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RKH spaces of Brownian type defined by Cesàro-Hardy operators

José E. Galé, Pedro J. Miana, Luis Sánchez--Lajustici

Abstract

We study reproducing kernel Hilbert spaces introduced as ranges of generalized Cesàro-Hardy operators, in one real variable and in one complex variable. Such spaces can be seen as formed by absolutely continuous functions on the positive half-line (or paths of infinite length) of fractional order, in the real case. A theorem of Paley-Wiener type is given which connects the real setting with the complex one. These spaces are related with fractional operations in the context of integrated Brownian processes. We give estimates of the norms of the corresponding reproducing kernels.

RKH spaces of Brownian type defined by Cesàro-Hardy operators

Abstract

We study reproducing kernel Hilbert spaces introduced as ranges of generalized Cesàro-Hardy operators, in one real variable and in one complex variable. Such spaces can be seen as formed by absolutely continuous functions on the positive half-line (or paths of infinite length) of fractional order, in the real case. A theorem of Paley-Wiener type is given which connects the real setting with the complex one. These spaces are related with fractional operations in the context of integrated Brownian processes. We give estimates of the norms of the corresponding reproducing kernels.
Paper Structure (10 sections, 17 theorems, 142 equations)

This paper contains 10 sections, 17 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Generalized Cesàro-Hardy operators
  3. Cesàro-Hardy operators on the real line
  4. Cesàro-Hardy operators on the (right-hand) complex half-plane
  5. Range spaces of Cesàro-Hardy operators
  6. RKH-Sobolev spaces
  7. RKH-Sobolev spaces and Brownian motion
  8. Hardy-Sobolev spaces
  9. Estimating the kernel
  10. Averaging Brownian motion

Key Result

Proposition 2.1

For $\alpha>0$, $1\le p<\infty$, $q$ such that $(1/p)+(1/q)=1$, $f\in L_p(\mathbb{R}^+)$ and $g\in L_q(\mathbb{R}^+)$ we have where the convergence of integrals are in the Bochner sense.

Theorems & Definitions (42)

  • Proposition 2.1
  • Corollary 2.2
  • proof
  • Definition 2.3
  • Remark 2.4
  • Corollary 2.5
  • Corollary 2.6
  • proof
  • Proposition 3.1
  • proof
  • ...and 32 more