Approximations in spaces of locally integrable functions - Stepanets, O. I. (Ukrains’kyi Matematychnyi Zhurnal, 1994)
We study approximations of functions from the sets L^ψ_βN, which are determined by convolutions of the following form: f(x) = A_0 + _-∞^+∞φ(x + t) ψ_β(f)dt, φ N,ψ_β L(-∞, +∞) where 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 L_p determined by a locally integrable norm ||·||_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.