32. The complex form of Fourier series.

Let function f(x) be integrated in [-π,π]. Let’s write Fourier series for it:

(1), where(2)

Let use Euler’s formula. Then, .

Because that , . Let’s change by

these expressions and in (1). Then we have got

(3) Let’s denote ,,(4). Then m-partial sum of the series (1) can be written as(5). And, finally, we canwrite(6). Series of this kind is called the complex form of Fourier series for function f(x). The converges of series (6) must be understood as the existence of limit whenof symmetric sums (5).The coefficientsnoted by (4) is called the complex Fourier coefficients of function f(x). It may be proved that( ) (7) Really, taking into account Euler’s formulas (4) for positive indexes we have got

And for negative indexes

q.e.d

Let’s note that for real function f(x) Fourier coefficients are mutually conjugated.

Complex form of Fourier series can be written for any interval (-l,l)

The number is called wave numbers of function f(x), and expression

is called harmonic. The set of wave numbers is called spectrum. The convergence

conditions for Fourier series keeps also for complex form too.

33. The extremal property of the partial sums of Fourier series.

Let f(x) be function given in [a,b]. We shall say that f(x) is integrated with square if function itself and its square are integrated in [a,b]. Any limited function is function integrated with square. Let f(x) and g(x) two functions given and integrated in [a,b]. The mean square deviation of this function in [a,b] means value ρ(f,g) expressed as

(1) This value satisfies to many properties of the distance between the points. Let’s investigate the next problem: integrated with square function f(x) is given in [a,b] and orthogonal system is also given. It’s necessary to find the linear combinationthat has the minimal mean square deviation of function f(x) in [a,b]. Letand.

So (2). Obviously,, where c_k - Fourier coefficients of f(x) by the orthogonal system {φ_n(x)}. Because that (3). So, because this system is orthogonal we have.

And, so (4). Next

. It’s clear that value will be minimal if, it means. The main result:the mean square deviation will be minimal if the coefficients of polynom p_n(x) is Fourier coefficients, and so polynom p_n(x) is partial sum of Fourier series by system{φ_n(x)}.

(5)

This expression means with growth of n the partial Fourier sums supplies more exact approaching representation – extremal property of partial Fourier sum.

34. Bessel inequality. Parswval-Steklov equality

From (5):

Bessel inequality follows: Fourier coefficients for any function f(x) integrated with square satisfies to estimation:

(6). Really, by equality (5) follows , where n – any natural number. The partial sums of series (6) aren’t limited and monotonically grow. So, we can go to limit when n approaches toand we’ll have got (6). The orthogonal systemis called closed if for any integrated with square function the equality(7).This expression is called Parseval-Steklov equality. If we orthogonal system the last expressions may be rewrote as

(Bessel inequality)

(Parseval-Steklov equality)