Approximation by Certain Linear Convergence Techniques

Abstract

The present thesis titled as ``Approximation by Certain Linear Convergence Techniques'' involves the construction of new positive linear operators and examines their approximation properties as well as their applications in multidisciplinary fields. The thesis is divided into eight chapters that majorly covers three important aspects of approximation theory.

Firstly, we give the brief introduction of approximation theory and the basic results that are the inspiration for the development of this theory. We also provide basic definitions as well as approximation tools such as moduli of continuity, modulus of smoothness, Peetre's $K$-functional for the univariate operators, complete and partial modulus of continuities for function of two variables to check the convergence of positive linear operators for the given function. Also, we list the brief account of the related work of various authors for the positive linear operators and found some gaps according which the research work in this thesis has been carried out.

The first direction of the thesis is to focus on the order of approximation of existing positive linear operators. We know the convergence and the order of approximation of the positive linear operators are two important attributes in approximation theory to approximate any function. Using the well-known Korovkin theorem, we can easily verify the convergence of these operators but improving the order of approximation is a critical attribute. We describe a recursive method designed to improve the approximation results of known operators, which results in increased accuracy and efficiency. We presented three modifications of $\alpha$-Bernstein P\u{a}lt\u{a}nea operators with linear, quadratic and cubic order of approximation and study some approximation results concerning the rate of convergence, error estimation and Voronovskaja type formulas for the new modifications.

The next aspect for the thesis is to introduce new positive linear operators that outperform classical operators in terms of approximation properties. These new operators are designed to approximate functions effectively on both finite and infinite domains by using certain parameters to introduce the flexibility in these positive linear operators. It helps to increase the applicability and utility of positive linear operators in various mathematical and engineering contexts.

We define the bivariate operators as well as Generalized Boolean Sum (GBS) operators associated with these operators for the first order modification of $\alpha$-Bernstein P\u{a}lt\u{a}nea operators. In order to approximate Lebesgue integrable functions, we introduce the Durrmeyer-variant of Lupa\c{s} type operators by using Pochhammer $k$-symbol in one as well as two dimensional space. Also, we give the univariate and bivariate versions of the Bernstein-Lototsky operators that are able to preserve any polynomial with some certain conditions by introducing a real parameter $\rho>0$. We define the new operators to approximate integrable functions by using $\alpha$-Baskakov operators and a non-negative parameter defined on infinite domain. We study the approximation results of these operators by using well-known tools of the approximation theory including convergence and error estimates in terms of moduli of continuity. The results of all these positive linear operators have been verified by graphical illustrations for certain examples. Also, the introduced bivariate versions of the operators can be extended to approximate the functions of several variables. This feature is especially beneficial in real-world situations when functions rely on more than one variable.

In the last part of the thesis, we introduce the applications of positive linear operators in the realm of B\'{e}zier curves. With the widespread use of computers in all industries, these curves have become critical to study. These curves utilize positive linear operators such as Bernstein and their generalizations. We can include some parameters to obtain greater control over these curves. These factors assist in reducing time and expense while improving the curves' accuracy and adaptability.

We generalize the B\'ezier curves by using two parameters to get the better control on shape of the curves. Firstly, we construct the generalized B\'{e}zier curves and B\'{e}zier surfaces depending upon the parameter $\alpha.$ Secondly, we explore the applications of $q$-calculus in polynomial basis functions and curve modeling. We define the $q$-variant of Bernstein-Chlodowsky basis polynomials and introduce generalization of B\'{e}zier curves by utilizing these basis polynomials. We study the properties of these curves and surfaces and show that the introduced parameter provides us the flexibility to modify the curves as well as surfaces by giving some numerical examples with the help of MATLAB. Also, we provide an exact approach to calculate the control points for the given $\alpha$-B\'{e}zier curve.

In summary, the thesis makes substantial contributions to the field of approximation theory by improving existing approximation techniques, introducing new operators with superior properties, and extending the applications of these operators to the construction and modification of B\'{e}zier curves and surfaces. These advancements hold promise for enhancing various practical applications in computational mathematics, computer graphics, and related areas.