7.2 Orthonormal Systems. General Fourier Series

The system of complex-valued functions { ? _n( x)} defined on the interval I = [ a, b] is said to be orthonormal iff

for every m, n ? N.

For example, the system is orthonormal on [ ? ?, ?].

Let { ? _n( x)} be an orthonormal system and f( x) be an absolutely integrable function on [ a, b]. Then the numbers

are well-defined and are called the Fourier coeficients of f with respect to { ? _n}. We write as before

where the series is called the Fourier series of f with respect to { ? _n}.

The partial sums of the Fourier series have a minimal property in L ²-sense.

Theorem 7.8

Let f ? L ²[ a,b] and { ? _n} be an orthonormal system in L ²[ a, b] . Then the function

attains its minimum at the point ( c ₁ , , c _n) , where c _k is the k-th Fourier coefficient of f.

Proof

By the orthonormality of the system { ? _n} we have

(7.34)

By (7.34) it follows that ?( ? ₁ , , ? _n) is minimal at ( c ₁ , , c _n) and

(7.35)