Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Approximation of the Bergman norm by the norms of the direct product of two Szegő spaces


Author: Saburou Saitoh
Journal: Proc. Amer. Math. Soc. 75 (1979), 226-230
MSC: Primary 30C40; Secondary 30E10
DOI: https://doi.org/10.1090/S0002-9939-1979-0532141-1
MathSciNet review: 532141
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let G be a bounded regular region in the plane. Let $ H_2^{1/2}(G)$ denote the Szegő space of G composed of analytic functions on G with finite norm

$\displaystyle {\left\{ {\frac{1}{{2\pi }}\int_{\partial G} {\vert f(z){\vert^2}\vert dz\vert} } \right\}^{1/2}} < \infty .$

We set $ f(z) = \Sigma _{j = 1}^\infty {\varphi _j}(z){\psi _j}(z)\;({\varphi _j},{\psi _j} \in H_2^{1/2}(G))$. Then, we determine a necessary and sufficient condition for $ f(z)$ to make the equality

$\displaystyle \frac{1}{\pi }\int {\int\limits_G {\vert f(z){\vert^2}\;dx\;dy = ... ... _j}({z_2}} )\overline {{\psi _k}({z_2})} \vert d{z_2}\vert} } } } \right\}} } $

hold. The minimum is taken here over all analytic functions $ \Sigma _{j = 1}^\infty {\varphi _j}({z_1}){\psi _j}({z_2})$ on $ G \times G$ satisfying $ f(z) = \Sigma _{j = 1}^\infty {\varphi _j}(z){\psi _j}(z)$ on G.

References [Enhancements On Off] (What's this?)

  • [1] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337-404. MR 0051437 (14:479c)
  • [2] D. A. Hejhal, Theta functions, kernel functions and Abeian integrals, Mem. Amer. Math. Soc. 129 (1972). MR 0372187 (51:8403)
  • [3] S. Saitoh, The exact Bergman kernel and the kernels of Szegő type, Pacific J. Math. 71 (1977), 545-557. MR 0454007 (56:12258)
  • [4] -, The Bergman norm and the Szegő norm, Trans. Amer. Math. Soc. (to appear). MR 525673 (80h:30011)
  • [5] -, The Dirichlet norm and the norm of Szegő type (submitted).
  • [6] M. Schiffer and D. C. Spencer, Functional of finite Riemann surfaces, Princeton Univ. Press, 1954. MR 0065652 (16:461g)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 30C40, 30E10

Retrieve articles in all journals with MSC: 30C40, 30E10


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1979-0532141-1
Keywords: Bergman and Szegő kernels, the direct product of two Szegő spaces, Dirichlet norm, approximation of the Bergman norm, generalized isoperimetric inequalities
Article copyright: © Copyright 1979 American Mathematical Society

American Mathematical Society