By N. K. Bary

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

**Example text**

Then t0 can be found, — njn < t0 < π/η, such that Ψη(ίο) < 0. 1) We suppose that 0 < t0 < njn (in the case when — n\n < t < 0, the reasoning must be carried out not for [0, 2n] but for [— 2 π , 0]; otherwise everything is exactly the same). 1)]. Meanwhile, we will show that it has not less than 2« + 1 roots. Indeed, we have for any k kn\ ( kn\ ( and since |Γ„(χ)| < M, then %pn{knjn) has the sign (— l) fc+1 or = 0. In particular, ψη(η/η) > 0 and therefore in the interval [tQ,njn] the function xpn{f) has a root.

This holds for all values of n (n = 1, 2 , . . ) . The orthogonality of the system {rn(x)} ob- 52 BASIC THEORY OF T R I G O N O M E T R I C SERIES tained in the interval [0,1] follows from the fact that if m # n (let m < ή), then the function rn(x) in every interval when rm(x) is constant takes the value + 1 just as many times as the value — 1 and the lengths of the intervals in which it is constant are all equal. Thus we are satisfied that 1 = 0 S rm{x)rn{x)dx o (m φ ή). Since for any n we have r2n(x) = 1 everywhere, apart from a finite number of points, then the system {rn(x)} is normalf.

T t Strictly speaking, these formulae were already known to Euler, but Fourier began to use them systematically; therefore they are traditionally called Fourier formulae and the corresponding series Fourier series. t t t For references to the text or formulae from the same chapter, the number of the chapter is omitted. 2). The numbers cn are called the complex Fourier coefficients of the function f (x). § 6. Problems in the theory of Fourier series; Fourier-Lebesgue series In §§ 4 and 5 we have solved only the problem of how the coefficients of a trigonometric series should be determined if we know that it converges uniformly to some function/(x).