Fourier Theory on the Complex Plane I: Conjugate Pairs of Fourier Series and Inner Analytic Functions
arXiv:1409.2582
Abstract
A correspondence between arbitrary Fourier series and certain analytic functions on the unit disk of the complex plane is established. The expression of the Fourier coefficients is derived from the structure of complex analysis. The orthogonality and completeness relations of the Fourier basis are derived in the same way. It is shown that the limiting function of any Fourier series is also the limit to the unit circle of an analytic function in the open unit disk. An alternative way to recover the original real functions from the Fourier coefficients, which works even when the Fourier series are divergent, is thus presented. The convergence issues are discussed up to a certain point. Other possible uses of the correspondence established are pointed out.
arXiv comment44 pages, including 19 pages of appendices with explicit calculations and examples, 2 figures; fixed a few typos and made a few improvements; updated cross-references; made a few further improvements in the text
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Cited by in corpus (11)
- Fourier Theory on the Complex Plane II: Weak Convergence, Classification and Factorization of Singularities
- 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
- Complex Analysis of Real Functions II: Singular Schwartz Distributions
- Complex Analysis of Real Functions III: Extended Fourier Theory
- Fourier Theory on the Complex Plane III: Low-Pass Filters, Singularity Splitting and Infinite-Order Filters
- 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
- Low-Pass Filters, Fourier Series and Partial Differential Equations