Approximations in spaces of locally integrable functions - Stepanets, O. I. (Ukrains’kyi Matematychnyi Zhurnal, 1994)
We study approximations of functions from the sets $\hat{L}^{ψ}_{β}\mathfrak{N}$, which are determined by convolutions of the following form:$$f(x) = A_0 + \int\limits_{-∞}^{+∞}φ(x + t) \hat{ψ}_{β}(f)dt, \quad φ \in \mathfrak{N},\quad \hat{ψ}_{β} \in L(-∞, +∞)$$where $\mathfrak{N}$ is a fixed subset of functions with locally integrable $p$-th powers $(p ≥ 1)$. As an approximating aggregate, we use so-called Fourier operators, which are entire functions of the exponential type $≤ σ$that turn into trigonometric polynomials if the function $φ(·)$ 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 $||·||_{\hat{p}}$. 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.