Fourier series | Orr Shalit

Recall Theorem 6 from Notes 3:

Theorem 6: For every $latex f in C_{per}([0,1]) cap C^1([0,1])$, the Fourier series of $latex f$ converges uniformly to $latex f$.

It is natural to ask how much can we weaken the assumptions of the theorem and still have uniform convergence, or how much can we weaken and still have pointwise convergence. Does the Fourier series of a continuous (and periodic) function always converge? In this post we will use the principle of uniform boundedness to see that the answer to this question is a very big NO.

Once again, we begin with some analytical preparations. Continue reading Advanced Analysis, Notes 10: Banach spaces (application: divergence of Fourier series) →

Consider the cube $latex K := [0,1]^k subset mathbb{R}^k$. Let $latex f$ be a function defined on $latex K$. For every $latex n in mathbb{Z}^k$, the $latex n$th Fourier coefficient of $latex f$ is defined to be

$latex hat{f}(n) = int_{K} f(x) e^{-2 pi i n cdot x} dx ,$

where for $latex n = (n_1, ldots, n_k)$ and $latex x = (x_1, ldots, x_k) in K$ we denote $latex n cdot x = n_1 x_1 + ldots n_k x_k$. The sum

$latex sum_{n in mathbb{Z}^k} hat{f}(n) e^{2 pi i n cdot x} $

is called the Fourier series of $latex f$. The basic problem in Fourier analysis is whether one can reconstruct $latex f$ from its Fourier coefficients, and in particular, under what conditions, and in what sense, does the Fourier series of $latex f$ converge to $latex f$.

One week into the course, we are ready to start applying the structure theory of Hilbert spaces that we developed in the previous two lectures, together with the Stone-Weierstrass Theorem we proved in the introduction, to obtain easily some results in Fourier series.

Continue reading Advanced Analysis, Notes 3: Hilbert spaces (application: Fourier series) →