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Hardy Space

The Hardy space H^p(D), for 0<p<infty, is the class of holomorphic functions on the unit disk D satisfying the growth condition

||f||_(H^p)=sup_(0<r<1)[1/(2pi)int_0^(2pi)|f(re^(itheta))|^pdtheta]^(1/p)<infty,

where ||f||_(H^p) is the Hardy norm.

On the unit ball B_d subset C^d, the analogous space H^2 consists of holomorphic functions for which the integrals of |f(rzeta)|^2 against normalized surface measure on partialB_d are bounded uniformly for 0<r<1. The existence of a uniformly bounded orthonormal basis of homogeneous polynomials is Rudin's homogeneous basis problem (Wang et al. 2026).

See also

Bergman Space, Takenaka-Malmquist System

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References

Duren, P. L. Theory of Hp Spaces. New York: Academic Press, 1970.Garnett, J. Bounded Analytic Functions. New York: Academic Press, 1981.Koosis, P. Introduction to Hp Spaces, 2nd ed. Cambridge, England: Cambridge University Press, 1998.Krantz, S. G. "Hardy Spaces." §12.3 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 152-154, 1999.Wang, P.; Zhang, F.; and Zhu, Z. "Rudin's Problem on Uniformly Bounded Homogeneous Orthonormal Bases." 21 Oct 2026. https://arxiv.org/abs/2610.29946.

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Hardy Space

Cite this as:

Weisstein, Eric W. "Hardy Space." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HardySpace.html

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