By N. K. Bary

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**Extra resources for A Treatise on Trigonometric Series. Volume 1**

**Sample text**

1) is true if instead of the functions fn(x) being all bounded it is supposed that \fn(x)\ < Φ(χ) almost everywhere in E, where Φ(χ) is a positive function summable in E. (See Natanson, ref. A 23, p. 166f). 3. THEOREM 3 (Fatou's lemma). If a sequence of measurable and non-negative functions fx(x), f2(x), . . , / „ ( x ) , . . 2) j F(x) dx < infi jfn(x) dx\ E \ \E (see Natanson, réf. A23, p. 155). THEOREM 4. Iffi(x),f2(x), . . , / „ ( x ) , . . 1) holds in the sense that both terms of the equality become equal to + oo.

Therefore I b \j[f(x)-fn(x)]dg Ia and the theorem is proved. < ε var g for n > N9 [a,b] § 17. Helly's two theorems HELLY'S FIRST THEOREM. Let {f(x)} be some family of functions of bounded variation in [a, b]. If these functions themselves and their complete variations are all bounded^ in [a,b], that is \f(x)\ < M and (a < x < b), b V a{f) < M then from the family {f(x)} it is possible to extract the sequence fn(x), converging at every point of [a, b] to some function

Thus the trigonometric polynomial ψη(ί) possesses 2n + 1 roots and we have already seen that this leads to a contradiction. The lemma is thus proved. Let us return to proving the theorem. Let |Γ„(Λ:)| < M. We will denote max |Γ„(χ)| = μ. Let x0 be a point such that T'n(x0) = μ (if — T'n{x0) = μ, then it is sufficient to discuss — Tn(x)). Then according to the preceding lemma π π Tn(x0 + t) > μοοζηί, < t < —, whence π π ~2n ~2n I T'n(x0 + t)dt = Tn ix0 + -^Λ - Tn (x0 - -^Λ >μ cosntdt = 2 ~ . 2« Thus n Tn (xo + £ ) - Γ.