Infinitely Divisible Matrices, Kernels, and Functions |
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Page 4
... All func- tions and products of matrices will be understood in the sense of Schur , i.e. , f ( A ) = ( f ( a ,; ) ) ” , and AoB = ( a b ij''i , j = 1 Schur Product Theorem : ijˇij'i , j = 1 and we recall the > Theorem 1.1.1 ( vid . [ 2 , p ...
... All func- tions and products of matrices will be understood in the sense of Schur , i.e. , f ( A ) = ( f ( a ,; ) ) ” , and AoB = ( a b ij''i , j = 1 Schur Product Theorem : ijˇij'i , j = 1 and we recall the > Theorem 1.1.1 ( vid . [ 2 , p ...
Page 7
... Schur theorem ) and so the function f ( x ) + Tg ( x ) has the same property for all T > 0 . τ But then we have just shown that for T > 0 , necessarily P ( T ) = ( f ( a ) + Ta2 ) ( f ' ( a ) + τnan - 1 ) ... Schur product theorem cannot be 7.
... Schur theorem ) and so the function f ( x ) + Tg ( x ) has the same property for all T > 0 . τ But then we have just shown that for T > 0 , necessarily P ( T ) = ( f ( a ) + Ta2 ) ( f ' ( a ) + τnan - 1 ) ... Schur product theorem cannot be 7.
Page 53
... Schur product theorem in this case , all a = 0 . while part f ) follows from repeated use of Theorem 1.1.10 , since implies that A YO on L. Since AYO , а A XO 0 azi 2 i + j ai + j a2j 0 , and so a21a2j > 0 for i + j 2 to get a a2 > a ...
... Schur product theorem in this case , all a = 0 . while part f ) follows from repeated use of Theorem 1.1.10 , since implies that A YO on L. Since AYO , а A XO 0 azi 2 i + j ai + j a2j 0 , and so a21a2j > 0 for i + j 2 to get a a2 > a ...
Common terms and phrases
a₁ an+1 an+2 C₁ choice of arguments completely monotonic function completely monotonic sequence consistent choice continuous function continuous kernel Conversely Corollary define difference quotient divisible characteristic function divisible completely monotonic divisible positive definite du(e du(t du(x dv(t dµ(t dµ(x equivalent finite functions f graph Green's function Hermitian Hilbert space implies incidence matrix inequality infinitely divisible characteristic infinitely divisible completely infinitely divisible kernels infinitely divisible positive integer interpolation problem irreducible component Lemma Let K(P,Q Let K(x,y Loewner nodes nonnegative measure nonnegative quadratic form nth roots nx n matrix Pick's theorem principally infinitely divisible probability measure Proof real valued representation formula Schur product theorem semigroup symmetric matrix teristic function Theorem 2.2 three point property unique unit disc univalent analytic function zeros αμ фє