q-Peano Kernel and its Applications

Journal of Approximation Theory, 1976

Applied Numerical Mathematics, 2021

We establish the uniform convergence of the control polygons generated by repeated degree elevation of q-Bézier curves (i.e., polynomial curves represented in the q-Bernstein bases of increasing degrees) on [0, 1], q > 1, to a piecewise linear curve with vertices on the original curve. A similar result is proved for q < 1, but surprisingly the limit vertices are not on the original curve, but on the q −1-Bézier curve with control polygon taken in the reverse order. We introduce a q-deformation (quantum Lorentz degree) of the classical notion of Lorentz degree for polynomials and we study its properties. As an application of our convergence results, we introduce a notion of q-positivity which guarantees that the q-Lorentz degree is finite. We also obtain upper bounds for the quantum Lorentz degrees. Finally, as a by-product we provide a generalization to polynomials positive on q-lattices of the univariate Pólya theorem concerning polynomials positive on the non-negative axis.

Mathematical Foundations of Computing, 2021

In the present paper, we shall investigate the pointwise approximation properties of the q−analogue of the Bernstein-Schurer operators and estimate the rate of pointwise convergence of these operators to the functions f whose q−derivatives are bounded variation on the interval [0, 1 + p]. We give an estimate for the rate of convergence of the operator (Bn,p,qf) at those points x at which the one sided q−derivatives D + q f (x) and D − q f (x) exist. We shall also prove that the operators (Bn,p,qf) (x) converge to the limit f (x). As a continuation of the very recent and initial study of the author deals with the pointwise approximation of the q−Bernstein Durrmeyer operators [12] at those points x at which the one sided q−derivatives D + q f (x) and D − q f (x) exist, this study provides (or presents) a forward work on the approximation of q-analogue of the Schurer type operators in the space of DqBV .

Journal of Mathematics, 2022

In recent years, the usage of the q -derivative and symmetric q -derivative operators is significant. In this study, firstly, many known concepts of the q -derivative operator are highlighted and given. We then use the symmetric q -derivative operator and certain q -Chebyshev polynomials to define a new subclass of analytic and bi-univalent functions. For this newly defined functions’ classes, a number of coefficient bounds, along with the Fekete–Szegö inequalities, are also given. To validate our results, we give some known consequences in form of remarks.

2000

Journal of Mathematical Analysis and Applications, 2008

The q− difference analog of the classical ladder operators is derived for those orthogonal polynomials arising from a class of indeterminate moments problem.

Mathematical and Computational Applications, 2020

In this paper, we introduce the complementary q-Lidstone interpolating polynomial of degree 2 n , which involves interpolating data at the odd-order q-derivatives. For this polynomial, we will provide a q-Peano representation of the error function. Next, we use these results to prove the existence of solutions of the complementary q-Lidstone boundary value problems. Some examples are included.

Bulletin of the "Transilvania" University of Braşov, 2022

In the present paper we shall investigate the pointwise approximation properties of the q analogue of the Bernstein operators and estimate the rate of pointwise convergence of these operators to the functions f whose qderivatives are bounded variation on the interval [0, 1]. We give an estimate for the rate of convergence of the operator (B n,q f) at those points x at which the one sided q-derivatives D + q f (x), D − q f (x) exist. We shall also prove that the operators B n,q f converge to the limit f. As a continuation of the very recent study of the author on the q-Bernstein Durrmeyer operators [10], the present study will be the first study on the approximation of q analogous of the discrete type operators in the space of D q BV .

This paper includes some new investigations and results for post quantum calculus, denoted by (p, q)-calculus. A chain rule for (p, q)-derivative is given. Also, a new (p, q)-analogue of the exponential function is introduced and its properties including the addition property for (p, q)-exponential functions are investigated. Several useful results involving (p, q)binomial coefficients and (p, q)-antiderivative are discovered. At the final part of this paper, (p, q)-analogue of some elementary functions including trigonometric functions and hyperbolic functions are considered and some properties and relations among them are analyzed extensively.

Symmetry, Integrability and Geometry: Methods and Applications, 2011