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This article is cited in 1 scientific paper (total in 1 paper)
Approximation properties of Fourier sums for $2\pi$-periodic piecewise linear continuous functions
G. G. Akniev Daghestan scientific center of RAS
Abstract:
In various areas of applications the problem of approximation of continuous function $f = f(x)$, whose values are known in nodes of some grid $\Omega_m = \{\xi_i\}_{i=0}^{m}$ arises.
Usually, to solve this problem the polynomial spline $l_m^r(x)$ of given degree $r$ is used [ahlberb_splines,zavyalov_splain_functions,stechkin_splines_in_math], which in simplest case $r = 1$ is polyline $l_m = l_m(x) = l_m^1(x)$, that coincides in grid nodes with function $f$.
If we want to store this polyline, we should store all the pairs $(\xi_0, y_0), \ldots, (\xi_m, y_m)$, where $y_i = f(\xi_i)$ $(i = 0, \ldots, m)$, which may take o lot of storage space if number of nodes is big. In this connection the interim problem of compression of information $(\xi_0, y_0), \ldots, (\xi_m, y_m)$ so that we can restore original polyline with given precision arises.
To solve such problems the so called spectral method is usually applied, which is based on expansion of function $l_m$ in a series of a given orthonormal system and saving a minimal amount of the obtained expansion coefficients provided an ability to restore this function with given precision.
In the present paper we attempted to solve this problem for $2\pi$-periodic continuous polylines by expansion them into trigonometric Fourier series.
Keywords:
Fourier sums, polyline, function approximation.
Received: 07.04.2016 Revised: 18.05.2016 Accepted: 19.05.2016
Citation:
G. G. Akniev, “Approximation properties of Fourier sums for $2\pi$-periodic piecewise linear continuous functions”, Daghestan Electronic Mathematical Reports, 2016, no. 5, 13–19
Linking options:
https://www.mathnet.ru/eng/demr21 https://www.mathnet.ru/eng/demr/y2016/i5/p13
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Abstract page: | 126 | Full-text PDF : | 45 | References: | 28 |
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