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On Theory of Informational Functions Presence
) 4
2 22 c
cx c
x ≥∗ ℘
∗ ℘ ωσωσ (16)
Proof. Let's enter designations
Ex ˆˆˆ ∗−℘≡ ωα , Ecxc ˆˆˆ ωβ ∗−℘≡ (17)
appropriately
( ) ( )AxxAxxx ~,2ˆ,~,2ˆ2 ∗∗==∗ ℘ ωαωαωσ
(18)
( ) ( )AxxAxxcx c
~,2ˆ,~,2ˆ2 ∗∗==∗ [...] ) =∗+∗ ∫ ℜ
+≥∗ ℘
∗ ℘
2 ~,ˆ~,ˆ22 dxAxxAxxcx
c x ωβωαωσωσ
=∗ ∫ ℜ
∗= 2
~,ˆˆ~, dxAxxAxx ωβαω
( ) ( ) =∗ ∫ ℜ
−++ +∗= ⎥⎦
⎤ ⎢⎣ ⎡
2 ~,ˆˆˆˆ
2
1 ˆˆˆˆ
2
1~, dxAxxAxx ωαββααββαω
(21)
( ) ( ) 2
~,ˆˆˆˆ~, 4 12
~,ˆˆˆˆ~ [...] ∗+∗ ∫ ℜ
∗+∗= dxAxxAxxdxAxxAxx ~,2ˆ~,~,2ˆ~, ωβωωαω
dxAxxAxxdxAxxAxx ~,ˆ~,ˆ~,ˆ~,ˆ1 ∗+∗ ∫ ℜ
+∗+∗ ∫ ℜ
+= ωβωβωαωα (19)
Using Cauchy-Buniakovski inequality:
∫ ∫ ∫≥ 2
)()( 2
)( 2
)( dxxgxfdxxgdxxf (20)
And suppose …
