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Hilbert Space of Complex-Valued Harmonic Functions in the Unit Disc

Tseganesh Getachew Gebrehana, Hunduma Legesse Geleta

Abstract

We investigate an extended version of Hilbert space of analytic functions called Hilbert space of complex-valued harmonic functions. It is found that functions in Hilbert space of complex-valued harmonic functions exhibit many properties analogous to its analytic counter part such as complex-valued harmonic function analogous of norm, equivalent norms, reproducing kernels, growth estimates and Littlewood-Paley Identity Theorem. In conclusion we prove that many results in Hilbert space of analytic functions also hold in larger Hilbert space of complex-valued harmonic functions.

Hilbert Space of Complex-Valued Harmonic Functions in the Unit Disc

Abstract

We investigate an extended version of Hilbert space of analytic functions called Hilbert space of complex-valued harmonic functions. It is found that functions in Hilbert space of complex-valued harmonic functions exhibit many properties analogous to its analytic counter part such as complex-valued harmonic function analogous of norm, equivalent norms, reproducing kernels, growth estimates and Littlewood-Paley Identity Theorem. In conclusion we prove that many results in Hilbert space of analytic functions also hold in larger Hilbert space of complex-valued harmonic functions.

Paper Structure

This paper contains 4 sections, 7 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Norm on $\mathnormal{H}_{h}^{2}(\mathbb{D})$
  3. Equivalent Norms
  4. Growth estimates and Kernels

Key Result

Theorem 2.1

Let $f$ be a complex-valued harmonic function on the unit disc $\mathbb{D}$ given by where $h(z)=\sum_{n=0}^{\infty} a_{n}z^{n}$ and $g(z)=\sum_{n=0}^{\infty} {b_{n}z^{n}}$ are analytic. Suppose Then $||.||_{\mathnormal{H}_{h}^{2}}: {\mathnormal{H}_{h}^{2}}(\mathbb{D}) \rightarrow \mathbb{R}$ defined by $||f||_{\mathnormal{H}_{h}^{2}}=({\sum_{n=0}^{\infty}|a_{n}|^{2} + |b_{n}|^{2}})^{\frac{1}{2}

Theorems & Definitions (14)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • proof
  • Theorem 3.3
  • proof
  • ...and 4 more