Approximations in spaces of locally integrable functions

  • O. I. Stepanets

Abstract

We study approximations of functions from the sets $\hat{L}^{\psi}_{\beta}\mathfrak{N}$, which are determined by convolutions of the following form: $$f(x) = A_0 + \int\limits_{-\infty}^{+\infty}\varphi(x + t) \hat{\psi}_{\beta}(f)dt, \quad \varphi \in \mathfrak{N},\quad \hat{\psi}_{\beta} \in L(-\infty, +\infty)$$ where $\mathfrak{N}$ is a fixed subset of functions with locally integrable $p$-th powers $(p \geq 1)$. As an approximating aggregate, we use so-called Fourier operators, which are entire functions of the exponential type $\leq \sigma$ that turn into trigonometric polynomials if the function $\varphi(\cdot)$ is periodic (in particular, they may be the Fourier sums of the function approximated). Approximations are studied in the spaces $\hat{L}_p$ determined by a locally integrable norm $||\cdot||_{\hat{p}}$. Analogs of the Lebesgue and Favard inequalities, well-known in the periodic case, are obtained and used for finding order-exact estimates of the corresponding best approximations and estimates of approximations by Fourier operators, which are order-exact and, in some important cases, they arc also exact in the sense of constants with principal terms of these estimates.
Published
25.05.1994
How to Cite
StepanetsO. I. “Approximations in Spaces of Locally Integrable Functions”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 46, no. 5, May 1994, pp. 597–625, https://umj.imath.kiev.ua/index.php/umj/article/view/5722.
Section
Research articles