Introduction: Trigonometric polynomials. Complex-valued Fourier polynomials and sums of exponentials. Periodic functions. Topological groups. Characters. Finite and finitely generated abelian groups. The unit circle in the complex plane and the additive group of real numbers as topological groups. Dual groups. Inner products. Orthogonality, orthonormality and linear independence of characters. Fourier coefficients. Fourier series and Fourier transform in concrete and abstract settings.
Convergence results on the unit circle: Fourier series and partial sums. Bessel's inequality. Translations and convolutions. Homogeneous Banach spaces. Integrals valued in Banach spaces and their applications to convolutions. Summability kernels and convergence result for their convolutions. Cesàro averages of a sequence. Dirichlet and Fejér kernels. Their applications to partial sums of Fourier series and their Cesàro averages.
Fejér's theorem and applications: Uniform convergence for continuously differentiable functions. Density of trigonometric functions in the space of continuous functions on the unit circle. Unique determination of functions in L1 by their Fourier series. Convergence of Fourier series in L2. Parseval's identity. Fourier characterization of functions in L2. Riemann-Lebesgue lemma.
For exam-related problems look in TCD past examination papers and Mathematics department examination papers.
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