Almost everywhere divergent Fourier series with respect to arbitrary bounded orthonormal system of the form {ϕ(n k x)}

1981 ◽  
Vol 37 (1-3) ◽  
pp. 35-38
Author(s):  
I. Joó
Keyword(s):  
Fourier Series ◽  
Orthonormal System ◽  
Almost Everywhere ◽  
Divergent Fourier Series

scholarly journals Generalized localization for spherical partial sums of multiple Fourier series

2019 ◽  
Vol 489 (1) ◽  
pp. 7-10
Author(s):  
R. R. Ashurov
Keyword(s):  
Fourier Series ◽  
Multiple Fourier Series ◽  
Partial Sums ◽  
Localization Principle ◽  
Open Set ◽  
Almost Everywhere

In this paper the generalized localization principle for the spherical partial sums of the multiple Fourier series in the L2-class is proved, that is, if f L2 (ТN) and f = 0 on an open set ТN then it is shown that the spherical partial sums of this function converge to zero almost - ​everywhere on . It has been previously known that the generalized localization is not valid in Lp (TN) when 1 p 2. Thus the problem of generalized localization for the spherical partial sums is completely solved in Lp (TN), p 1: if p 2 then we have the generalized localization and if p 2, then the generalized localization fails.

Modeling the Quasigeoid Heights on Local Areas of the Earth Surface by the Results of Expansion into a Generalized Fourier Series

2020 ◽  
Vol 28 (4) ◽  
pp. 82-94
Author(s):  
V.F. Kanushin ◽  
◽  
I.G. Ganagina ◽  
D.N. Goldobin ◽  
◽  
...  
Keyword(s):  
Fourier Series ◽  
Orthonormal System ◽  
Local Area ◽  
Spherical Functions ◽  
Trigonometric Functions ◽  
Computing Technology ◽  
Generalized Fourier Series ◽  
The Earth ◽  
Local Areas ◽  
Discrete Values

The article presents two methods of modeling discrete heights of a quasigeoid on a local area of the earth’s surface using a gen-eralized Fourier series. The first method is based on modeling the characteristics of the earth’s gravitational field on a plane and involves the use of a two-dimensional Fourier transform by an orthonormal system of trigonometric functions. The second method consists in the expansion of the quasigeoid heights in a Fourier series by an orthonormal system of spherical functions on a local area of the earth’s surface. The errors of approxima-tion of the obtained discrete values of the quasigeoid heights on the local territory are analyzed. It is shown that with the modern computing technology, the most accurate and technologically simple way to model the quasigeoid heights on local areas is to expand them into a Fourier series by an orthonormal system of spherical functions.

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