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25994 results in Abstract analysis

Page 91 of 1300

  • PEM series 2 volume 65 issue 3 Cover and Front matter

  • Journal:
    Proceedings of the Edinburgh Mathematical Society / Volume 65 / Issue 3 / August 2022
    Published online by Cambridge University Press:
    18 October 2022, pp. f1-f2
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  • The Halász–Székely barycenter

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  • Jairo Bochi, Godofredo Iommi, Mario Ponce
  • Journal:
    Proceedings of the Edinburgh Mathematical Society / Volume 65 / Issue 4 / November 2022
    Published online by Cambridge University Press:
    13 October 2022, pp. 881-911
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  • We introduce a notion of barycenter of a probability measure related to the symmetric mean of a collection of non-negative real numbers. Our definition is inspired by the work of Halász and Székely, who in 1976 proved a law of large numbers for symmetric means. We study the analytic properties of this Halász–Székely barycenter. We establish fundamental inequalities that relate the symmetric mean of a list of non-negative real numbers with the barycenter of the measure uniformly supported on these points. As consequence, we go on to establish an ergodic theorem stating that the symmetric means of a sequence of dynamical observations converge to the Halász–Székely barycenter of the corresponding distribution.

  • Polynomial null solutions to bosonic Laplacians, bosonic Bergman and Hardy spaces

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  • Chao Ding, Phuoc-Tai Nguyen, John Ryan
  • Journal:
    Proceedings of the Edinburgh Mathematical Society / Volume 65 / Issue 4 / November 2022
    Published online by Cambridge University Press:
    13 October 2022, pp. 958-989
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  • A bosonic Laplacian, which is a generalization of Laplacian, is constructed as a second-order conformally invariant differential operator acting on functions taking values in irreducible representations of the special orthogonal group, hence of the spin group. In this paper, we firstly introduce some properties for homogeneous polynomial null solutions to bosonic Laplacians, which give us some important results, such as an orthogonal decomposition of the space of polynomials in terms of homogeneous polynomial null solutions to bosonic Laplacians, etc. This work helps us to introduce Bergman spaces related to bosonic Laplacians, named as bosonic Bergman spaces, in higher spin spaces. Reproducing kernels for bosonic Bergman spaces in the unit ball and a description of bosonic Bergman projection are given as well. At the end, we investigate bosonic Hardy spaces, which are considered as generalizations of harmonic Hardy spaces. Analogs of some well-known results for harmonic Hardy spaces are provided here. For instance, connections to certain complex Borel measure spaces, growth estimates for functions in the bosonic Hardy spaces, etc.

Page 91 of 1300