Approximation of conjugate functions by trigonometric polynomials in weighted Orlicz spaces

Abstract

Abstract. We investigate the approximation of a conjugate function by the Fejér sums of the Fourier series of the conjugate function and obtain the estimate between the derivatives of the conjugate functions and the derivatives of the conjugate trigonometric polynomials in the weighted Orlicz spaces with Muckenhoupt weights. We prove inverse theorem of approximation theory for the derivatives conjugate functions in the weighted Orlicz spaces.Mathematics subject classification (2010): 41A10, 42A10, 41A25, 46E30.

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.…”

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

“…Note that the estimates obtained in this work depend on sequence of the best approximation E n ( f ) M, ω . Similar problems in different spaces have been investigated by several researchers (see, e.g., [1,2,11,27,42,43,[48][49][50][52][53][54][55][56][57]59,60]). …”

“…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]). …”

Trigonometric Approximation by Modulus of Smoothness in Lp,α (X)

“…Here denote by ( ) , ( = 0,1, … ) the degree of best approximation of ∈ , ( ) by trigonometric polynomials of degree less than n , i.e., ( ) , = {‖ − ‖ , , ∈ } , where denotes the class of trigonometric polynomials of degree n The problems of approximation theory in the weighted and non-weighted space have been investigated in [1] and [6] weight respectively. The approximation problems by trigonometric polynomials in different spaces have been investigated by several authors see, for example, [3], [7], [8] and [9]. In this work we study the approximation problems of unbounded functions by trigonometric polynomials in the weighted space , ( ).…”