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

By N. K. Bary

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A23, p. 290). The following theorem holds: — THEOREM. ) be a sequence of continuous functions converging uniformly in [a,b]. 3) jf(x)dg. «->oo a a In fact, for any ε we can find N such that \f(x) — f„(x)\ < ε for n > N and a < x < b. 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

Bernstein's inequality cannot be strengthened, since supposing that we see that Tn(x)M = max \Tn(x)\ = M, and cosnx, max |Γ;(χ)[ = nM. § 24. Trigonometric polynomial of best approximation L e t / ( x ) be a continuous function with period 2jr. We will denote by Tn{x) any trigonometric polynomial of order not higher than n. Let A(Tn) = max\f(x)-Tn(x)\. 1) t We note that if ψη(ί) has the same root in two neighbouring intervals, then this root is multiple. MODULI OF CONTINUITY AND SMOOTHNESS 37 Let us consider the lower bound of the values Δ (Tn) where Tn refers to all the trigonometric polynomials Tn(x) in turn.

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. 28 THEORY OF SETS A N D THEORY OF FUNCTIONS However, in a summable function there need not be a single point of continuity. The following theorem is no less true. LEBESGUE'S THEOREM.