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 Mat. Sb., 2016, Volume 207, Number 7, Pages 131–158 (Mi msb8509)

Approximation of functions in variable-exponent Lebesgue and Sobolev spaces by de la Vallée-Poussin means

I. I. Sharapudinovab

a Daghestan Scientific Centre of the Russian Academy of Sciences, Makhachkala

Abstract: We consider the space $L^{p(\cdot)}_{2\pi}$ formed by $2\pi$-periodic real measurable functions $f$ for which the integral $\displaystyle\int_{-\pi}^{\pi}|f(x)|^{p(x)} dx$ exists and is finite, where $p(x)$, $1\le p(x)$, is a $2\pi$-periodic measurable function (a variable exponent). If $p(x)\le \overline p<\infty$, then the space $L^{p(\cdot)}_{2\pi}$ can be endowed with the structure of Banach space with the norm
$$\|f\|_{p(\cdot)}=\inf\{\alpha>0:\int_{-\pi}^{\pi}|\frac{f(x)}{\alpha}|^{p(x)} dx\le1\}.$$
In the space $L^{p(\cdot)}_{2\pi}$ we distinguish a subspace $W^{r,p(\cdot)}_{2\pi}$ of Sobolev type. We investigate the approximation properties of the de la Vallée-Poussin means for trigonometric Fourier sums for functions in the space $W^{r,p(\cdot)}_{2\pi}$. In particular, we prove that if the variable exponent $p=p(x)$ satisfies the Dini-Lipschitz condition $|p(x)-p(y)|\ln\frac{2\pi}{|x-y|}\le c$ and if $f\in W^{r,p(\cdot)}_{2\pi}$, then the de la Vallée-Poussin means $V_m^n(f)=V_m^n(f,x)$ with $n\le am$ satisfy
$$\|f-V_m^n(f)\|_{p(\cdot)}\le \frac{c_r(p,a)}{n^r}\Omega(f^{(r)}, \frac1n)_{p(\cdot)},$$
where $\Omega(g,\delta)_{p(\cdot)}$ is a modulus of continuity of the function $g\in L^{p(\cdot)}_{2\pi}$ defined in terms of the Steklov functions. It is proved that if $1<p(x)$, $r\ge1$, $f\in W^{r,p(\cdot)}_{2\pi}$ and the Dini-Lipschitz condition holds, then
$$|f(x)-V_m^n(f,x)|\le\frac{c_r(p)}{m+1}\sum_{k=n}^{n+m}\frac{E_k(f^{(r)})_{p(\cdot)}}{(k+1)^{r-{{1}/{p(x)}}}},$$
where $E_k(g)_{p(\cdot)}$ stands for the best approximation to $g\in L^{p(\cdot)}_{2\pi}$ by trigonometric polynomials of order $k$.
Bibliography: 19 titles.

Keywords: Lebesgue and Sobolev spaces with variable exponents, approximation of functions by de la Vallée-Poussin means.

 Funding Agency Grant Number Russian Foundation for Basic Research 16-01-00486-a This work was supported by the Russian Foundation for Basic Research (grant no. 16-01-00486-a).

DOI: https://doi.org/10.4213/sm8509

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English version:
Sbornik: Mathematics, 2016, 207:7, 1010–1036

Bibliographic databases:

UDC: 517.538
MSC: Primary 42A10; Secondary 46E30, 46E35

Citation: I. I. Sharapudinov, “Approximation of functions in variable-exponent Lebesgue and Sobolev spaces by de la Vallée-Poussin means”, Mat. Sb., 207:7 (2016), 131–158; Sb. Math., 207:7 (2016), 1010–1036

Citation in format AMSBIB
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This publication is cited in the following articles:
1. D. M. Israfilov, E. Yirtici, “On some properties of convolutions in variable exponent Lebesgue spaces”, Complex Anal. Oper. Theory, 11:8 (2017), 1817–1824
2. D. M. Israfilov, A. Testici, “Simultaneous approximation in Lebesgue space with variable exponent”, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb., 44:1 (2018), 3–18
3. D. M. Israfilov, A. Testici, “Approximation by matrix transforms in weighted Lebesgue spaces with variable exponent”, Results Math., 73:1 (2018), 8, 25 pp.
4. D. Israfilov, A. Testici, “Multiplier and approximation theorems in Smirnov classes with variable exponent”, Turkish J. Math., 42:3 (2018), 1442–1456
5. D. M. Israfilov, A. Testici, “Some inverse and simultaneous approximation theorems in weighted variable exponent Lebesgue spaces”, Anal. Math., 44:4 (2018), 475–492
6. D. M. Israfilov, E. Gursel, E. Aydin, “Maximal convergence of Faber series in Smirnov classes with variable exponent”, Bull. Braz. Math. Soc. (N.S.), 49:4 (2018), 955–963
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