MDPI - Publisher of Open Access Journals

12 pages, 284 KiB

Open AccessArticle

Genuine q-Stancu-Bernstein–Durrmeyer Operators

by Pembe Sabancıgil

Symmetry 2023, 15(2), 437; https://doi.org/10.3390/sym15020437 - 7 Feb 2023

Cited by 2 | Viewed by 1365

In the present paper, we introduce the genuine q-Stancu-Bernstein–Durrmeyer operators

Z_{n}^{q, α} (f; x)

. We calculate the moments of these operators,

Z_{n}^{q, α} (t^{j}; x)

for [...] Read more.

In the present paper, we introduce the genuine q-Stancu-Bernstein–Durrmeyer operators

Z_{n}^{q, α} (f; x)

. We calculate the moments of these operators,

Z_{n}^{q, α} (t^{j}; x)

for

j = 0, 1, 2,

which follows a symmetric pattern. We also calculate the second order central moment

Z_{n}^{q, α} ({(t - x)}^{2}; x) .

We give a Korovkin-type theorem; we estimate the rate of convergence for continuous functions. Furthermore, we prove a local approximation theorem in terms of second modulus of continuity; we obtain a local direct estimate for the genuine q-Stancu-Bernstein–Durrmeyer operators in terms of Lipschitz-type maximal function of order

β

and we prove a direct global approximation theorem by using the Ditzian-Totik modulus of second order. Full article

19 pages, 521 KiB

Open AccessArticle

On a New Construction of Generalized q-Bernstein Polynomials Based on Shape Parameter λ

by Qing-Bo Cai and Reşat Aslan

Symmetry 2021, 13(10), 1919; https://doi.org/10.3390/sym13101919 - 12 Oct 2021

Cited by 24 | Viewed by 2465

Abstract

This paper deals with several approximation properties for a new class of q-Bernstein polynomials based on new Bernstein basis functions with shape parameter

λ

on the symmetric interval

[- 1, 1]

. Firstly, we computed some moments and central [...] Read more.

This paper deals with several approximation properties for a new class of q-Bernstein polynomials based on new Bernstein basis functions with shape parameter

λ

on the symmetric interval

[- 1, 1]

. Firstly, we computed some moments and central moments. Then, we constructed a Korovkin-type convergence theorem, bounding the error in terms of the ordinary modulus of smoothness, providing estimates for Lipschitz-type functions. Finally, with the aid of Maple software, we present the comparison of the convergence of these newly constructed polynomials to the certain functions with some graphical illustrations and error estimation tables. Full article

(This article belongs to the Special Issue New Directions in Theory of Approximation and Related Problems)

► Show Figures

Figure 1

20 pages, 20323 KiB

Open AccessArticle

Degree Reduction of Q-Bézier Curves via Squirrel Search Algorithm

by Xiaomin Liu, Muhammad Abbas, Gang Hu and Samia BiBi

Mathematics 2021, 9(18), 2212; https://doi.org/10.3390/math9182212 - 9 Sep 2021

Cited by 4 | Viewed by 2616

Abstract

Q-Bézier curves find extensive applications in shape design owing to their excellent geometric properties and good shape adjustability. In this article, a new method for the multiple-degree reduction of Q-Bézier curves by incorporating the swarm intelligence-based squirrel search algorithm (SSA) is proposed. We formulate the degree reduction as an optimization problem, in which the objective function is defined as the distance between the original curve and the approximate curve. By using the squirrel search algorithm, we search within a reasonable range for the optimal set of control points of the approximate curve to minimize the objective function. As a result, the optimal approximating Q-Bézier curve of lower degree can be found. The feasibility of the method is verified by several examples, which show that the method is easy to implement, and good degree reduction effect can be achieved using it. Full article

(This article belongs to the Special Issue Modern Geometric Modeling: Theory and Applications II)

► Show Figures

Figure 1

18 pages, 897 KiB

Open AccessArticle

Certain Identities Associated with (p,q)-Binomial Coefficients and (p,q)-Stirling Polynomials of the Second Kind

by Talha Usman, Mohd Saif and Junesang Choi

Symmetry 2020, 12(9), 1436; https://doi.org/10.3390/sym12091436 - 31 Aug 2020

Cited by 17 | Viewed by 3256

Abstract

The q-Stirling numbers (polynomials) of the second kind have been investigated and applied in a variety of research subjects including, even, the q-analogue of Bernstein polynomials. The

(p, q)

-Stirling numbers (polynomials) of the second kind have been [...] Read more.

The q-Stirling numbers (polynomials) of the second kind have been investigated and applied in a variety of research subjects including, even, the q-analogue of Bernstein polynomials. The

(p, q)

-Stirling numbers (polynomials) of the second kind have been studied, particularly, in relation to combinatorics. In this paper, we aim to introduce new

(p, q)

-Stirling polynomials of the second kind which are shown to be fit for the

(p, q)

-analogue of Bernstein polynomials. We also present some interesting identities involving the

(p, q)

-binomial coefficients. We further discuss certain vanishing identities associated with the q-and

(p, q)

-Stirling polynomials of the second kind. Full article

(This article belongs to the Special Issue Recent Advances in Number Theory and Their Applications)

11 pages, 251 KiB

Open AccessArticle

On p-adic Integral Representation of q-Bernoulli Numbers Arising from Two Variable q-Bernstein Polynomials

by Dae San Kim, Taekyun Kim, Cheon Seoung Ryoo and Yonghong Yao

Symmetry 2018, 10(10), 451; https://doi.org/10.3390/sym10100451 - 1 Oct 2018

Cited by 2 | Viewed by 2566

Abstract

The q-Bernoulli numbers and polynomials can be given by Witt’s type formulas as p-adic invariant integrals on

Z_{p}

. We investigate some properties for them. In addition, we consider two variable q-Bernstein polynomials and operators and derive several properties [...] Read more.

The q-Bernoulli numbers and polynomials can be given by Witt’s type formulas as p-adic invariant integrals on

Z_{p}

. We investigate some properties for them. In addition, we consider two variable q-Bernstein polynomials and operators and derive several properties for these polynomials and operators. Next, we study the evaluation problem for the double integrals on

Z_{p}

of two variable q-Bernstein polynomials and show that they can be expressed in terms of the q-Bernoulli numbers and some special values of q-Bernoulli polynomials. This is generalized to the problem of evaluating any finite product of two variable q-Bernstein polynomials. Furthermore, some identities for q-Bernoulli numbers are found. Full article

(This article belongs to the Special Issue Current Trends in Symmetric Polynomials with their Applications)

9 pages, 232 KiB

Open AccessArticle

On p-Adic Fermionic Integrals of q-Bernstein Polynomials Associated with q-Euler Numbers and Polynomials †

by Lee-Chae Jang, Taekyun Kim, Dae San Kim and Dmitry Victorovich Dolgy

Symmetry 2018, 10(8), 311; https://doi.org/10.3390/sym10080311 - 1 Aug 2018

Cited by 3 | Viewed by 2477

Abstract

We study a q-analogue of Euler numbers and polynomials naturally arising from the p-adic fermionic integrals on

Z_{p}

and investigate some properties for these numbers and polynomials. Then we will consider p-adic fermionic integrals on

Z_{p}

of the [...] Read more.

We study a q-analogue of Euler numbers and polynomials naturally arising from the p-adic fermionic integrals on

Z_{p}

and investigate some properties for these numbers and polynomials. Then we will consider p-adic fermionic integrals on

Z_{p}

of the two variable q-Bernstein polynomials, recently introduced by Kim, and demonstrate that they can be written in terms of the q-analogues of Euler numbers. Further, from such p-adic integrals we will derive some identities for the q-analogues of Euler numbers. Full article

(This article belongs to the Special Issue Current Trends in Symmetric Polynomials with their Applications)

Search Results (6)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Saved Queries

Search Filter Reset All

Years

Feature Papers

Subjects

Journals

Article Types

Countries / Regions

Search Results (6)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI