Dive into a comprehensive 3.5-hour tutorial on Fourier Series, covering essential concepts from Fourier Coefficients and the Riemann Lebesgue Theorem to advanced topics like Fejer's Theorem and Parseval's Identity. Explore trig identities, orthogonal basis, even and odd functions, the Dirichlet Kernel, and partial sums. Learn about integration of Fourier Series, least squares, Bessel's Inequality, and convergence conditions. Gain insights into the Weierstrass M Test, Cauchy–Schwarz Inequality, and Jordan's Decomposition Theorem. Master the intricacies of functions with bounded variation and the 2nd Mean Value Theorem for Integrals in this in-depth mathematical exploration.
Overview
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.
Taught by
statisticsmatt
