Advanced Calculus fi..

Chapter 7 Fourier Series and Orthogonal Functions 497If f (x) is piecewise continuous in the interval a 5 x 5 b and {$,(x)} is an orthogonalsystem in this interval, then the serieswhereis called the Fourier series off with respect to the system {&(x)). The numbersel, c2, . . . are called the Fourier coefficients of f (x) with respect to the system{4n(x>l.The preceding formulas can be simplified if one assumes that the constant Bn isalways 1, that is, that. t iThis can always be achieved by dividing the original 4,(x) by appropriate constants.When the condition B, = 1 is satisfied for all n, the system of functions @n(x)is called normalized. A system that is both normalized and orthogonal is calledorthonormal. This is illustrated by the functions:1 cosx sinx-cosnx sinnx, Jn, , - -7 fi 9 fi 9 ""Although the general theory is simpler for'normalized systems, the advantages f&applications are slight, and we shall not use normalization in what follows.The operations with orthogonal systems are strikingly similar to those withvectors. In fact, we can consider the piecewise continuous functions for a 5 x 5 bas a sort of vector space, as in Section 1.18. The sum or difference of two suchfunctions f (x), g(x) is again piecewise continuous, as is the product cf of f (x) bya constant or scalar c. Equation (7.38) suggests a definition of inner product (orscalar product): !:One can then define a norm (or absolute value):The zero function 0* is a function that is 0 except at a finite number of points; ingeneral, in this vector theory of functions we consider two functions that differ onlyat a finite number of points to be the same function. It is now a straightforwardexercise to verify that all the axioms for a Euclidean vector space, except the one