N. K. Bary's A Treatise on Trigonometric Series. Volume 1 PDF

By N. K. Bary

ISBN-10: 1483199169

ISBN-13: 9781483199160

Show description

Read or Download A Treatise on Trigonometric Series. Volume 1 PDF

Similar mathematics_1 books

Eugene L. Allgower, Klaus Böhmer, Mei Zhen (auth.), Prof.'s Bifurcation and Symmetry: Cross Influence between PDF

Symmetry is a estate which happens all through nature and it's for that reason traditional that symmetry might be thought of whilst trying to version nature. in lots of instances, those types also are nonlinear and it's the research of nonlinear symmetric versions that has been the foundation of a lot fresh paintings. even if systematic reviews of nonlinear difficulties should be traced again a minimum of to the pioneering contributions of Poincare, this is still a space with hard difficulties for mathematicians and scientists.

Get International Mathematical Olympiads, 1955-1977 PDF

The foreign Olympiad has been held every year considering the fact that 1959; the U. S. all started partaking in 1974, while the 16th foreign Olympiad used to be held in Erfurt, G. D. R. In 1974 and 1975, the nationwide technological know-how origin funded a 3 week summer season work out with Samuel L. Greizer of Rutgers collage and Murray Klamkin of the college of Alberta because the U.

Additional resources for A Treatise on Trigonometric Series. Volume 1

Sample text

Therefore and 3 Baryl J \f(x + A) - f(x)\ dx < 4Mh o ω(1)(<5,/) < sup AM\h\ = 4M<5 = 0{S). CHAPTER I BASIC CONCEPTS A N D T H E O R E M S I N THE T H E O R Y OF T R I G O N O M E T R I C SERIES § 1. )? ^ If such a series converges for all x in — oo < x < + oo, then it represents a function possessing a period of 2π. Therefore, if a function is to be represented by a trigonometric series, either periodic functions with period 2π are considered or a function is taken which is given in an interval of length 2π and is then expanded periodically, that is, it is required that/(jc + 2π) = f{x) for any x.

THEOREM 5. 3) summable then the series ]£ un(x) converges almost everywhere in E to the non-negative function f{x). n In fact, supposing that Sn(x) = £ uk(x), we see that S±(x) < S2(x) < ··· < Sn(x) A:=0 < ···. Supposing that/(x) = lim Sn(x) where f(x) is finite or infinite, we have from Theorem 4 "^00 lim j Sn(x) dx = j f(x) dx. n-> oo E But since E n lim J Sn(x)dx = lim J] J iik(^) rfx < + oo, «-»oo E it follows that jf(x)dx k=Q E < + oo and then/(x) is summable. § 15. <*'= o. 1) It is clear that any point where/(x) is continuous is a Lebesgue point; in fact, if there exists any ε > 0 and the function/^) is continuous at the point x, there can be found δ such that \f{t) - f(x) \ < ε for | h \ <ô and then for all such values of x x+h J J \f(t) - f(x)\ dt < ε, which provides the necessary proof.

1) holds in the sense that both terms of the equality become equal to + oo. Indeed, iff(x) is summable, then this assertion immediately follows from Theorem 2. Even if f(x) is not summable, then, supposing (f)N = f{x) at f{x) < N and (f)N = Natf(x) > N, we see that j (f)N dx -» oo as N -> oo. If a function (fn)N is defined E f The theorem was proved here on the assumption that/„(;t) converges in measure t o / ( x ) , but since any kind of sequence which converges almost everywhere also converges in measure (see § 13), then our statement is all the more accurate.

Download PDF sample

A Treatise on Trigonometric Series. Volume 1 by N. K. Bary


by Mark
4.2

Rated 4.39 of 5 – based on 49 votes