Problem of correctness of the best approximation in the space of continuous functions

1978 ◽  
Vol 23 (3) ◽  
pp. 190-195
Author(s):  
A. V. Kolushov
Keyword(s):  
Best Approximation ◽  
Continuous Functions ◽  
Space Of Continuous Functions ◽  
The Best Approximation

Alternating Chebyshev Approximation with A Non-Continuous Weight Function

1976 ◽  
Vol 19 (2) ◽  
pp. 155-157 ◽  
Author(s):  
Charles B. Dunham
Keyword(s):  
Weight Function ◽  
Best Approximation ◽  
Closed Interval ◽  
Chebyshev Approximation ◽  
Continuous Functions ◽  
Space Of Continuous Functions ◽  
Negative Function ◽  
Chebyshev Problem ◽  
Image Position ◽  
Continuous Weight

Let [α, β] be a closed interval and C[α, β] be the space of continuous functions on [α, β], For g a function on [α, β] defineLet s be a non-negative function on [α, β]. Let F be an approximating function with parameter space P such that F(A, .)∊ C[α, β] for all A∊P. The Chebyshev problem with weight s is given f ∊ C[α, β], to find a parameter A* ∊ P to minimize e(A) = ||s * (f - F(A, .))|| over A∊P. Such a parameter A* is called best and F(A*,.) is called a best approximation to f.

scholarly journals Upper Estimates of the angle best approximations of generalized Liouville-Weyl derivatives

2021 ◽  
Vol 103 (3) ◽  
pp. 54-67
Author(s):  
A.E. Jetpisbayeva ◽  
◽  
A.A. Jumabayeva ◽  
Keyword(s):  
Best Approximation ◽  
Three Dimensional ◽  
Continuous Functions ◽  
Trigonometric Polynomials ◽  
Dimensional Case ◽  
Best Approximations ◽  
Approximation Of Functions ◽  
The Best Approximation

In this article we consider continuous functions f with period 2π and their approximation by trigonometric polynomials. This article is devoted to the study of estimates of the best angular approximations of generalized Liouville-Weyl derivatives by angular approximation of functions in the three-dimensional case. We consider generalized Liouville-Weyl derivatives instead of the classical mixed Weyl derivative. In choosing the issues to be considered, we followed the general approach that emerged after the work of the second author of this article. Our main goal is to prove analogs of the results of in the three-dimensional case. The concept of general monotonic sequences plays a key role in our study. Several well-known inequalities are indicated for the norms, best approximations of the r-th derivative with respect to the best approximations of the function f. The issues considered in this paper are related to the range of issues studied in the works of Bernstein. Later Stechkin and Konyushkov obtained an inequality for the best approximation f^(r). Also, in the works of Potapov, using the angle approximation, some classes of functions are considered. In subsection 1 we give the necessary notation and useful lemmas. Estimates for the norms and best approximations of the generalized Liouville-Weyl derivative in the three-dimensional case are obtained.

scholarly journals On approximation of continuous functions by piecewise-constant ones in integral metrics

1998 ◽  
Vol 6 ◽  
pp. 128
Author(s):  
O.V. Chernytska
Keyword(s):  
Upper Bound ◽  
Best Approximation ◽  
Continuous Functions ◽  
Moduli Of Continuity ◽  
Piecewise Constant ◽  
Integral Metrics ◽  
The Best Approximation ◽  
Piecewise Constant Functions ◽  
Approximation Of Continuous Functions ◽  
Decreasing Functions

We obtain upper bound of the best approximation of the classes $H^{\omega} [a, b]$ by piecewise-constant functions over uniform split in metrics of $L_{\varphi}[a, b]$ spaces, which are generated by continuous non-decreasing functions $\varphi$ that are equal to zero in zero. We study the classes of functions $\varphi$, for which the obtained bound is exact for all convex moduli of continuity.

Best Trigonometric Approximation, Fractional Order Derivatives and Lipschitz Classes

1977 ◽  
Vol 29 (4) ◽  
pp. 781-793 ◽  
Author(s):  
P. L. Butzer ◽  
H. Dyckhoff ◽  
E. Görlich ◽  
R. L. Stens
Keyword(s):  
Fractional Order ◽  
Best Approximation ◽  
Continuous Functions ◽  
Trigonometric Approximation ◽  
Trigonometric Polynomials ◽  
Lipschitz Classes ◽  
The Best Approximation ◽  
Image Position ◽  
Best Trigonometric Approximation

Let C2π denote the space of 2π-periodic continuous functions and πn the set of trigonometric polynomials of degree ≦ n, where n ϵ P = {0, 1, … } . Given θ > 0, the well-known theorem of Stečkin and its converse state that the best approximation of an ƒ ϵ C2π with respect to the max-norm satisfies

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