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

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

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

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

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

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

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