N- , , .
. L2[a, b] :
||s(t)|| = .
, [a, b] , ( ) . ( , ) , , [a, b]. Ñ:
||s(t)||Ñ = , ||sn||Ñ = .
( ) :
dÑ(s(t), v(t)) = , dÑ(sn, vn) =
- .
:
ás(t), v(t)ñÑ = s(t)v(t) dt = ||s(t)||Ñ ||v(t)||Ñ cos j.
ásn, vnñÑ =(1/N) sn vn = ||sn||Ñ ||sn||Ñ cos j.
( ) , ( (2.1.8) ). .
, .
. (orthogonal), :
áu(t), v(t)ñ = u(t)v(t) dt = 0.
, ( j = 90), (, r = cos j = 0), (Euv = 0).
2.3.1 . ( ), - ( ).
. 2.3.1. .
, , .. , , .
. {vk, k = 1, 2, , N} N- :
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ávm, vnñ = (2.3.1)
. (2.3.1) :
ávm, vnñ = dmn, (2.3.1')
dmn , (2.3.1).
:
s = c1v1 + c2v2 + + cNvN,
k s :
ck = ás, vkñ.
L2[a, b] {u0(t), u1(t), u2(t), }, - , {uk(t), k=0, 1, 2, }, .. :
áum(t), un(t)ñ = um(t) un(t) dt = 0, m = 1, 2,...; n = 1, 2,...; m ¹ n.
[a, b] (orthonormal functions), m=n , .. :
áum(t), um(t)ñ = ||um(t)||2 = (um(t))2 dt = 1, ||um(t)|| = 1, m = 1, 2,....
:
um(t)un*(t) dt = dm,n.
, .. .
. s(t) Î H ( ), [a, b], uk(t):
s(t) = ckuk(t). (2.3.2)
k um(t) m t, :
s(t)um(t) dt = ck umuk dt.
ui(t), m = k ukuk dt =1, , , m = k , .. :
ck = s(t)uk(t) dt. (2.3.2)
, k - s(t) uk(t), .. s(t) , u(t), - . (2.3.2) N, N ck .
, x(t) y(t) , , , :
áx(t), y(t)ñ = x(t)y(t) dt = [ anun(t)] [ bmum(t)] dt = an×bn. (2.3.3)
x(t) y(t) (2.3.3):
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cos j = an×bn /(||x(t)||×||y(t)||).
, .
, , :
- ;
- ;
- ;
- .
, s(t), [a, b], , , ..:
|s(t)| dt < ¥, |s'(t)| dt < ¥.
. , , , , . , (, , .), . , , : .
[-p, p] :
{1, sin t, sin 2t, , sin kt}, k = 1, 2, 3, (2.3.4)
:
á1, sin ktñ =(1/2p) sin kt dt = (1/2kp) [cos kp - cos(-kp)] = 0, k = 1, 2, 3,
ásin mt, sin ntñ =(1/2p) sin mt sin nt dt =(1/4p) {cos (m+n)t cos (m-n)t} dt =
= = 0, m ¹ n.
, (2.3.4) . :
||sin kt||2=(1/2p) sin2 kt dt= (1/4p) (1-cos 2kt) dt= =1/2.
||sin kt|| = 1/ , k = 1, 2, 3,
, (2.3.4) (. 2.3.2):
{1, uk(t) = sin kt}, k = 1, 2, 3, (2.3.4')
. 2.3.2. .
:
{1, uk(t) = cos kt}, k = 1, 2, 3, , (2.3.5)
- :
{1, uk(t) = sin kt, uk(t) = cos kt}, k = 1, 2, 3, (2.3.6)
exp(pt) p = jf ( ) p = s+jf ( ),
exp(jwt) = cos(wt) + j sin(wt), exp(-jwt) = cos(wt) - j sin(wt),
cos(wt) = [(jwt)+exp(-jwt)]/2, sin(wt) = [(jwt)-exp(-jwt)]/2j
- . "" , , , .. f = 1/|t|. , .. . , , "", , f - 1/||.
, , , 1. 0,5 0,5 . 2.3.3. . : wal(k,), k = 0,1,2, , = t/T ( ).
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: cal(n,) = wal(2n,), , sal(n,) = wal (2n-1,), .
. 2.3.3. .
.
. , s(t) (2.3.2). (2.3.2) (2.2.2):
Es = s2(t) dt = cmcnumun dt = cmcn umun dt. (2.3.7)
m = n. :
Es = s2(t) dt = cn2, (2.3.8)
.. , . .