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

ISBN-10: 1483199169

ISBN-13: 9781483199160

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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.