GENERALIZED TRIGONOMETRIC FUNCTIONS AND INTERPOLATING CONVOLUTIONS
DOI:
https://doi.org/10.18372/2310-5461.29.10092Keywords:
generalized trigonometric functions, generalized Dirac-functions, test functions, divergent trigonometric series, convolution, interpolationAbstract
We consider a class of generalized trigonometric functions (distributions) given by the divergent trigonometric series; coefficients of these series have a certain order of growth. These functions have no values in usual sense and manifest themselves only as convolution with the test functions. Classes of test functions are formed by even periodical functions that are either defined by chosen Fourier coefficients or have certain differential properties. Therefore generalized trigonometric functions are analogous to well known Dirac -functions. The operations of differentiation and integration of these generalized trigonometric functions are considered. The class of generalized trigonometric functions of zero order with coefficients forming -periodical sequences is investigated in details. Such functions under certain conditions that are received in this paper have interpolating properties. The results of calculations for the test example are given; these results are well correlated to those predicted by theory.
References
Харди Г. Расходящиеся ряды / Г. Харди. — М. : Изд-во иностранной литературы, 1951. — 504 с.
Денисюк В. П. Узагальнені тригонометричні функції та їх застосування / В. П. Денисюк // Віс-ник астрономічної школи, 2015. — № 2. — С. 164–168.
Колмогоров А. Н. Элементы теории функций и функционального анализа / А. Н. Колмогоров, С. В. Фомин. — М. : Физматгиз, 1968. — 496 с.
Денисюк В.П. Фундаментальні функції та тригонометричні сплайни / В. П. Денисюк. — К. : ЗАТ «ВІПОЛ», 2015. — 296 с.
Денисюк В. П. Сплайни та сигнали. — К.:, ЗАТ «ВІПОЛ», 2007. — 228 с.
