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 Concentration of the error between a function and its polynomial of best uniform approximation

1998 ◽  
Vol 41 (3) ◽  
pp. 447-463 ◽  
Author(s):  
Maurice Hasson
Keyword(s):  
Uniform Approximation ◽  
Algebraic Polynomial ◽  
General Class ◽  
Supremum Norm ◽  
Best Uniform Approximation ◽  
Class A ◽  
Image Position ◽  
Class Of Functions

Let f be a continuous real valued function defined on [−1, 1] and let En(f) denote the degree of best uniform approximation to f by algebraic polynomial of degree at most n. The supremum norm on [a, b] is denoted by ∥.∥[a, b] and the polynomial of degree n of best uniform approximation is denoted by Pn. We find a class of functions f such that there exists a fixed a ∈(−1, 1) with the following propertyfor some positive constants C and N independent of n. Moreover the sequence is optimal in the sense that if is replaced by then the above inequality need not hold no matter how small C > 0 is chosen.We also find another, more general class a functions f for whichinfinitely often.

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.

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).

scholarly journals Approximation by Generalised Polynomials with Integral Coefficients

1977 ◽  
Vol 20 (1) ◽  
pp. 129-131
Author(s):  
J. Tzimbalario
Keyword(s):  
Supremum Norm ◽  
Real Numbers ◽  
Image Position

Let C[0,1] be the space of all continuous real valued functions defined in [0,1] with the supremum norm1The subspace of C[0,1] consisting of all functions f(x) for which f(0) and f(l) are integers will be denoted by C0[0,1], Let be a sequence of real numbers satisfying:2

Characterization of some classes of operators on spaces of vector-valued continuous functions

1985 ◽  
Vol 97 (1) ◽  
pp. 137-146 ◽  
Author(s):  
Fernando Bombal ◽  
Pilar Cembranos
Keyword(s):  
Banach Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Supremum Norm ◽  
Representing Measure ◽  
Bounded Linear ◽  
Second Dual ◽  
Image Position ◽  
Vector Valued

Let K be a compact Hausdorff space and E, F Banach spaces. We denote by C(K, E) the Banach space of all continuous. E-valued functions defined on K, with the supremum norm. It is well known ([6], [7]) that every operator (= bounded linear operator) T from C(K, E) to F has a finitely additive representing measure m of bounded semi-variation, defined on the Borel σ-field Σ of K and with values in L(E, F″) (the space of all operators from E into the second dual of F), in such a way thatwhere the integral is considered in Dinculeanu's sense.

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 $z,\ |z|\le 1$.

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 \([- 1,1]\).

Exact Inequalities for the Norms of Factors of Polynomials

1994 ◽  
Vol 46 (4) ◽  
pp. 687-698 ◽  
Author(s):  
Peter B. Borwein
Keyword(s):  
Chebyshev Polynomial ◽  
Algebraic Polynomial ◽  
Supremum Norm

AbstractThis paper addresses a number of questions concerning the size of factors of polynomials. Let p be a monic algebraic polynomial of degree n and suppose q1q2 … qi is a monic factor of p of degree m. Then we can, in many cases, exactly determine Here ‖ . ‖ is the supremum norm either on [—1, 1] or on {|z| ≤ 1}. We do this by showing that, in the interval case, for each m and n, the n-th Chebyshev polynomial is extremal. This extends work of Gel'fond, Mahler, Granville, Boyd and others. A number of variants of this problem are also considered.

scholarly journals Decay estimates in the supremum norm for the solutions to a nonlinear evolution equation

2014 ◽  
Vol 144 (3) ◽  
pp. 557-566 ◽  
Author(s):  
Petri Juutinen
Keyword(s):  
Evolution Equation ◽  
Asymptotic Behaviour ◽  
Decay Rate ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Supremum Norm ◽  
Decay Estimates ◽  
Lateral Boundary ◽  
Boundary Values ◽  
Image Position

We study the asymptotic behaviour, as t → ∞, of the solutions to the nonlinear evolution equationwhere ΔpNu = Δu + (p−2) (D2u(Du/∣Du∣)) · (Du/∣Du∣) is the normalized p-Laplace equation and p ≥ 2. We show that if u(x,t) is a viscosity solution to the above equation in a cylinder Ω × (0, ∞) with time-independent lateral boundary values, then it converges to the unique stationary solution h as t → ∞. Moreover, we provide an estimate for the decay rate of maxx∈Ω∣u(x,t) − h(x)∣.

scholarly journals Maximal ideal spaces of Banach algebras of derivable elements

1970 ◽  
Vol 11 (3) ◽  
pp. 310-312 ◽  
Author(s):  
R. J. Loy
Keyword(s):  
Banach Algebra ◽  
Maximal Ideal ◽  
Present Note ◽  
Banach Algebras ◽  
Commutative Banach Algebra ◽  
Supremum Norm ◽  
Simple Consequence ◽  
Maximal Ideals ◽  
Image Position

Let A be a commutative Banach algebra, D a closed derivation defined on a subalgebra Δ of A, and with range in A. The elements of Δ may be called derivable in the obvious sense. For each integer k ≦.l, denote by Δk the domain of Dk (so that Dgr;1 = Δ); it is a simple consequence of Leibniz's formula that each Δk is an algebra. The classical example of this situation is A = C(O, 1) under the supremum norm with D ordinary differentiation, and here Δk = Ck(0, 1) is a Banach algebra under the norm ∥.∥k: Furthermore, the maximal ideals of Ak are precisely those subsets of Δk of the form M ∩ Δk where M is a maximal ideal of A, and = M, the bar denoting closure in A. In the present note we show how this extends to the general case.

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