), s>0,$$ where $a_{k}(f)$ and $b_{k}(f)$ are the Fourier coefficients of $f$. We establish the exact-order estimates of uniform approximations by the Zygmund sums $Z^{s}_{n-1}$ of $2\pi$-periodic continuous functions from the classes $C^{\psi}_{\beta,p}$. These classes are defined by the convolutions of functions from the unit ball in the space $L_{p}$, $1\leq p<\infty$, with generating fixed kernels $$\Psi_{\beta}(t)\sim\sum_{k=1}^{\infty}\psi(k)\cos\left(kt+\frac{\beta\pi}{2}\right), \Psi_{\beta}\in L_{p'}, \beta\in \mathbb{R}, \frac1p+\frac{1}{p'}=1.$$ We additionally assume that the product $\psi(k)k^{s+1/p}$ is generally monotonically increasing with the rate of some power function, and, besides, for $1< p<\infty$ it holds that $\sum_{k=n}^{\infty}\psi^{p'}(k)k^{p'-2}<\infty$, and for $p=1$ the following condition $\sum_{k=n}^{\infty}\psi(k)<\infty$ is true. It is shown, that under these conditions Zygmund sums $Z^{s}_{n-1}$ and Fejér sums $\sigma_{n-1}=Z^{1}_{n-1}$ realize the order of the best uniform approximations by trigonometric polynomials of these classes, namely for $1<p<\infty$ $${E}_{n}(C^{\psi}_{\beta,p})_{C}\asymp{\cal E}\left(C^{\psi}_{\beta,p}; Z_{n-1}^{s}\right)_{C}\asymp\Big(\sum_{k=n}^{\infty}\psi^{p'}(k)k^{p'-2}\Big)^{1/p'}, \ \frac{1}{p}+\frac{1}{p'}=1,$$ and for $p=1$ $$ {E}_{n}(C^{\psi}_{\beta,1})_{C}\asymp{\cal E}\left(C^{\psi}_{\beta,1}; Z_{n-1}^{s}\right)_{C}\asymp {\left\{{\begin{array}{l l} \sum\limits_{k=n}^{\infty}\psi(k), & \cos \frac{\beta\pi}{2}\neq 0,\\ \psi(n)n, &\cos \frac{\beta\pi}{2}= 0, \end{array}} \right.} $$ where $${E}_{n}(C^{\psi}_{\beta,p})_{C}:=\sup_{f\in C^{\psi}_{\beta,p}}\inf\limits_{t_{n-1}\in\mathcal{T}_{2n-1}}\|f(\cdot)-t_{n-1}(\cdot)\|_{C}, $$ and $\mathcal{T}_{2n-1}$ is the subspace of trigonometric polynomials $t_{n-1}$ of order $n-1$ with real coefficients, $${\cal E}\left(C^{\psi}_{\beta,p}; Z_{n-1}^{s}\right)_{C}:=\mathop{\sup}\limits_{f\in C^{\psi}_{\beta,p}}\|f(\cdot)-Z^{s}_{n-1}(f;\cdot)\|_{C}.$$

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How to Cite
Serdyuk A., Hrabova U. Order Estimates of the Uniform Approximations by Zygmund Sums on the Classes of Convolutions of Periodic Functions. Carpathian Math. Publ. 2021, 13 (1), 68-80.