加权Bergman空间的若干逼近定理
|
摘要
设H2(Ω,φ)为区域Ω上相对于权φ的Bergman空间.给出若Ω为有限个Carathéodory区域之交且φ在Ω-次调和,那么Ω-上的全纯函数在H2(Ω,φ)中稠密,证明了当Ω=Cn且φ是近似圆形时,多项式在H2(Ω,φ)中稠密.
Let H2(Ω,φ)be the Bergman space with respect toφon the domainΩ.It is proved that holomorphic functions onΩare dense in H2(Ω,φ)whenΩis the intersection of a finite number of Carathéodory domains and φ is a subharmonic function onΩ.IfΩ = Cn andφis approximately circular polynomials are dense in H2(Ω,φ).
Let H2(Ω,φ)be the Bergman space with respect toφon the domainΩ.It is proved that holomorphic functions onΩare dense in H2(Ω,φ)whenΩis the intersection of a finite number of Carathéodory domains and φ is a subharmonic function onΩ.IfΩ = Cn andφis approximately circular polynomials are dense in H2(Ω,φ).
引文
[1]Carleman T.ber die approximation analytischer funktionen durch lineare aggregate von vorgegebenen potenzen[M].Stockholm:Almqvist&Wiksell,1923.
[2]Bers L.An approximation theorem[J].d’Analyse Mathématique,1965,14:1.
[3]Hedberg L I.Weighted mean square approximation in plane regions and generators of an algebra of analytic functions[J].Arkiv fr Matematik,1965,5(6):541.
[4]Havin V P.Approximation in the mean by analytic functions[J].Doklady Akademii,Nauk SSSR,1968,178:1025.
[5]Farrell O J.On approximation to an analytic function by polynomials[J].Bulletin of the American Mathematic Society,1934,40:908.
[6]Hrmander L,Wermer J.Uniform approximation on compact sets in Cn[J].Mathematica Scandinavica,1968,23:5.
[7]Taylor B A.On weighted polynomial approximation of entire functions[J].Pacific Journal of Mathematics,1971,36:523.
[8]Sibony N.Approximation polynomiale pondérée dans un domaine d’holomorphie de Cn[J].Annales de l’institut Fourier,1976,26:71.
[9]Wohlgelernter D.Weighted L2 approximation of entire functions[J].Transactions of The American Mathematical Society,1975,202:211.
[10]Tsuji M.Potential theory in modern function theory[M].Tokyo:Maruzen Co,Ltd,1959.
[11]Boas H P.The Lu Qi-Keng conjecture fails generically[J].Proceedings of the American Mathematical Society,1996,124:2021.
[12]Walsh J L.Interpolation and approximation by rational functions in the complex domain[M].New York:American Mathematical Society,1969.
[13]Gaier D.Lectures on complex approximation[M].Boston:Stuttgart,1980.
[14]Goluzin G M.Geometric theory of functions of a complex variable[M].New York:American Mathematical Society,1969.
[15]Chen B Y,Zhang J H.On Bergman completeness and Bergman Stability[J].Mathematische Annalen,2000,318:517.
[16]Diederich K,Ohsawa T.An estimate for the Bergman distance on pseudoconvex domains[J].Annals of Mathematic,1995,141:181.
[17]Donnelly H,Fe.erman C.L2-cohomology and index theorem for the Bergman metric[J].Annals of Mathematic,1983,118:593.
[18]Berndtsson B.The extension theorem of Ohsawa-Takegoshi and the theorem of Donnelly-Fe.erman[J].Annales de l’institut Fourier,1996,46:1083.
[19]Hewitt E,Stromberg K.Real and abstract analysis,a modem treatment of the theory of functions of a real variable[M].New York:Springer-Verlag,1965.
[2]Bers L.An approximation theorem[J].d’Analyse Mathématique,1965,14:1.
[3]Hedberg L I.Weighted mean square approximation in plane regions and generators of an algebra of analytic functions[J].Arkiv fr Matematik,1965,5(6):541.
[4]Havin V P.Approximation in the mean by analytic functions[J].Doklady Akademii,Nauk SSSR,1968,178:1025.
[5]Farrell O J.On approximation to an analytic function by polynomials[J].Bulletin of the American Mathematic Society,1934,40:908.
[6]Hrmander L,Wermer J.Uniform approximation on compact sets in Cn[J].Mathematica Scandinavica,1968,23:5.
[7]Taylor B A.On weighted polynomial approximation of entire functions[J].Pacific Journal of Mathematics,1971,36:523.
[8]Sibony N.Approximation polynomiale pondérée dans un domaine d’holomorphie de Cn[J].Annales de l’institut Fourier,1976,26:71.
[9]Wohlgelernter D.Weighted L2 approximation of entire functions[J].Transactions of The American Mathematical Society,1975,202:211.
[10]Tsuji M.Potential theory in modern function theory[M].Tokyo:Maruzen Co,Ltd,1959.
[11]Boas H P.The Lu Qi-Keng conjecture fails generically[J].Proceedings of the American Mathematical Society,1996,124:2021.
[12]Walsh J L.Interpolation and approximation by rational functions in the complex domain[M].New York:American Mathematical Society,1969.
[13]Gaier D.Lectures on complex approximation[M].Boston:Stuttgart,1980.
[14]Goluzin G M.Geometric theory of functions of a complex variable[M].New York:American Mathematical Society,1969.
[15]Chen B Y,Zhang J H.On Bergman completeness and Bergman Stability[J].Mathematische Annalen,2000,318:517.
[16]Diederich K,Ohsawa T.An estimate for the Bergman distance on pseudoconvex domains[J].Annals of Mathematic,1995,141:181.
[17]Donnelly H,Fe.erman C.L2-cohomology and index theorem for the Bergman metric[J].Annals of Mathematic,1983,118:593.
[18]Berndtsson B.The extension theorem of Ohsawa-Takegoshi and the theorem of Donnelly-Fe.erman[J].Annales de l’institut Fourier,1996,46:1083.
[19]Hewitt E,Stromberg K.Real and abstract analysis,a modem treatment of the theory of functions of a real variable[M].New York:Springer-Verlag,1965.