Complex Analysis of Real Functions I: Complex-Analytic Structure and Integrable Real Functions
arXiv:1708.06182
Abstract
A complex-analytic structure within the unit disk of the complex plane is presented. It can be used to represent and analyze a large class of real functions. It is shown that any integrable real function can be obtained by means of the restriction of an analytic function to the unit circle, including functions which are non-differentiable, discontinuous or unbounded. An explicit construction of the analytic functions from the corresponding real functions is given. The complex-analytic structure can be understood as an universal regulator for analytic operations on real functions.
26 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
- Fourier Theory on the Complex Plane IV: Representability of Real Functions by their Fourier Coefficients
Cited by in corpus (6)
- Complex Analysis of Real Functions II: Singular Schwartz Distributions
- 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