By Michel Herve
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Extra info for Analytic and plurisubharmonic functions in finite and infinite dimensional spaces. Course given at the University of Maryland, Spring 1970
0 k =1 0 (xk − τk )δk −1 b (τ1 , . . , τm ) dτ1 . . dτm = 0. 35 ) Now the following theorem is applicable. Theorem of Mikusinski and Ryll-Nardzewski (). Let the functions a(x1 , . , xm ) and b(x1 , . . , xm ) be defined for xk ≥ 0, k = 1, . . , m and have the form µ µ a (x1 , . . , xm ) = xλ1 1 . . xλmm a ˜ (x1 , . . , xm ) ; b (x1 , . . , xm ) = x1 1 . . xmm ˜b (x1 , . . , xm ) with λk > −1, µk > −1, k = 1, . . , m and continuous functions a ˜ (x1 , . . , xm ), ˜b (x1 , . . , xm ) for xk ≥ 0, k = 1, .
0) turns into the identity operator in Cα . 13 ) takes the form m,0 γ1 , . . , γs , γs+1 + δs+1 , . . , γm + δm γ1 , . . , γs , γs+1 , . . , γm γs+1 + δs+1 , . . , γm + δm m−s,0 = Gm−s,m−s σ γs+1 , . . , γm Gm,m σ and yields the following corollary. 2. Let s (1 ≤ s ≤ m − 1) of the components γk , k = 1, . . g. 0 = δ1 = . . = δs < δs+1 ≤ . . ≤ δm . ,δm ) s+1 ( k )s+1 Iβ,m 1 f (x) = Iβ,m−s f (x) 1 γs+1 + δs+1 , . . , γm + δm γs+1 , . . 3) f xσ β dσ. 4) γ +δ,−δ of negative order δ = −δ < 0 are defined as where the “integrations” R−δ , Iβ differintegral operators usually referred to as fractional derivatives.
0 a (x1 − τ1 , . . , xm − τm ) b (τ1 , . . , τm ) dτ1 . . dτm = 0 0 yields that a ≡ 0 or b ≡ 0 at least. δ −1 Let us choose now a (x1 , . . , xm ) = k xkk . The conditions δk > 0, k = 1, . . , m Γ(δk ) yield δk − 1 > −1, k = 1, . . , m. The function b (x1 , . . , xm ) has the form m 1 γk + p xk β f˜ (x1 , . . , xm ) β m µ xk k ˜b (x1 , . . , xm ) with µk > −1, k = 1, . . , m. = k =1 k =1 since p > max [−β (γk + 1)] and µk = γk + βp . Then due to the above cited theorem, it k follows that b (x1 , .
Analytic and plurisubharmonic functions in finite and infinite dimensional spaces. Course given at the University of Maryland, Spring 1970 by Michel Herve