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Some Generalized Hardy Spaces

  • L. D. Meeker (a1)

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This paper is concerned with generalizations of the classical Hardy spaces (8, p. 39) and the question of boundary values for functions of these various spaces. The general setting is the “big disk” Δ discussed by Arens and Singer in (1, 2) and by Hoffman in (7). Analytic functions are defined in (1). Classes of such functions corresponding to the Hardy Hp spaces are considered and shown to possess boundary values in (2). Contrary to the classical case, such functions do not form a Banach space; hence they are not the functional analytic analogue of the classical spaces. In (3) quasi-analytic functions are defined while in (4) Hardy spaces of such functions are considered and are shown to have boundary values and to form a Banach space.

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References

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1. Arens, R., A Banach algebra generalization of conformai mappings of the disk, Trans. Amer. Math. Soc., 81 (1956), 501513.
2. Arens, R. and Singer, I. M., Generalized analytic functions, Trans. Amer. Math. Soc., 81 (1956), 379393.
3. Berberian, S., Measure and integration (Toronto, 1962).
4. de Leeuw, K., The Fubini theorem and convolution formula for regular measures, Math. Scand. 11 (1962), 117122.
5. de Leeuw, K. and Glicksberg, I., Quasi-invariance and analyticity of measures on compact groups, Acta Math., 109 (1963), 179205.
6. Helson, H. and Lowdenslager, D., Prediction theory and Fourier Series in several variables II, Acta Math., 106 (1961), 175213.
7. Koffman, H., Boundary behavior of generalized analytic functions, Trans. Amer. Math. Soc., 87 (1958), 447466.
8. Koffman, H., Banach spaces of analytic functions (Englewood Cliffs, 1962).
9. Rudin, W., Fourier analysis on groups (London, 1962).
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Some Generalized Hardy Spaces

  • L. D. Meeker (a1)

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