On norms of trigonometric polynomials and approximation of differentiable functions by linear averages of their Fourier series. II

1970 ◽  
pp. 157-180
Author(s):  
S. A. Teljakovskiĭ
Keyword(s):  
Fourier Series ◽  
Trigonometric Polynomials ◽  
Differentiable Functions

scholarly journals Integral presentation of deviations of rectangular linear means of Fourier series on classes of periodic differentiable functions

2018 ◽  
Vol 32 ◽  
pp. 92-103
Author(s):  
Oleg Novikov ◽  
Olga Rovenska
Keyword(s):  
Fourier Series ◽  
Real Variable ◽  
Upper Bounds ◽  
Integral Representations ◽  
Linear Operators ◽  
Trigonometric Polynomials ◽  
Periodic Functions ◽  
Differentiable Functions ◽  
Fourier Sums ◽  
Linear Means

The paper deals with the problems of approximation in a uniform metric of periodic functions of many variables by trigonometric polynomials, which are generated by linear methods of summation of Fourier series. Questions of asymptotic behavior of the upper bounds of deviations of linear operators generated by the use of linear methods of summation of Fourier series on the classes of periodic differentiable functions are studied in many works. Methods of investigation of integral representations of deviations of polynomials on the classes of periodic differentiable functions of real variable originated and received its development through the works of S.M. Nikol'skii, S.B. Stechkin, N.P.Korneichuk, V.K. Dzadik, A.I. Stepanets, etc. Along with the study of approximation by linear methods of classes of functions of one variable, are studied similar problems of approximation by linear methods of classes of functions of many variables. In addition to the approximative properties of rectangular Fourier sums, are studied approximative properties of other approximation methods: the rectangular sums of Valle Poussin, Zigmund, Rogozinsky, Favar. In this paper we consider the classes of \(\overline{\psi}\)-differentiable periodic functions of many variables, allowing separately to take into account the properties of partial and mixed \(\overline{\psi}\)-derivatives, and given by analogy with the classes of \(\overline{\psi}\)-differentiable periodic functions of one variable. Integral representations of rectangular linear means of Fourier series on classes of \(\overline{\psi}\)-differentiable periodic functions of many variables are obtained. The obtained formulas can be useful for further investigation of the approximative properties of various linear rectangular methods on the classes \(\overline{\psi}\)-differentiable periodic functions of many variables in order to obtain a solution to the corresponding Kolmogorov-Nikolsky problems.

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

2021 ◽  
Vol 19 (1) ◽  
pp. 1047-1055
Author(s):  
Zhihua Zhang
Keyword(s):  
Fourier Series ◽  
Trigonometric Polynomial ◽  
Algebraic Polynomial ◽  
Fourier Coefficients ◽  
Random Processes ◽  
Qualitative Theory ◽  
Trigonometric Polynomials ◽  
Algebraic Polynomials ◽  
Fourier Approximation ◽  
Random Differential Equations

Abstract 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 m m -order differentiable function f f on [ 0 , 1 0,1 ], we will construct an m m -degree algebraic polynomial P m {P}_{m} depending on values of f f and its derivatives at ends of [ 0 , 1 0,1 ] such that the Fourier coefficients of R m = f − P m {R}_{m}=f-{P}_{m} decay fast. Since the partial sum of Fourier series R m {R}_{m} is a trigonometric polynomial, we can reconstruct the function f f well by the combination of a polynomial and a trigonometric polynomial. Moreover, we will extend these results to the case of random processes.

scholarly journals On the Uniform Convergence of the Fourier Series by the System of Polynomials Generated by the System of Laguerre Polynomials

2020 ◽  
Vol 20 (4) ◽  
pp. 416-423
Author(s):  
Ramis M. Gadzhimirzaev ◽  
Keyword(s):  
Fourier Series ◽  
Sobolev Space ◽  
Weight Function ◽  
Uniform Convergence ◽  
Laguerre Polynomials ◽  
Inner Product ◽  
Sobolev Type ◽  
Absolutely Continuous ◽  
Differentiable Functions ◽  
Proof Of Convergence

Let w(x) be the Laguerre weight function, 1 ≤ p < ∞, and Lpw be the space of functions f, p-th power of which is integrable with the weight function w(x) on the non-negative axis. For a given positive integer r, let denote by WrLpw the Sobolev space, which consists of r−1 times continuously differentiable functions f, for which the (r−1)-st derivative is absolutely continuous on an arbitrary segment [a, b] of non-negative axis, and the r-th derivative belongs to the space Lpw. In the case when p = 2 we introduce in the space WrL2w an inner product of Sobolev-type, which makes it a Hilbert space. Further, by lαr,n(x), where n = r, r + 1, ..., we denote the polynomials generated by the classical Laguerre polynomials. These polynomials together with functions lαr,n(x) = xn / n! , where n = 0, 1, r − 1, form a complete and orthonormal system in the space WrL2w. In this paper, the problem of uniform convergence on any segment [0,A] of the Fourier series by this system of polynomials to functions from the Sobolev space WrLpw is considered. Earlier, uniform convergence was established for the case p = 2. In this paper, it is proved that uniform convergence of the Fourier series takes place for p > 2 and does not occur for 1 ≤ p < 2. The proof of convergence is based on the fact that WrLpw ⊂ WrL2w for p > 2. The divergence of the Fourier series by the example of the function ecx using the asymptotic behavior of the Laguerre polynomials is established.

scholarly journals On strong approximation by modified Meyer-K"onig and Zeller operators

2006 ◽  
Vol 37 (2) ◽  
pp. 123-130
Author(s):  
L. Rempulska ◽  
M. Skorupka
Keyword(s):  
Fourier Series ◽  
Strong Approximation ◽  
Differentiable Functions

We introduce certain modified Meyer-K"onig and Zeller operators $ M_{n;r} $ in the space of  $r $-th times differentiable functions $ f $ and we study strong differences $ H_{n;r}^q(f) $ for them.  This note is motivated by results on strong approximation connected with Fourier series ([7]).

Generalization of the Fourier Series Approach to Model Hourly Energy Use in Commercial Buildings

1999 ◽  
Vol 121 (1) ◽  
pp. 54-62 ◽  
Author(s):  
A. Dhar ◽  
T. A. Reddy ◽  
D. E. Claridge
Keyword(s):  
Fourier Series ◽  
Thermal Energy ◽  
Energy Use ◽  
Functional Form ◽  
Commercial Buildings ◽  
Trigonometric Polynomials ◽  
Generalized Fourier Series ◽  
On Line ◽  
Primary Variable ◽  
Fourier Series Analysis

Development of the accurate models for hourly energy use in commercial buildings has important ramifications for (I) retrofit savings analysis, (ii) diagnostics, (iii) on-line control and (iv) acquiring physical insights into the operating patterns of the buildings. Electric and thermal energy uses in commercial buildings, being strongly periodic, are eminently suitable for Fourier series analysis. Earlier studies assumed trigonometric polynomials with the hour of the day as the primary variable and one week as the period. This model, though suitable on the whole, was poor during certain weekday periods and during weekends. This paper presents a generalized Fourier series approach which, while ensuring a wider range of applicability, also yields superior regression fits partly because of the care taken to separate days of the year during which commercial buildings are operated differently and partly because of the rational functional form of regression model proposed. The validity of the approach is verified with year-long data of twenty-two monitored buildings.

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