函数集的完全性

1.:敲g(x)篇〔一二,二]上之非降的有界缝差两数,业具有性鬓(K)s‘二一0,一。(:);f--:.,。g。尹(:)!d:一郁匕,(‘一”,”;dg)篇在〔一二,司上定羲业且满足修件:,一{户,(柳dg(·)}青<一,>l的可测蝮值函数族{f(幻}.封龄一徊乙“(一二,侧d刃中之子族凌B(幻},若由f(劣)(乙,(一二,二:dg),夕>1生+上夕q=1,及f--:ha”“’“““’一0纷{B(x)}之任何B(哟成立必滇致f(幻在〔一二,司上规乎虚虚等焚零则释{B(x)}在乙“(一二,侧dg)中完全. 函数族的完全性是舆函数横造的一些简题很有阴保的.徙【l]我们知道{e‘”}豁。是在乙,(一二,州dg),,>1,中完全的,…

COMPLETENESS OF SETS OF FUNCTIONS

If the real function g(x)is non-decreasing and of bounded variation on-π,π].The set {B_n(x)}_1~∞(?)∠~p(-π,π;dg),p>1,is complete ∠~p(-π,π;dg), p>1,if(?)and f(x)∈∠~q(-π,π;dg),1/p+1/q=1,imly that f(x)=0 almost everywhere on -π,π].In this paper,we suppose the following conditions are usually satisfied:g(x-0)=g(x);(?) If 0<λ_1<λ_2…and let λ(r)be the number of λ_n0,then(?),θ ∈ E,is complete∠~p(-π,π;dg),p>1,where E(?)-π,π]is some set of points havingpositive Lebesgue measure.7)If G(z)∈(M_1)is analytic in |z|<1 and continuous on |z|≤1,let us denot the modulus of continuity of G(e~(ix))by ω(t),then either(?);(?)θ ∈ E, is complete ∠~p(-π,π;dg),p>1.If(?),01,where ψ_E(t)=mes E_t,and E_t is the closed set of pointswith distance≤t from E.8)Let G(z)∈(M_1)be an analytic function defined in unit circle.If|α_n|,n=0,1,2,…,lim α_n=α,|a|<1; |α_n-α|≤1-|α|; ∑|α_n--α_(n+1)|<∞,then(?),p>1.9)Let(?)and μ(r) be the number ofβ_n≤r.For an entire function G(z)∈(M_1),if(?)then(?)is complete ∠~p(-π,π;dg),p>1.10)If{α_n}possesses a limite point α,|α|≠1,∞,then(?)complete ∠~p(-π,π;dg),p>1.If{α_n}possesses the properties describedin 8),then(?)is complete ∠~p(-π,π;dg)p>1.In the second part,for the class ∠~p(0,l),we have considered theproblem of the completeness of a set Of functions of the following forms:{G(α_ne~(ix))};(?);{G(α_nx)};(?);(?);(?)Naturally,some similar results are obtained.The completeness of above sets of functions with respect to the classesH_2(D)or E_2(D)is also discussed.