## Abstract

We begin the present section with some simple definitions (probably already known to most readers). For is the complex conjugate of exists for all then (f

*m*,*n*integers the so-called*Kronecker delta**δ*_{ mn }is defined by*δ*_{ mn }= 1 if*m*=*n*and*δ*_{ mn }= 0 if*m*≠*n*. For our second definition, let (*f*_{ n }:*n*= 0, ±1, ±2,...) be a set of real or complex functions, defined on the subset*Δ*of ℝ^{ k }. The set (*f*_{ n }) is called an*orthonormal set*(or*orthonormal system*) on*Δ*if$$f_n^ - $$

(1)

*f*_{ n }and*dx*stands for*dx*_{1}…*dx*_{ k }Of course, the definition makes sense only if the integral of*f*_{ m }$${f_m}f_n^ - $$

(3)

*m*,*n*. The definition is analogous if*m*and*n*are restricted to 0,1,2,… or to 1,2,… . If it is only given that$$\int {_\Delta } fmf_n^ - dx = 0form \ne n,$$

(4)

_{n}) is said to be an*orthogonal system*on*Δ*. We immediately mention an example. For*n*= 0, ±1, ±2,…, let*e*_{ n }(*x*) = (2π)^{-1/2einx}on ℝ. The system (*e*_{ n }:*n*= 0, ±1, ±2,…) is orthonormal on any interval [*a*,*a*+ 2π], i.e., on any interval of length 2π in ℝ. The proof is immediate by observing that$$2\pi \left\{ {{e_m}\left( x \right)\overline {{e_n}\left( x \right)} } \right\} = {e^{i\left( {m - n} \right)x + i\sin (m - n)x}}.$$

(5)

## Keywords

Fourier Series Fourier Coefficient Trigonometric Polynomial Trigonometric Series Finite Variation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer-Verlag Berlin Heidelberg 1989