Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials

Fourier approximation plays a key role in qualitative theory of deterministic and random differential equations. In this paper, we will develop a new approximation tool. For an mm-order differentiable function ff on [0,10,1], we will construct an mm-degree algebraic polynomial Pm{P}_{m} depending on...

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Autor principal:	Zhang Zhihua
Formato:	article
Lenguaje:	EN
Publicado:	De Gruyter 2021
Materias:	fourier approximation trigonometric polynomial differential equations 41-xx 42-xx 65-xx Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/c093d6b3d1754b3999d514535459f019
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spelling	oai:doaj.org-article:c093d6b3d1754b3999d514535459f0192021-12-05T14:10:53ZDeterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials2391-545510.1515/math-2021-0089https://doaj.org/article/c093d6b3d1754b3999d514535459f0192021-09-01T00:00:00Zhttps://doi.org/10.1515/math-2021-0089https://doaj.org/toc/2391-5455Fourier approximation plays a key role in qualitative theory of deterministic and random differential equations. In this paper, we will develop a new approximation tool. For an mm-order differentiable function ff on [0,10,1], we will construct an mm-degree algebraic polynomial Pm{P}_{m} depending on values of ff and its derivatives at ends of [0,10,1] such that the Fourier coefficients of Rm=f−Pm{R}_{m}=f-{P}_{m} decay fast. Since the partial sum of Fourier series Rm{R}_{m} is a trigonometric polynomial, we can reconstruct the function ff well by the combination of a polynomial and a trigonometric polynomial. Moreover, we will extend these results to the case of random processes.Zhang ZhihuaDe Gruyterarticlefourier approximationtrigonometric polynomialdifferential equations41-xx42-xx65-xxMathematicsQA1-939ENOpen Mathematics, Vol 19, Iss 1, Pp 1047-1055 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	fourier approximation trigonometric polynomial differential equations 41-xx 42-xx 65-xx Mathematics QA1-939
spellingShingle	fourier approximation trigonometric polynomial differential equations 41-xx 42-xx 65-xx Mathematics QA1-939 Zhang Zhihua Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
description	Fourier approximation plays a key role in qualitative theory of deterministic and random differential equations. In this paper, we will develop a new approximation tool. For an mm-order differentiable function ff on [0,10,1], we will construct an mm-degree algebraic polynomial Pm{P}_{m} depending on values of ff and its derivatives at ends of [0,10,1] such that the Fourier coefficients of Rm=f−Pm{R}_{m}=f-{P}_{m} decay fast. Since the partial sum of Fourier series Rm{R}_{m} is a trigonometric polynomial, we can reconstruct the function ff well by the combination of a polynomial and a trigonometric polynomial. Moreover, we will extend these results to the case of random processes.
format	article
author	Zhang Zhihua
author_facet	Zhang Zhihua
author_sort	Zhang Zhihua
title	Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
title_short	Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
title_full	Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
title_fullStr	Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
title_full_unstemmed	Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
title_sort	deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
publisher	De Gruyter
publishDate	2021
url	https://doaj.org/article/c093d6b3d1754b3999d514535459f019
work_keys_str_mv	AT zhangzhihua deterministicandrandomapproximationbythecombinationofalgebraicpolynomialsandtrigonometricpolynomials
_version_	1718371626412670976

Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials

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