Trigonometric Approximation of Functions from $$L_{2\pi }^{p(x)}$$

Abstract

We generalize and improve the results of A. Guven, D. Israfilov, Xh. Z. Krasniqi and T. N. Shakh-Emirov. We consider the general methods of summability of Fourier series of functions from L p(x) 2π with p (x) ≥ 1. For estimate of the error of approximation of functions by the matrix means we use a modulus of continuity constructed by the Steklov functions of the increments of considered functions without of absolute values.

On the degree of approximation of continuous functions by a linear transformation of their Fourier series

“…These results are extended further in [4], then they are generalized in [5], and are treated again in [6] using some other means (see also [9]). Very recently, W. Lenski and B. Szal [8], used (for trigonometric approximation in the variable space Ä Ô(Ü) and Ñ = Ò)…”

“…Motivated by the above condition (for Ò Ñ = 1 Ò+1 and Ñ = Ò this condition has been introduced in [8]) we introduce another new condition…”

“…(1.6), respectively. Also, it was shown in [8] that if ( + 1) ¬ Ò ¾ À Î Ë, then condition (£) holds true. Moreover, it easy to verify that condition (££) implies ( + 1) ¬ Ò ¾ Ê Î Ë.…”

On the degree of approximation of continuous functions by a linear transformation of their Fourier series

“…These results are extended further in [4], then they are generalized in [5], and are treated again in [6] using some other means (see also [9]). Very recently, W. Lenski and B. Szal [8], used (for trigonometric approximation in the variable space L p(x) and m = n) the following condition…”

“…Motivated by the above condition (for a n,m = 1 n+1 and m = n this condition has been introduced in [8]) we introduce another new condition…”