Fourier series for singular measures in higher dimensions
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911783843790848 |
|---|---|
| author | Berner, Chad Herr, John E. Jorgensen, Palle E. T. Weber, Eric S. |
| author_facet | Berner, Chad Herr, John E. Jorgensen, Palle E. T. Weber, Eric S. |
| contents | For multi-variable finite measure spaces, we present in this paper a new framework for non-orthogonal $L^2$ Fourier expansions. Our results hold for probability measures $μ$ with finite support in $\mathbb{R}^d$ that satisfy a certain disintegration condition that we refer to as ``slice-singular''. In this general framework, we present explicit $L^{2}(μ)$-Fourier expansions, with Fourier exponentials having positive Fourier frequencies in each of the d coordinates. Our Fourier representations apply to every $f \in L^2(μ)$, are based on an extended Kaczmarz algorithm, and use a new recursive $μ$ Rokhlin disintegration representation. In detail, our Fourier series expansion for $f$ is in terms of the multivariate Fourier exponentials $\{e_n\}$, but the associated Fourier coefficients for $f$ are now computed from a Kaczmarz system $\{g_n\}$ in $L^{2}(μ)$ which is dual to the Fourier exponentials. The $\{g_n\}$ system is shown to be a Parseval frame for $L^{2}(μ)$. Explicit computations for our new Fourier expansions entail a detailed analysis of subspaces of the Hardy space on the polydisk, dual to $L^{2}(μ)$, and an associated d-variable Normalized Cauchy Transform. Our results extend earlier work for measures $μ$ in one and two dimensions, i.e., $d=1 (μ$ singular), and $d=2 (μ$ assumed slice-singular). Here our focus is the extension to the cases of measures $μ$ in dimensions $d >2$. Our results are illustrated with the use of explicit iterated function systems (IFSs), including the IFS generated Menger sponge for $d=3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_15950 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fourier series for singular measures in higher dimensions Berner, Chad Herr, John E. Jorgensen, Palle E. T. Weber, Eric S. Functional Analysis 42B05 (Primary), 46C07 (Secondary) For multi-variable finite measure spaces, we present in this paper a new framework for non-orthogonal $L^2$ Fourier expansions. Our results hold for probability measures $μ$ with finite support in $\mathbb{R}^d$ that satisfy a certain disintegration condition that we refer to as ``slice-singular''. In this general framework, we present explicit $L^{2}(μ)$-Fourier expansions, with Fourier exponentials having positive Fourier frequencies in each of the d coordinates. Our Fourier representations apply to every $f \in L^2(μ)$, are based on an extended Kaczmarz algorithm, and use a new recursive $μ$ Rokhlin disintegration representation. In detail, our Fourier series expansion for $f$ is in terms of the multivariate Fourier exponentials $\{e_n\}$, but the associated Fourier coefficients for $f$ are now computed from a Kaczmarz system $\{g_n\}$ in $L^{2}(μ)$ which is dual to the Fourier exponentials. The $\{g_n\}$ system is shown to be a Parseval frame for $L^{2}(μ)$. Explicit computations for our new Fourier expansions entail a detailed analysis of subspaces of the Hardy space on the polydisk, dual to $L^{2}(μ)$, and an associated d-variable Normalized Cauchy Transform. Our results extend earlier work for measures $μ$ in one and two dimensions, i.e., $d=1 (μ$ singular), and $d=2 (μ$ assumed slice-singular). Here our focus is the extension to the cases of measures $μ$ in dimensions $d >2$. Our results are illustrated with the use of explicit iterated function systems (IFSs), including the IFS generated Menger sponge for $d=3$. |
| title | Fourier series for singular measures in higher dimensions |
| topic | Functional Analysis 42B05 (Primary), 46C07 (Secondary) |
| url | https://arxiv.org/abs/2402.15950 |