21. Coefficient determination by Euler-Fourier method.

Let’s function f(x) is integrated in . Then we suppose that expanding (5):takes place and integrate it term by term from –π to π; we obtainBut, as easily to see(6). Because that all terms under sum sign are equal to zero. And, finally, we have got(7). For calculation of coefficient a_m let’s multiply both parts of equality (5) by cosmx and again integrate it term by term in the same interval:

. The first term at right disappears on account of (6). Next we have

(8)

(9) if only n≠m. And finally

.(10)

So, all integrals under the sum signs turn into zero excluding one just having coefficient a_m . Then this coefficient may be determined as .(11)Analogically, multiplying expanding (5) by sinmx and integrating it term by term, we can determine coefficient b_m: (12) . Here we use besides (6) and (8) easily checked ratios:(13) if n≠m and.(14)

Formulas (7),(11) and (12) are known by name Euler-Fourier formulas; coefficients calculated by this formulas are called Fourier coefficients of given function, and composed with its’ help trigonometric series (5) is called Fourier series of function.

We use term by term integrating of series but this operation is applicable only if series is uniformly convergent. Because that we can only affirm that if function f(x) with period T expands in uniformly convergent series (5) then this series will necessarily be its Fourier series. If uniformly convergence is not established then we can’t affirm anything about series but only that series is “born” by function. This bond between series and function usually is denoted as (15) (without equality sign).

22.Orthogonal systems of functions.

We shall call two functions defined in [a,b] orthogonal in this interval if their product have integral equal to zero:. Let’s consider system of functions {φ_n(x)} defined in interval [a,b] and integrated there together with their squares. If functions of this system are orthogonal by pairs(15) then system is called orthogonal system of functions. Always we shall suppose that(16), so system doesn’t contain functions equals to zero. If conditionis hold then system is called normal. If this condition doesn’t take place then we always can pass to systemwhich obviously will be normal. Let’s Consider some examples.

1. The most important example of orthogonal system is just trigonometric series:

(17) in interval [-π,π]. Its orthogonality follows from ratios (6),(8),(9) and (13). However, it isn’t normal, this follows from (10) and (14). Multiplying trigonometric functions by proper factors it’s easy to get the normal system

(17*)

Let some orthogonal system {φ_n(x)}is defined in [a,b]. Our purpose is to expand defined in [a,b] function f(x) in series by functions φ of the kind:(20).

For determining of coefficients of this expanding we multiply both parts (22) by φ_m(x), then we integrate term by term: . Since the system is orthogonal all integrals at right will be equal to zero excluding one – see (15) and (16)/ And we obtain(21).Series (20) with coefficients (21) is called generalized Fourier series for function f(x) in relation to system {φ_n(x)}. If system is normal then coefficients have the easiest form: (21*).As for previous case the bond between obtained series and function f(x) is only formal -(20*). The convergence of this series is need to be investigated.

23. Expanding of functions in Fourier series Dirichlet’s integral.

Let function periodic f(x) (T=2π) be absolutely integrable function in interval [-π,π] and therefore in any finite interval. Let’s calculate its Fourier coefficients:

(1) and compose Fourier series (2) For investigation of series’ (2) behavior in some definite point x=x₀ let’s write convenient expression for its partial sum . Next let’s substitute integral expressions (1) for a_m and b_m and put constants cosmx₀, sinmx₀ under the integral sign .

Let’s use ratio ( it’s easy to prove if left part will be multiplied byand every productwill be changed to). Then we have gotand, finally,(3). This important integral is called Dirichlet’s integral. Since function f(x) is periodic with period T=2π then interval of integration may be changed by intervalfor example:

. Let’s use change and transform integral to the kind of , then, dividing integral into two integralsand transforming second integral by sign changing also to interval [0,π] we have got a final form for n-th partial sum of Fourier series(4).

Thus our task is to investigate the behavior just this integral contained parameter n.