Calculus: Analytic Geometry and Calculus, with Vectors

612

E and is such that the two integrals

(12.34) fE f(x) dx, fE if(x)I' dx

Series

both exist as Riemann integrals or as Cauchy extensions of Riemann

integrals. Supposing that n is a positive integer, we seek constants

c1, c2,. .., cfor which the integral in the left member of the formula

= x dx -

n

(12.35) fE f() - ckOk(x)'2 dx fEIf(>IZ IakIZ

k=1 k=1

n)1

kI I c., - a,, l2

will attain the least possible value. The first big step in the theory is

made by working out the formula (12.35) in which the constants a1, a2,

are the Fourier coefficients of f defined by the formulast

(12.36) ak =fE f(x)¢k(x) dx (k = 1,2, ).

The series

(12.361) aioi(x) + a202(x) + a303(x), ...

in which the coefficients are defined by (12.36), is called the Fourier series

of f. It is very easy to see that the c's for which the right side (and

hence also the left side) of (12.35) is a minimum are those for which

Ck = ak. Putting Ck = ak in (12.35) gives the key formula

(12.362) 1EI f(x) - 1 ak0k(x) i2 dx = fE If(x)I2 dx - IakI2.

k=1 k=1

Since the left side of (12.35) cannot be negative, we obtain the first and

then the second of the inequalities

n `m

(12.363) IakI2 = fE If(x)I2 dx, IakI2 fE if(x)I2 dx.

k-1 k=1

The second inequality is called the Bessel (1784-1846) inequality. For

most purposes, the important orthonormal sets 01, 02, are those

for which the members of (12.362) converge to 0 asn - w so that

(12.364) lim fE f(x) - ' akcbk(x) I2 dx= 0,

k=1

IakI2 = fE If(x)I2dx.

k-1

t These formulas were known by Euler. Fourier contributed very little to the theory

of Fourier coefficients and Fourier series. The things bear his name, not because he

invented them, but because he advertised their formal usefulness in problems of mathe-

matical physics.