Abstract
We show the results, corresponding to theorem of Lal (2009), on the rate of pointwise approximation of functions from the pointwise integral Lipschitz classes by matrix summability means of their Fourier series as well as the theorems on norm approximations.
Highlights
Let Lp 1 ≤ p < ∞ be the class of all 2π-periodic real-valued functions integrable in the Lebesgue sense with pth power over Q −π, π with the norm1/p f : f · Lp f t pdt, Q and consider the trigonometric Fourier series Sf x : ao f 2 ∞aν f cos νx bν f sin νx, ν11.2 and conjugate oneSf x : bν f cos νx − aν f sin νx Journal of Function Spaces and Applications with the partial sums Skf and Skf, respectively
1.15 holds for all n, where K αn denotes the sequence of constants appearing in the inequalities
As a measure of approximation of functions by the above means, we use the generalized pointwise moduli of continuity of f in the space Lp defined for β ≥ 0 by the formulas wx, βf δ Lp : wx, βf δ Lp : 1δ δ1 βp 0
Summary
Sf x : bν f cos νx − aν f sin νx Journal of Function Spaces and Applications with the partial sums Skf and Skf, respectively. A sequence c : cn of nonnegative numbers will be called the Head Bounded Variation Sequence, or briefly c ∈ HBVS, if it has the property m−1 Followed by Leindler 3 , a sequence c : cn of nonnegative numbers tending to zero is called the Mean Rest Bounded Variation Sequence, or briefly c ∈ MRBVS, if it has the property
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