Tag Archives: Fourier series

Math2111: Chapter 1: Fourier series. Recommended reading: Motivation of formulae for Fourier series and a comparison to Taylor series

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 motivate the formulae for Fourier series. For simplicity we ignore here all questions concerning convergence, which are dealt with in Section 3 and the additional material.

Fourier series

As pointed out in the post on Section 2 at the beginning, in order to represent a function {}f we need an orthonormal bases with respect to the inner product

\displaystyle \langle f, g \rangle = \int_{-\pi}^\pi f(x) g(x) \, \mathrm{d} x.

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Math2111: Chapter 1: Fourier series. Section 6: Heat equation

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.

In this part we discuss applications of Fourier series to solving a certain type of partial differential equation (pde). In more detail, we discuss the heat equation. The aim is to show how Fourier series naturally come up in the solution of this equation. Continue reading

Math2111: Chapter 1: Fourier series. Additional Material: L_2 convergence of Fourier series.

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.

In this post I present some ideas which shed light on the question why one can expect the Fourier series to converge to the function (under certain assumptions). Continue reading

Math2111: Chapter 1: Fourier series. Section 4: Examples and general periodic functions

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.

In this part we calculate the Fourier series for some given functions f:[-\pi, \pi] \to \mathbb{R}. We also define Fourier series for functions f:[a, b] \to \mathbb{R}.

Examples of Fourier series

Example (sawtooth wave function) Find the Fourier series of the function

\displaystyle f(x) = x \quad \mbox{for } -\pi < x < \pi,

\displaystyle f(x) = f(x+2\pi) \quad \mbox{for } x \in \mathbb{R}.

Further we define f((2n + 1) \pi) = 0 for n \in \mathbb{Z}.

The graph of the function can be found here. Continue reading

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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Math2111: Chapter 1: Fourier series. Section 1: Background information

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 start the lectures with \rhd Fourier series which go back to \rhd Joseph Fourier.

Fourier discovered what we now call Fourier series when he studied the heat equation Continue reading