On the trigonometric approximation of functions in a weighted Lipschitz class
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ORIGINAL RESEARCH PAPER
On the trigonometric approximation of functions in a weighted Lipschitz class Uaday Singh1 Received: 18 June 2018 / Accepted: 26 August 2020 Forum D’Analystes, Chennai 2020
Abstract Mishra et al. (Applied Mathematics and Computation 237:252–263, 2014) obtained the degree of approximation of f~, conjugate of a 2p periodic function f belonging to the weighted Lipschitz class WðLr ; b; nÞðr 1Þ, through the Cesa`ro-No¨rlund means of the conjugate Fourier series of f, which is an extension of the results of Lal (Applied Mathematics and Computation 209:346–350, 2009) and Singh et al. (International Journal of Mathematics and Mathematical Sciences 2012:1–12, 2012). Zhang (Applied Mathematics and Computation 247:1139–1140, 2014) has pointed out that the conclusions of the theorem of Mishra et al. (Applied Mathematics and Computation 237:252–263, 2014) hold only for constant functions, and thus the results are trivial. In this paper, we redefine the problems of Lal (Applied Mathematics and Computation 209:346–350, 2009) and Mishra et al. (Applied Mathematics and Computation 237:252–263, 2014) for f 2 WðLr ; b; nÞðr 1Þ and its conjugate f~. We obtain the degree of approximation through more general summability means under weighted norm which in turn resolve the issues raised by Zhang (Applied Mathematics and Computation 247:1139–1140, 2014). We also derive some corollaries from our results. Keywords Degree of approximation Weighted norm Minkowski inequality Fourier series Conjugate function
Mathematics Subject Classification 41A10
& Uaday Singh [email protected] 1
Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, India
123
U. Singh
1 Preliminaries and introduction For f 2 Lr ½0; 2p, the space of 2p - periodic rth power integrable (in the sense of Lebesgue) functions, the norm is defined as follows: 8 Z 1=r 2p > > < 1 jf ðxÞjr dx ; r1 2p 0 k f kr ¼ > > : ess sup jf ðxÞj; r ¼ 1: 0 x 2p
We denote by WðLr ; bÞ, the weighted Lr ½0; 2p with weight function sinbr ðx=2Þ; b 0 and the norm 8 Z 1=r > 1 2p r > b < jf ðxÞ sin ðx=2Þj dx ; r1 2p 0 ð1Þ k f kr;b ¼ > > : ess sup jf ðxÞ sinb ðx=2Þj; r ¼ 1: 0 x 2p
The following subclasses of Lr ½0; 2p-space are also well known in the literature [2, 4–9]. Lipa ¼ ff : ½0; 2p ! R : f ðx þ tÞ f ðxÞ ¼ Oðta Þ; t [ 0; a [ 0g;
ð2Þ
Lipða; rÞ ¼ ff 2 Lr ½0; 2p : kf ðx þ tÞ f ðxÞkr ¼ Oðta Þ; t [ 0; r 1; a [ 0g; ð3Þ Lipðn; rÞ ¼ ff 2 Lr ½0; 2p : kf ðx þ tÞ f ðxÞkr ¼ OðnðtÞÞ; t [ 0; r 1g;
ð4Þ
where nðtÞ is a positive increasing function with limþ nðtÞ ¼ 0. The weighted of the t!0
class Lipðn; rÞ denoted by WðLr ; b; nÞ is defined as WðLr ; b; nÞ ¼ ff 2 WðLr ; bÞ : kf ðx þ tÞ f ðxÞkr;b ¼ OðnðtÞÞ; t [ 0; r 1; b 0g;
ð5Þ
The weighted class WðLr ; b; nÞ in Eq. (5) is a modified version of its earlier definition [2, p. 347 and references therein] in the sense that the weight function sinbr ðxÞ is replaced by sinbr ðx=2Þ(Cf. [1], [6, p. 6871] and
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