Approximation by trigonometric polynomials in weighted Orlicz spaces

Abstract

Abstract. We investigate the approximation properties of the partial sums of the Fourier series and prove some direct and inverse theorems for approximation by polynomials in weighted Orlicz spaces. In particular we obtain a constructive characterization of the generalized Lipschitz classes in these spaces.

Approximation of functions by de la Vallée-Poussin sums in weighted Orlicz spaces

“…The problems of approximation theory in weighted and nonweighted Lebesgue spaces, weighted and nonweighted Orlicz spaces have been investigated by several authors (see, e.g., [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][30][31][32]36,37,45,46]). …”

On approximation in Weighted Orlicz spaces

“…In the case r = 0, this result was obtained in [11]. In the nonweighted Orlicz spaces, under some restrictive conditions on the function M , Theorem 1 was obtained in [12].…”

“…When r is an even natural number, the first part of Theorem 1 was proved in [11]. The second part of this theorem, in the case r = 0 was proved also in [11].…”

“…This modulus of smoothness is well defined, because the linear operator σ h is bounded in the space L M,ω (see [11]). We define the shift operator σ h and the modulus of smoothness…”

Approximation of periodic functions in weighted Orlicz spaces

“…Approximation by trigonometric polynomials and other related problems in the Orlicz and weighted Orlicz spaces were studied in [2,[8][9][10][11][12]17,18,21,23,24].…”