scholarly journals INEQUALITIES FOR ALGEBRAIC POLYNOMIALS ON AN ELLIPSE

2020 ◽  
Vol 6 (2) ◽  
pp. 87
Author(s):  
Tatiana M. Nikiforova
Keyword(s):  
Approximation Theory ◽  
Upper Bound ◽  
Algebraic Polynomial ◽  
Uniform Norm ◽  
Algebraic Polynomials

The paper presents new solutions to two classical problems of approximation theory. The first problem is to find the polynomial that deviates least from zero on an ellipse. The second one is to find the exact upper bound of the uniform norm on an ellipse with foci \(\pm 1\) of the derivative of an algebraic polynomial with real coefficients normalized on the segment \([- 1,1]\).

scholarly journals Approximation by Boolean sums of linear operators: Telyakovskiǐ-type estimates

1990 ◽  
Vol 42 (2) ◽  
pp. 253-266 ◽  
Author(s):  
Jia-Ding Cao ◽  
Heinz H. Gonska
Keyword(s):  
Upper Bound ◽  
Algebraic Polynomial ◽  
Present Note ◽  
Type Inequality ◽  
Linear Operators ◽  
Polynomial Operators ◽  
Boolean Sums ◽  
General Conditions

In the present note we study the question: “Under which general conditions do certain Boolean sums of linear operators satisfy Telyakovskiǐ-type estimates?” It is shown, in particular, that any sequence of linear algebraic polynomial operators satisfying a Timan-type inequality can be modified appropriately so as to obtain the corresponding upper bound of the Telyakovskiǐ-type. Several examples are included.

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 Point-wise estimates for the derivative of algebraic polynomials

2021 ◽  
Vol 56 (2) ◽  
pp. 208-211
Author(s):  
A. V. Savchuk
Keyword(s):  
Algebraic Polynomial ◽  
Bernstein Inequality ◽  
Sufficient Condition ◽  
Algebraic Polynomials

We give a sufficient condition on coefficients $a_k$ of an algebraic polynomial $P(z)=\sum\limits_{k=0}^{n}a_kz^k$, $a_n\not=0,$ such that the pointwise Bernstein inequality $|P'(z)|\le n|P(z)|$ is true for all $z,\ |z|\le 1$.

Convex Polynomial Approximation in the Uniform Norm: Conclusion

2005 ◽  
Vol 57 (6) ◽  
pp. 1224-1248 ◽  
Author(s):  
K. A. Kopotun ◽  
D. Leviatan ◽  
I. A. Shevchuk
Keyword(s):  
Convex Function ◽  
Polynomial Approximation ◽  
Uniform Norm ◽  
Degree Of Approximation ◽  
Modulus Of Smoothness ◽  
Jackson Type ◽  
Algebraic Polynomials ◽  
Type Estimate ◽  
Open Problems ◽  
Open Questions

AbstractEstimating the degree of approximation in the uniform norm, of a convex function on a finite interval, by convex algebraic polynomials, has received wide attention over the last twenty years. However, while much progress has been made especially in recent years by, among others, the authors of this article, separately and jointly, there have been left some interesting open questions. In this paper we give final answers to all those open problems. We are able to say, for each r-th differentiable convex function, whether or not its degree of convex polynomial approximation in the uniform norm may be estimated by a Jackson-type estimate involving the weighted Ditzian–Totik kth modulus of smoothness, and how the constants in this estimate behave. It turns out that for some pairs (k, r) we have such estimate with constants depending only on these parameters. For other pairs the estimate is valid, but only with constants that depend on the function being approximated, while there are pairs for which the Jackson-type estimate is, in general, invalid.

Approximation in weighted generalized grand smirnov classes

2017 ◽  
Vol 54 (4) ◽  
pp. 471-488 ◽  
Author(s):  
Daniyal M. Israfilov ◽  
Ahmet Testici
Keyword(s):  
Complex Plane ◽  
Approximation Theory ◽  
Connected Domain ◽  
Modulus Of Smoothness ◽  
Algebraic Polynomials ◽  
Lipschitz Classes ◽  
Carleson Curve ◽  
Smirnov Classes ◽  
Direct And Inverse Theorems ◽  
Inverse Theorems

Let G be a finite simple connected domain in the complex plane C, bounded by a Carleson curve Γ := ∂G. In this work the direct and inverse theorems of approximation theory by the algebraic polynomials in the weighted generalized grand Smirnov classes εp),θ(G,ω) and , 1 < p < ∞, in the term of the rth, r = 1, 2,..., mean modulus of smoothness are proved. As a corollary the constructive characterizations of the weighted generalized grand Lipschitz classes are obtained.

Markov's and Bernstein's Inequalities on Disjoint Intervals

1981 ◽  
Vol 33 (1) ◽  
pp. 201-209 ◽  
Author(s):  
Peter B. Borwein
Keyword(s):  
Algebraic Polynomial ◽  
Supremum Norm ◽  
Algebraic Polynomials ◽  
Image Position

In 1889, A. A. Markov proved the following inequality:INEQUALITY 1. (Markov [4]). If pn is any algebraic polynomial of degree at most n thenwhere ‖ ‖A denotes the supremum norm on A.In 1912, S. N. Bernstein establishedINEQUALITY 2. (Bernstein [2]). If pn is any algebraic polynomial of degree at most n thenfor x ∈ (a, b).In this paper we extend these inequalities to sets of the form [a, b] ∪ [c, d]. Let Πn denote the set of algebraic polynomials with real coefficients of degree at most n.THEOREM 1. Let a < b ≦ c < d and let pn ∈ Πn. Thenfor x ∈ (a, b).

scholarly journals On Best Polynomial Approximations in the Spaces Sp and Widths of Some Classes of Functions

2016 ◽  
Vol 7 ◽  
pp. 19-32
Author(s):  
Alexander N. Shchitov
Keyword(s):  
Approximation Theory ◽  
Upper Bound ◽  
Special Form ◽  
Fourier Coefficients ◽  
Jackson Type ◽  
Polynomial Approximations

In the article are studied some problems of approximation theory in the spaces Sp (1 ≤ p < ∞) introduced by A.I. Stepanets. It is obtained the exact values of extremal characteristics of a special form which connect the values of best polynomial approximations of functions en-1(f)Sp with expressions which contain modules of continuity of functions f(x) є Sp. We have obtained the asymptotically sharp inequalities of Jackson type that connect the best polynomial approximations en-1(f)Sp with modules of continuity of functions f(x) є Sp (1 ≤ p < ∞). Exact values of Kolmogorov, linear, Bernstein, Gelfand and projection n-widths in the spaces Sp are obtained for some classes of functions f(x) є Sp. The upper bound of the Fourier coefficients are found for some classes of functions.

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