用广义多项式逼近在正数轴上的函数

<正> 导言伯恩斯坦曾经证明:设 F(x)是偶的整函数,其泰勒系数不是负数,并且它的性(род,genus)大于零.如果 f(x)在(—∞,∞)上连续,并且适合

APPROXIMATION TO FUNCTIONS ON THE POSITIVE REAL AXIS BY GENERALIZED POLYNOMIALS

We generalize in this paper some results of S.Mandelbrojt and S.Ag-mon.Theorem 1.Let F(x)be a positive function(x≥0)and log F(x)be aconvex function of logx(x>0).Let{υ_}n(n=1,2,…)be a sequence ofcomplex numbers such that:A)0<|ν_1|<|ν_2|<…|ν_n|<…;B)(?),where K is a positive integer;C)(?)D){ν_n}lie in a half-strip in the s=σ+it plane:|t|<α,σ>0,whereα is a positive contant;E)(?)Suppose that(?)(1)where 00,we can find a polynomial of the formP(x)=a_0+a_1x~(ν_1)+a_2x~(ν_3)+…+a_nx~(ν_n),(2)such that for x≥0,(?)β)If f(x)is a function continuous on0,∞),and if (?)then given any ε>0,we can find a polynomial P(x)of the form(2)suchthat for x≥0,|f(x)-P(x)|<εF(x).γ)If f(x)∈L~p0,∞)(1≤p<∞),then given any ε>0,we can finda polynomial P(x)of the form(2)such that(?)The results of S.Mandelbrojt and S.Agmon on uniqueness of the solu-tion of the moment problem can be generalized as follows:Theorem 2.Let{ν_n}(n=1,2,…)be a strictly increasing sequence ofpositive numbers verifying the conditions B),C)and E)mentioned above.Let{m_n}(n=0,1,2,…)be a sequence of positive numbers.PutB(σ)=sup(ν_nσ-log m_n)(ν_0=0).If(?)(3)where 0