Direct and inverse approximation theorems in the weighted Orlicz-type spaceswith a variable exponent

Abstract

In weighted Orlicz type spaces S p, µ with a variable summation exponent, the direct and inverse approximation theorems are proved in terms of best approximations of functions and moduli of smoothness of fractional order. It is shown that the constant obtained in the inverse approximation theorem is in a certain sense the best. Some applications of the results are also proposed. In particular, the constructive characteristics of functional classes defined by such moduli of smoothness are given. Equivalence between moduli of smoothness and certain Peetre K-functionals is shown in the spaces S p, µ .

Best Approximation of the Conjugate Functions in Morrey Spaces

“…Note that similar estimates in the different spaces for modulus of continuity were proved in [1], [2], [10], [24], [27], [41] and [43].…”

“…We prove an inverse theorem of approximation theory in Morrey spaces. Similar problems in different spaces have been investigated in [1][2][3], [7], [10][11][12][13], [19][20][21], [24][25][26][27], [29], [31], [34] and [39][40][41][42][43]. Our main results are the following Theorem 2.…”

Direct and inverse approximation theorems in the Besicovitch-Museilak-Orlicz spaces of almost periodic functions

“…If all the functions M k are identical (namely, M k (t) ≡ M(t), k ∈ Z), the spaces S M coincide with the ordinary Orlicz type spaces S M [15]. If M k (t) = µ k t p k , p k ≥ 1, µ k ≥ 0, then S M coincide with the weighted spaces S p, µ with variable exponents [2].…”

Exponential approximation in variable exponent Lebesgue spaces on the real line

“…See the books [16,18,51] for more references. Nowadays, many mathematician solved many problems for the approximation of function in these type spaces defined on [0, 2π] ⊂ R (see e.g., [7,8,26,30,31,34], [1,2,3,11,12], [5,6,9,13,14], [22,24,25,28,32,33,36], [37,38,44,49,50,56]). In this paper, we propose generalized our last results in [10] which we obtained a direct and inverse theorems for approximation by entire functions of finite degree in variable exponent Lebesgue spaces on the whole real axis R with (1.1) sup…”