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A Modern Introduction to Fuzzy Mathematics
book

A Modern Introduction to Fuzzy Mathematics

by Apostolos Syropoulos, Theophanes Grammenos
July 2020
Intermediate to advanced content levelIntermediate to advanced
384 pages
11h 25m
English
Wiley
Content preview from A Modern Introduction to Fuzzy Mathematics

AFuzzy Approximation

Classical approximation theory of real‐valued continuous functions by algebraic or trigonometric polynomials has been a subject of research for more than two centuries (see, e.g. [72, 164, 194, 286] for a rigorous as well as instructive presentation). Here, we are going to give a brief account of some basic results of approximation theory put in a fuzzy setting.

A.1 Weierstrass and Stone–Weierstrass Approximation Theorems

Let us recall two of the most fundamental questions in classical approximation theory:

  1. Can every continuous function, images, images, be arbitrarily well approximated by algebraic polynomials?
  2. Can every continuous periodic function with period images be arbitrarily well approximated by trigonometric polynomials?

Both questions, that are actually intrinsically interrelated, were answered affirmatively by Karl Theodor Wilhelm Weierstrass in 1885 through perhaps the most significant result in approximation theory known as Weierstrass approximation theorem which, in its basic form, reads:

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Publisher Resources

ISBN: 9781119445289Purchase book