The really new thing about Stone’s approach to the approximation theorem was the approach via lattices of continuous functions, although Lebesgue had noticed the importance of approximating the absolute-value function earlier. There is a segment of the mathematical community formed of people who are as likely to encounter a lattice in their work as an algebra and for whom the lattice version of...

The basic purpose of this article is to prove the important Weier-strass' theorem which states that a real valued continuous function f on a topological space T assumes a maximum and a minimum value on the compact subset S of T , i.e., there exist points x1, x2 of T being elements of S, such that f(x1) and f(x2) are the supremum and the innmum, respectively, of f(S), which is the image of S und...

One hundred years after the introduction of the Bernstein polynomial basis, we survey the historical development and current state of theory, algorithms, and applications associated with this remarkable method of representing polynomials over finite domains. Originally introduced by Sergei Natanovich Bernstein to facilitate a constructive proof of the Weierstrass approximation theorem, the leis...

1 2 Contents 0. One-variable calculus 1. The derivative 2. Inverse function and implicit function theorem 3. Fundamental local existence theorem for ODE 4. The Riemann integral in n variables 5. Integration on surfaces 6. Differential forms 7. Products and exterior derivatives of forms 8. The general Stokes formula 9. The classical Gauss, Green, and Stokes formulas 10. Holomorphic functions and...

We discuss and examine Weierstrass’ main contributions to approximation theory. §1. Weierstrass This is a story about Karl Wilhelm Theodor Weierstrass (Weierstraß), what he contributed to approximation theory (and why), and some of the consequences thereof. We start this story by relating a little about the man and his life. Karl Wilhelm Theodor Weierstrass was born on October 31, 1815 at Osten...

Abstract As a special case of general fuzzy numbers, the polygonal number can describe object by means an ordered representation finite real numbers. Different from numbers overcome shortcoming complex operations based on Zadeh’s traditional expansion principle, and maintain closeness arithmetic operation. Hence, it is feasible to use approximate number. First, extension theorem continuous func...