, k . k- .
. U(t) N U1 = U(t1),..., Ui= U(ti),..., UN = U(tN), t1...ti...tN. N U(t).
N- pN(U1,..., Ui,..., UN; t1,..., tN). N , t1,t2,...,tN (u1, u1+Δu1),..., (ui, ui+ ui),..., (uN, uN + uN), ui(1 ) , Ui, (. 1.12).
Δui, ,
N- , . N , .
. - .
p1(U1; t1) U(t) U1, t1. .
p2 = p2(U1, U2; t1, t2) U1 U2 t1 t2 , , . U(t) ,
. , . : , .
U(t) mu(t1), t1 U(t\) :
U(t1) mu(t1) t1 Du(t1):
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(t1) = U(t1) mu(t1) .
Du(t1) t1 su(t1):
(. 1.13, , ), .
U(t) t1 t2 Ru(t1,t2), .
t1 t2 U(t1) U(t2):
(1.71)
:
(1.69) (1.70) , t1 = t2 :
:
, .
U(t) V(t). :
1.9.
. , . .
, .
, .
, , . .
U , t = ti + τ (τ ).
, - + .
, , . , , .
, . U(t) , , τ = t2 - t1, ..
τ = 0:
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(1.79) (1.81) , U(t) .
. , , .
, , . , ,
(1.79), (1.81) , , .
. , , , .
,
u(t) U(t).
, . . (1.83) (1.85), .
(1.85) ().
1.10.
1.2 . , [21].
U(t), mu(t). (t) t1 (t1):
(t) , [. (1.1)], , jk(t) Ck, . (t):
k . , . . .
(1.7).
,
mu(f) (1.86) T<t<.T φk(t),
(1.87 ) (1.876) (1.86) U(t) ,
(1.87 ) ,
[mu(t), jk(t)], , , .
, (t),
(1.88) , {Ck} (Rkl = 0 k l, Rkl = 1 k = l):
, t1 = t2 = t U(t):
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, {k}. (1.87), , .
[21], , .
, , .
(1.87) () , ().
(1.2). , .
. .
1.11.
. Ru(t) (. 1.14) , [-, ], (1.15), 4T ( T<.t1, t2<T, -2Τ<τ<2Τ):
, Ru(t) ,
τ = t1 - t2,
(1.89) . , , :
(1.95) , . , (mu). .
, k (1.95) .
, , , :
, , (. 1.15).
, . . - <t< , .
. (1.91) . , , , (1.92), ,
, :
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S (wk) = Dk/(Δω) = 2DkT/ (k = 0,l, 2,..) (1.98)
S (wk) - , ωk.
(1.94) (1.98)
,
S (ωk)Δω Dk Ru(t), D[Ck] U(t), Suu(w)dw, , , , (ω, ω + dw). Suu(w), , U(t).
Ru(t) , (1.101) τ = t1 - t2:
G (w) = k/( w) T (1.95), U(t):
G(w)dw Suu(w)dw.
, .
. , (1.101) (1.102) Suu(w) , . , . , (1.102) :
Ru(t) ,
(1.105) , Suu(w) , . .
(1.101):
(1 101) (1.102), (1.105) (1.107) , (1.105) (1.107) . : Suu(ω), Ru(t) ( ), .
, Suu(w) , , Du U(t). , (1.107) τ = 0,
U(t) , Du , 1 :
,
, , (ω, ω + dw).
Suu(w) , , Suu(w) .
. Ρk(ω) (1.62) .
u (t) U(t) -T<t<.T. :
(1.63)
Ρ (ω) k.
, U(t) ,
t1 - t2 = τ.
(1.114) (1.113) T :
.
1.7. . .
Suu(ω) (. 1.16, ):
(1.107) U(t):
Ru(t) . 1.16,6. τ = 0 , , :
, (. 1.17, ). , , , Du .
w0 , - (. 1.17,6).
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, , . , . , . , . , . , , (), , .