Complex Analysis of Real Functions II: Singular Schwartz Distributions
arXiv:1708.07017
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 singular Schwartz distributions can be represented within that same structure, so long as one defines the limits involved in an appropriate way. In that previous paper it was shown that essentially all integrable real functions can be represented within the complex-analytic structure. The infinite collection of singular objects which we analyze here can thus be represented side by side with those real functions, thus allowing all these objects to be treated in a unified way.
23 pgs. Small formatting corrections and bibliography update
References in corpus (4)
- Fourier Theory on the Complex Plane II: Weak Convergence, Classification and Factorization of Singularities
- Fourier Theory on the Complex Plane I: Conjugate Pairs of Fourier Series and Inner Analytic Functions
- Fourier Theory on the Complex Plane V: Arbitrary-Parity Real Functions, Singular Generalized Functions and Locally Non-Integrable Functions
- Complex Analysis of Real Functions I: Complex-Analytic Structure and Integrable Real Functions
Cited by in corpus (5)
- Complex Analysis of Real Functions III: Extended Fourier Theory
- 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