Complex Analysis of Real Functions III: Extended Fourier Theory
arXiv:1708.07386
Abstract
In the context of the complex-analytic structure within the unit disk centered at the origin of the complex plane, that was presented in a previous paper, we show that the complete Fourier theory of integrable real functions is contained within that structure, that is, within the structure of the space of inner analytic functions on the open unit disk. We then extend the Fourier theory beyond the realm of integrable real functions, to include for example singular Schwartz distributions, and possibly other objects.
23 pgs. Small formatting corrections and bibliography update
References in corpus (7)
- Fourier Theory on the Complex Plane I: Conjugate Pairs of Fourier Series and Inner Analytic Functions
- Fourier Theory on the Complex Plane II: Weak Convergence, Classification and Factorization of Singularities
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- Fourier Theory on the Complex Plane IV: Representability of Real Functions by their Fourier Coefficients
- Complex Analysis of Real Functions I: Complex-Analytic Structure and Integrable Real Functions
- Complex Analysis of Real Functions II: Singular Schwartz Distributions
- Fourier Theory on the Complex Plane III: Low-Pass Filters, Singularity Splitting and Infinite-Order Filters
Cited by in corpus (4)
- Complex Analysis of Real Functions IV: Non-Integrable Real Functions
- Complex Analysis of Real Functions VI: On the Convergence of Fourier Series
- Complex Analysis of Real Functions V: The Dirichlet Problem on the Plane
- Complex Analysis of Real Functions VII: A Simple Extension of the Cauchy-Goursat Theorem