Complex Analysis of Real Functions VI: On the Convergence of Fourier Series
arXiv:1804.00769
Abstract
We define a compact version of the Hilbert transform, which we then use to write explicit expressions for the partial sums and remainders of arbitrary Fourier series. The expression for the partial sums reproduces the known result in terms of Dirichlet integrals. The expression for the remainder is written in terms of a similar type of integral. Since the asymptotic limit of the remainder being zero is a necessary and sufficient condition for the convergence of the series, this same condition on the asymptotic behavior of the corresponding integrals constitutes such a necessary and sufficient condition.
arXiv comment26 pgs. Small formatting corrections and bibliography update
References in corpus (6)
- 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
- Complex Analysis of Real Functions II: Singular Schwartz Distributions
- Complex Analysis of Real Functions III: Extended Fourier Theory