scholarly journals On approximation of nonperiodic functions by algebraic polynomials in $L_p$ metric ($0 < p < 1$)

2021 ◽  
pp. 35
Author(s):  
L.B. Khodak
Keyword(s):  
Algebraic Polynomials ◽  
Constructive Characteristic

In the paper, we consider approximations of nonperiodic functions defined on $[-1, 1]$ by algebraic polynomials in $L_p$ metric ($0 < p < 1$).In particular, for some classes we provide the constructive characteristic in the same metric.

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.

The average number of point of inflection of random algebraic polynomials

1998 ◽  
Vol 16 (4) ◽  
pp. 721-731
Author(s):  
M. Sambandham ◽  
Henry Gore ◽  
K. Farahmand
Keyword(s):  
Algebraic Polynomials ◽  
Random Algebraic Polynomials

On Tchebycheff Quadrature

1965 ◽  
Vol 17 ◽  
pp. 652-658 ◽  
Author(s):  
Paul Erdös ◽  
A. Sharma
Keyword(s):  
Algebraic Polynomials ◽  
History Of ◽  
Image Position

Tchebycheff proposed the problem of finding n + 1 constants A, x1, x2, . . , xn ( — 1 ≤ x1 < x2 < . . . < xn ≤ +1) such that the formula(1)is exact for all algebraic polynomials of degree ≤n. In this case it is clear that A = 2/n. Later S. Bernstein (1) proved that for n ≥ 10 not all the xi's can be real. For a history of the problem and for more references see Natanson (4).

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