We prove the Weierstrass-type approximation theorem that states every C1 curve in the 2-dimensional or 3-dimensional Euclidean space or in the 3-dimensional Minkowski space can be uniformly approximated by Pythagorean hodograph curves in the corresponding space. This abundance of PH curves is another theoretical confirmation of the usefulness and the versatility of the PH curves. We also addres...

We present the theorem of uniform approximation of continuous functions by polynomials in the Weierstrass framework for the construction of analysis based on the representation of functions by sums of power series and analytic functions, and his effort to bring new standards of rigor to analysis that accompanied his differences with Riemann. We also describe the most original proofs of the appr...

One of the greatest pleasures in mathematics is the surprising connections that often appear between apparently disconnected ideas and theories. Some particularly striking instances exist in the interaction between probability theory and analysis. One of the simplest is the elegant proof of the Weierstrass approximation theorem by S. Bernstein [2]: on the surface, this states that if f : [0, 1]...

Abstract. In this paper we study the set of functions G-valued which can be approximated by G-valued continuous functions in the norm L∞G (I, w), where I is a compact interval, G is a real and separable Hilbert space and w is certain G-valued weakly measurable weight. Thus, we obtain a new extension of celebrated Weierstrass approximation theorem. Also, we characterize the set of functions whic...

In standard books on calculus the existence of primitive functions continuous is proved, in one way or another, using Riemann sums. this note we present a completely different self-contained, however probably folkloristic, proof existence. Our combines, hand, so-called Stone Weierstrass theorem uniform approximation unit interval by polynomials, and, other classical result from limits different...

This paper rigorously establishes thut standard rnultiluyer feedforward networks with as f&v us one hidden layer using arbitrary squashing functions ure capable of upproximating uny Bore1 measurable function from one finite dimensional space to another to any desired degree of uccuracy, provided sujficirntly muny hidden units are available. In this sense, multilayer feedforward networks are u c...