scholarly journals The Trigonometric Polynomial Like Bernstein Polynomial

2014 ◽  
Vol 2014 ◽  
pp. 1-17 ◽  
Author(s):  
Xuli Han
Keyword(s):  
Trigonometric Polynomial ◽  
Bernstein Polynomial ◽  
Bernstein Polynomials ◽  
Trigonometric Polynomials ◽  
Polynomial Space ◽  
Polynomial Sequence ◽  
Symmetric Basis ◽  
Uniformly Convergent

A symmetric basis of trigonometric polynomial space is presented. Based on the basis, symmetric trigonometric polynomial approximants like Bernstein polynomials are constructed. Two kinds of nodes are given to show that the trigonometric polynomial sequence is uniformly convergent. The convergence of the derivative of the trigonometric polynomials is shown. Trigonometric quasi-interpolants of reproducing one degree of trigonometric polynomials are constructed. Some interesting properties of the trigonometric polynomials are given.

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 real zeros of a random trigonometric polynomial

1968 ◽  
Vol 64 (3) ◽  
pp. 721-730 ◽  
Author(s):  
Minaketan Das
Keyword(s):  
Trigonometric Polynomial ◽  
Mathematical Expectation ◽  
Random Variables ◽  
Trigonometric Polynomials ◽  
Real Zeros ◽  
Number Of Zeros ◽  
Number Of Real Zeros ◽  
Random Trigonometric Polynomial ◽  
Image Position ◽  
Mutually Independent

AbstractLet a1, a2,… be a sequence of mutually independent, normally distributed, random variables with mathematical expectation zero and variance unity; let b1, b2,… be a set of positive constants. In this work, we obtain the average number of zeros in the interval (0, 2π) of trigonometric polynomials of the formfor large n. The case when bk = kσ (σ > − 3/2;) is studied in detail. Here the required average is (2σ + 1/2σ + 3)½.2n + o(n) for σ ≥ − ½ and of order n3/2; + σ in the remaining cases.

scholarly journals A property of Bernstein-Schoenberg spline operators

1985 ◽  
Vol 28 (3) ◽  
pp. 333-340 ◽  
Author(s):  
T. N. T. Goodman ◽  
A. Sharma
Keyword(s):  
Bernstein Polynomial ◽  
Bernstein Polynomials ◽  
The Subject ◽  
The Many ◽  
Schoenberg Spline

Let Bnf; x) denote the Bernstein polynomial of degree n on [0,1] for a function f(x) defined on this interval. Among the many properties of Bernstein polynomials, we recall in particular that if f(x) is convex in [0,1] then (i) Bn(f;x) is convex in [0,1] and (ii) Bn(f;x)≧Bn+1(f;x), (n = l,2,…). Recently these properties have been the subject of study for Bernstein polynomials over triangles [1].

On the estimation of the Bernoulli regression function using Bernstein polynomials for group observations

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Petre Babilua
Keyword(s):  
Asymptotic Normality ◽  
Bernstein Polynomial ◽  
Bernstein Polynomials ◽  
Regression Function ◽  
Testing Hypothesis ◽  
Consistency And Asymptotic Normality

Abstract The estimate for the Bernoulli regression function is constructed using the Bernstein polynomial for group observations. The question of its consistency and asymptotic normality is studied. A testing hypothesis is constructed on the form of the Bernoulli regression function. The consistency of the constructed tests is investigated.

scholarly journals Change of Basis Transformation from the Bernstein Polynomials to the Chebyshev Polynomials of the Fourth Kind

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 120
Author(s):  
Abedallah Rababah ◽  
Esraa Hijazi
Keyword(s):  
Chebyshev Polynomial ◽  
Chebyshev Polynomials ◽  
Bernstein Polynomial ◽  
Bernstein Polynomials ◽  
Polynomial Basis ◽  
Bernstein Polynomial Basis ◽  
Fourth Kind

In this paper, the change of bases transformations between the Bernstein polynomial basis and the Chebyshev polynomial basis of the fourth kind are studied and the matrices of transformation among these bases are constructed. Some examples are given.

scholarly journals Some inequalities for trigonometric polynomials

1985 ◽  
Vol 39 (2) ◽  
pp. 216-226 ◽  
Author(s):  
Clément Frappier
Keyword(s):  
Trigonometric Polynomial ◽  
Trigonometric Polynomials ◽  
Classical Inequality

AbstractWe obtain various refinements and generalizations of a classical inequality of S. N. Bernstein on trigonometric polynomials. Some of the results take into account the size of one or more of the coefficients of the trigonometric polynomial in question. The results are obtained using interpolation formulas.

scholarly journals Approximate Solutions of Differential Equations by Using the Bernstein Polynomials

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Y. Ordokhani ◽  
S. Davaei far
Keyword(s):  
Differential Equations ◽  
Bernstein Polynomial ◽  
Numerical Solutions ◽  
Bernstein Polynomials ◽  
Approximate Solutions ◽  
High Accuracy ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Bernstein Polynomial Basis

A numerical method for solving differential equations by approximating the solution in the Bernstein polynomial basis is proposed. At first, we demonstrate the relation between the Bernstein and Legendre polynomials. By using this relation, we derive the operational matrices of integration and product of the Bernstein polynomials. Then, we employ them for solving differential equations. The method converts the differential equation to a system of linear algebraic equations. Finally some examples and their numerical solutions are given; comparing the results with the numerical results obtained from the other methods, we show the high accuracy and efficiency of the proposed method.

scholarly journals Lidstone–Euler interpolation and related high even order boundary value problem

CALCOLO ◽  
2021 ◽  
Vol 58 (2) ◽  
Author(s):  
Francesco Aldo Costabile ◽  
Maria Italia Gualtieri ◽  
Anna Napoli
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Bernstein Polynomials ◽  
Physical Models ◽  
Order Problem ◽  
Polynomial Sequence ◽  
Order Case ◽  
Uniqueness Of The Solution ◽  
Computational Aspects ◽  
Even Order

AbstractWe consider the Lidstone–Euler interpolation problem and the associated Lidstone–Euler boundary value problem, in both theoretical and computational aspects. After a theorem of existence and uniqueness of the solution to the Lidstone–Euler boundary value problem, we present a numerical method for solving it. This method uses the extrapolated Bernstein polynomials and produces an approximating convergent polynomial sequence. Particularly, we consider the fourth-order case, arising in various physical models. Finally, we present some numerical examples and we compare the proposed method with a modified decomposition method for a tenth-order problem. The numerical results confirm the theoretical and computational ones.

scholarly journals Factorization properties of subrings in trigonometric polynomial rings

2009 ◽  
Vol 86 (100) ◽  
pp. 123-131
Author(s):  
Tariq Shah ◽  
Ehsan Ullah
Keyword(s):  
Trigonometric Polynomial ◽  
Commutative Ring ◽  
Polynomial Rings ◽  
Trigonometric Polynomials ◽  
Ring Extension ◽  
Irreducible Elements ◽  
Euclidean Domain

We explore the subrings in trigonometric polynomial rings and their factorization properties. Consider the ring S' of complex trigonometric polynomials over the field Q(i) (see [11]). We construct the subrings S'1 , S'0 of S' such that S'1 ?S'0 ?S'. Then S'1 is a Euclidean domain, whereas S'0 is a Noetherian HFD. We also characterize the irreducible elements of S'1, S'0 and discuss among these structures the condition: Let A ?B be a unitary (commutative) ring extension. For each x ? B there exist x' ?U(B) and x'' ? A such that x = x'x''. .

scholarly journals Estimation of deviation of continuous $2\pi$-periodic functions from corresponding trigonometric polynomials of S.N. Bernstein's type

2021 ◽  
pp. 43
Author(s):  
N.Ya. Yatsenko
Keyword(s):  
Periodic Function ◽  
Trigonometric Polynomial ◽  
Modulus Of Continuity ◽  
Trigonometric Polynomials ◽  
Periodic Functions

We have established the estimation of deviation of continuous $2\pi$-periodic function $f(x)$ from the trigonometric polynomial of S.N. Bernstein's type that corresponds to it, by the modulus of continuity of the function $f(x)$.

Sign in / Sign up

Export Citation Format

Share Document