Fourier Series

This course presents the theory of Fourier series, including coefficients, orthogonal bases, Dirichlet and Fejér kernels, convergence conditions, integration, least squares, and Parseval's identity. It also reviews supporting results from analysis such as the Weierstrass M-test, Cauchy–Schwarz inequality, and Jordan's decomposition theorem.

Syllabus

Fourier Coefficients: Riemann Lebesgue Theorem (F1).
Fourier Series: Trig Indentities (F2).
Fourier Series: Preliminaries (F3).
Fourier Series: Orthogonal Basis (F4).
Fourier Series: f(x) is an even or odd function (F5).
Fourier Series: The Dirichlet Kernel (F6).
Fourier Series: Partial Sum (F7).
Fourier Series: Fejer's Kernel (F8).
Integration of a Fourier Series (F9).
Fourier Series: Least Squares. Bessel's Inequality. (F10).
Fourier Series: Remainder / Residual (F11).
Fourier Series: Conditions on f(x) and f'(x) for convergence. (F13).
Fourier Series: Fejer's Theorem (F14).
Fourier Series: Parseval's Identity (F15).
Weierstrass M Test.
Cauchy–Schwarz Inequality.
Show that f(t)=sin(t/2)^-1 - (t/2)^-1 is integrable in (0,d).
Jordan's Decomposition Theorem (Function with Bounded Variation).
2nd Mean Value Theorem for Integrals.