Hüseyin Aktuğlu

1 Introduction The Weierstrass Approximation Theorem [1] is a fundamental result in mathematical analysis, stating that any continuous function on a closed interval can be uniformly approximated by a polynomial function. This theorem has far-reaching implications for various areas of mathematics, including approximation theory, numerical analysis, and functional analysis.

1 Introduction One of the cornerstone theorems in approximation theory is the Weierstrass approximation [ 1 ] theorem, which proves that any continuous function defined on a closed interval can be uniformly approximated by polynomials. In 1912, Bernstein [ 2 ] introduced the Bernstein operators to provide a constructive proof of the Weierstrass theorem. Bernstein operators have become a classical tool in approximation theory.

1 INTRODUCTION The Weierstrass approximation theorem [ 1 ] is a fundamental result in mathematical analysis, stating that for a real-valued function f $$ f $$ that is continuous on a closed interval [ a , b ] $$ \left[a,b\right] $$ and for any given ϵ > 0 $$ \epsilon &gt;0 $$ , there exists an algebraic polynomial p $$ p $$ such that f ( x ) − p ( x ) < ϵ $$ \left&amp;#x0007C;f(x)-p(x)\right&amp;#x0007C;&lt;\epsilon $$ on [ a , b ] $$ \left[a,b\right] $$ .