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Construction of Principal Functions by Orthogonal Projection

Published online by Cambridge University Press:  20 November 2018

Mitsuru Nakai  and
Leo Sario
Mitsuru Nakai
Affiliation:
Nagoya University and University of California, Los Angeles
Leo Sario
Affiliation:
Nagoya University and University of California, Los Angeles
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Given a point set E on an open Riemann surface V we denote by H(E) the space of functions u harmonic in open sets O(u) containing E. Let V0 be a regular region of V with border α, and consider restrictions f to α of functions in H(α). For V1 = V — ⊽0, an operator L from H(α) to H(⊽1) is, by definition, normal if

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Type
Research Article
Information
Canadian Journal of Mathematics , Volume 18 , 1966 , pp. 887 - 896
Copyright
Copyright © Canadian Mathematical Society 1966

References

1. Ahlfors, L. V., Remarks on Riemann surfaces, Lectures on functions of a complex variable (Ann Arbor, 1955), pp. 45–48.Google Scholar
2. Ahlfors, L. V. and Sario, L., Riemann surfaces (Princeton, N.J., 1960).Google Scholar
3. Browder, F. E., Principal functions for elliptic systems of differential equations, Bull. Amer. Math. Soc, 71 (1965), 342–344.Google Scholar
4. Constantinescu, C. and Cornea, A., Ideale Ränder Riemannscher Flächen (Berlin, 1963).Google Scholar
5. Oikawa, K., A remark to Sario's lemma on harmonic functions, Proc. Amer. Math. Soc., 11 (1960), 425–428.Google Scholar
6. Oikawa, K., A constant related to harmonic functions, Japan. J. Math., 29 (1959), 111–113.Google Scholar
7. Rodin, B., Reproducing kernels and principal functions, Proc. Amer. Math. Soc., 13 (1962), 982–992.Google Scholar
8. Sario, L., Existence des fonctions d'allure donnée sur une surface de Riemann arbitraire, C. R. Acad. Sci. Paris, 229 (1949), 1293–1295.Google Scholar
9. Sario, L., A linear operator method on arbitrary Riemann surfaces, Trans. Amer. Math. Soc., 72 (1952), 281–295.Google Scholar
10. Sario, L., An integral equation and a general existence theorem for harmonic functions, Comment. Math. Helv., 38 (1964), 284–292.Google Scholar
11. Sario, L., Schiffer, M., and Glasner, M., The span and principal functions in Riemannian spaces, J. Analyse Math., 15 (1965), 115–134.Google Scholar
12. Sario, L. and Weill, G., Normal operators and uniformly elliptic self-adjoint partial differential equations, Trans. Amer. Math. Soc., 120 (1965), 225–235.Google Scholar
13. Weill, G., Capacity differentials on open Riemann surfaces, Pacific J. Math., 12 (1962), 769–776.Google Scholar