Generating Functions for the $q$-Bernstein Bases

Goldman, Ron; Simeonov, Plamen; Şimşek, Yılmaz

doi:10.1137/130921623

Cited by 29 publications

(35 citation statements)

References 8 publications

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“…The orthogonal complements of P m in P n with respect to the weighted inner product (11) and the weighted Euclidean inner product (12) are equal.…”

Section: Theoremmentioning

confidence: 99%

“…min Q∈Pm ||P − Q|| (15) has the same minimizer for the norm induced either by the inner product (11) or the inner product (12).…”

Section: Corollary 2 Given a Polynomial P Of Degree N The Approximamentioning

confidence: 99%

“…In this section we derive connections between orthogonal polynomials with respect to the two inner products (6) and (11). These relations are derived using two facts.…”

Section: Applications To Little Q-legendre and Q-hahn Polynomialsmentioning

confidence: 99%

See 2 more Smart Citations

Best polynomial degree reduction on q-lattices with applications to q-orthogonal polynomials

Ait‐Haddou

Goldman

2015

Applied Mathematics and Computation

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“…The orthogonal complements of P m in P n with respect to the weighted inner product (11) and the weighted Euclidean inner product (12) are equal.…”

Section: Theoremmentioning

confidence: 99%

“…min Q∈Pm ||P − Q|| (15) has the same minimizer for the norm induced either by the inner product (11) or the inner product (12).…”

Section: Corollary 2 Given a Polynomial P Of Degree N The Approximamentioning

confidence: 99%

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Best polynomial degree reduction on q-lattices with applications to q-orthogonal polynomials

Ait‐Haddou

Goldman

2015

Applied Mathematics and Computation

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“…By using the q ‐exponential function E q ( x ), the generating functions for the q ‐Bernstein basis functions is given by the following:

E_{q} false(x t false) G_{k} false(x, t; q false) = \frac{{false(x t false)}^{k}}{{false[k false]}_{q}!} E_{q} false(t false) = true \sum_{n = k}^{\infty} B_{k}^{n} false(x; q false) \frac{t^{n}}{{false[n false]}_{q}!},

where the function E q ( x ) is the q ‐exponential function, defined as follows:

E_{q} false(x false) = true \sum_{n = 0}^{\infty} \frac{x^{n}}{{false[n false]}_{q}!} .

…”

Section: Introductionmentioning

confidence: 99%

Generating functions for unification of the multidimensional Bernstein polynomials and their applications

Şimşek

2018

Math Methods in App Sciences

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The aim of this paper is to construct generating functions for m‐dimensional unification of the Bernstein basis functions. We give some properties of these functions. We also give derivative formulas and a recurrence relation of the m‐dimensional unification of the Bernstein basis functions with help of their generating functions. By combining the m‐dimensional unification of the Bernstein basis functions with m variable functions on simplex and cube, we give m‐dimensional unification of the Bernstein operator. Furthermore, by applying integrals method including the Riemann integral, the q‐integral, and the p‐adic integral to some identities for the (q‐) Bernstein basis functions, we derive some combinatorial sums including the Bernoulli numbers and Euler numbers and also the Stirling numbers and the Cauchy numbers (the Bernoulli numbers of the second kind).

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“…According to properties of -functions manyfunctions have been used in order to approximate a suitable function. For example, at [1], some identities and formulae for the -Bernstein basis function, including the partition of unity property, and formulae for representing the monomials were studied. In addition, a kind of approximation of a function in terms of Bernoulli polynomials is used in several approaches for solving differential equations, such as [2][3][4].…”

Section: Introductionmentioning

confidence: 99%

Approximation Properties of q-Bernoulli Polynomials

Momenzadeh

Kakangi

2017

Abstract and Applied Analysis

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We study the -analogue of Euler-Maclaurin formula and by introducing a new -operator we drive to this form. Moreover, approximation properties of -Bernoulli polynomials are discussed. We estimate the suitable functions as a combination of truncated series of -Bernoulli polynomials and the error is calculated. This paper can be helpful in two different branches: first we solve the differential equations by estimating functions and second we may apply these techniques for operator theory.

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Generating Functions for the $q$-Bernstein Bases

Cited by 29 publications

References 8 publications

Best polynomial degree reduction on q-lattices with applications to q-orthogonal polynomials

Best polynomial degree reduction on q-lattices with applications to q-orthogonal polynomials

Generating functions for unification of the multidimensional Bernstein polynomials and their applications

Approximation Properties of q-Bernoulli Polynomials

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