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26079 results in Abstract analysis

Page 1213 of 1304

  • 5 - Subharmonicity with Respect to a Uniform Algebra

  • T. W. Gamelin
  • Book:
    Uniform Algebras and Jensen Measures
    Published online:
    18 March 2010
    Print publication:
    01 March 1979, pp 54-82

  • Summary

    Let A be a uniform algebra on a compact space X, and let M denote the maximal ideal space of A. In this chapter, we continue the line of investigation begun in Chapters 1 and 2. We will introduce and treat various classes of “quasi-subharmonic” functions. The lower semi-continuous, quasi-subharmonic functions will be the log-envelope functions introduced in Chapter 2. The upper semi-continuous, quasi-subharmonic functions will be called simply “subharmonic”. The subharmonic functions in this context correspond to the subharmonic functions on an open subset of ¢, or to the plurisubharmonic functions on an open subset of ¢n.

    The main theorems of this chapter are Theorems 5.9 and 5.10, asserting that a locally subharmonic function is subharmonic, while a bounded, locally log-envelope function is a log-envelope function. Our exposition will be based on work of the author and N.Sibony[3,4].

    Quasi-subharmonic Functions

    Let u be a Borel function from a subset S of MA to [-∞, +∞]. We say that u is quasi-subharmonic on S if u(φ) ≤ ∫udσ for all φ ∈ S and all Jensen measures σ for φ supported on a compact subset of S It is understood implicitly that the negative part of u min(u,0), is integrable with respect to the Jensen measures σ for those φ ∈ S satisfying u(φ) > -∞.

    Evidently u is quasi-subharmonic on S if and only if u is quasi-subharmonic on each compact subset of S

    A function u from S to [-∞, +∞) is subharmonic if u is upper semi-continuous and quasi-subharmonic.

  • Boundary value problems for linear systems†

  • J. W. Neuberger
  • Journal:
    Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 83 / Issue 3-4 / 1979
    Published online by Cambridge University Press:
    14 November 2011, pp. 297-302
    Print publication:
    1979

  • Suppose H and K are Hilbert spaces and H′0, H′ are closed subspaces of H so that H′0 ⊂ H′. Denote by P the orthogonal projection of H onto H′0, denote by g an element of k and by C a bounded linear transformation from H to K so that CC* = I, the identity on K. Denote CPC* by M. Given w in H′ one has the problem of finding u in H′ so that

    There are given conditions on M (or certain operators related to M) which imply convergence of a certain iteratively generated sequence to a solution to this problem. The equation Cu = g represents an inhomogeneous system of linear differential equations (ordinary, partial or functional) and the condition P(u − w) = u − w is an abstract representation of inhomogeneous boundary conditions for u.

Page 1213 of 1304