Vector-valued Random Power Series On The Unit Ball Of C～n

Radom power series is an imjwrtant sludy in Analysis. In 1954,Salem and Zygmund had studied radom power series of Rn in [ 13]. In 1985,Duren had studied radom power series of C in [ 15]. In 1999,Shi Jihuai and Hu Pengyan had studied radom power series of Cn in[16].In this paper,we introduce the difinition of vector- valued random power series on the unit ball of C",and get the extension of Salem - Zygmund theorem by martingale and proof the convergence of vector - valued random power series. llien,we give die sufficient condition of vector - valued random power series in several complex variables which belong to vector - valued I lardy spaces. At last,we study the convergence of power series in relective sf aces.In chapter 2,we haveTheorem 2.1.1 ( vector - valued extension of Bernstein's theorem) Let Q be a vector - valued polynomial of degree n in then have a c when 6 E.then Where C is a constant which only has relation with n.Lenima 2.1.2 X be a Rademacher P type Banach space,1 p 2,and Cp is a P type constant,I r,,( 0 I is the Rademacher functional sequence of [0,l] ,then to arbitrary xn X,where mn and n- .Theorem 2.1.2 X be a Rademacher P type Banach space,1 p s2,and Cp is a P type constant, rn( t) is the Rademacher functional sequence of [0,l],,then to arbitrary A ,ifthenTheorem 2.2.1 (vector - valued Salem - Zyginund theorem) X be a Rademacher P type Banach space,1 s= p 2,and Cp is a P type constant, :ARNP. Let Qm(z,ea) =is a polymomial of degree m,P0 = 0,1,2, ,andthen for almost every,for almost every .Theorem 2.2.2 X be a Rademacher P type Banach space,1 )2,and Cp is a Ptype constant,sup is a polymomialof degree m,P0 = 0,1,2,...,andthen for almost every ,for almost every G dB. In chapter 3,we have Lemma 3.1 X be a Banach space,XG ARM',letthen for almost every :dBTheorem 3.1 X be a Rademacher P type Banach space,1 2,Cp is a P type constant,and X ARNP. If 3 [0,1) andthen for almost every the function have the propertyfor almost every Theorem 3.2 X be a Rademacher P type Banach space ,1p2,C ,isaP type constant,sunk,,and -Then for almost every ,the function have the propertyfor almost every In chapter 4,we haveTheorem 4.1.1 Theorem 4.2.1 X is a separable Hilbert space,ifthen for almost every (r 9fi,Theorem 4.2.2 X be a Rademacher P type Banach space,1 .p 2,and Cp is a P type constant,then for almost every for almost every :.llieorem 4.2.3 X be a llademachei Banudi sjwce,1 p2,and Cp is a P type constant,1 andthen for almost for almost every . In chapter 5,we have Theorem 5.1.2 X is a reflective Banach space.Letthen to all I e,I have the propertywhile n 2,then have a and have the propertyCorollary 5.1.1 X is a reflective Banach space. Letthen to almost every have the propertyTheorem 5.2.1 X is a reflective Banach space,letis a polynomial of degree then for almost every ,for almost every. Where max...