On the simultaneous approximation of a Hilbert transform and its derivatives on the real semiaxis

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• 作者：Maria Carmela De Bonis ; ¹ ; ^{mariacarmela.debonis@unibas.it} ; Donatella Occorsio¹ ; ^{donatella.occorsio@unibas.it}
• 关键词：Hadamard finite part integrals ; Lagrange interpolation ; Orthogonal polynomials ; Approximation by polynomials
• 刊名：Applied Numerical Mathematics
• 出版年：2017
• 出版时间：April 2017
• 年：2017
• 卷：114
• 期：Complete
• 页码：132-153
• 全文大小：901 K
• 卷排序：114

文摘

In this paper we propose a global method to approximate the derivatives of the weighted Hilbert transform of a given function fHp(fwα,t)=dpdtp⨍0+∞f(x)x−twα(x)dx=p!⨎0+∞f(x)(x−t)p+1wα(x)dx, where p∈{1,2,…}p∈{1,2,…}, t>0t>0, and wα(x)=e−xxαwα(x)=e−xxα is a Laguerre weight. The right-hand integral is defined as the finite part in the Hadamard sense. The proposed numerical approach is convenient when the approximation of the function Hp(fwα,t)Hp(fwα,t) is required. Moreover, if there is the need, all the computations can be performed without differentiating the density function f. Numerical stability and convergence are proved in suitable weighted uniform spaces and numerical tests which confirm the theoretical estimates are presented.