Tag Archives: continuous

Math2111: Chapter 1: Fourier series. Section 3: Fourier series and pointwise convergence

In this blog entry you can find lecture notes for Math2111, several variable calculus. See also the table of contents for this course. This blog entry printed to pdf is available here.

We consider now the Fourier coefficients of functions f:[-\pi, \pi] \to \mathbb{R} and discuss the convergence behaviour of Fourier series. We will see that the convergence behaviour depends on the smoothness of the function f. In the following we explain what we mean by smoothness of the function.

Piecewise continuity

Let f:[a,b] \to \mathbb{R} and let c \in [a,b]. We define the one-sided limits

\displaystyle f(c^{+}) = \lim_{x \to c^{+}} f(x)

and

\displaystyle f(c^{-}) = \lim_{x \to c^{-}} f(x).

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