\( \newcommand{\F}{\mathcal{F}} \newcommand{\ip}[2]{\langle #1 | #2\rangle} \)

Fourier Notes

This page collects some useful results from Fourier Analysis.

Fourier Series

Definition 1. Let \(f\in L^1(0,1)\) be a Riemann integrable function on \([0, 1]\) and define the Fourier coefficients by \[ \hat{f}(n) = \int_0^1 f(x) e^{-2\pi i nx} \,dx \qquad n\in\ZZ \]

Define the \(N\)'th partial sum of the Fourier series for \(f\) to be: \[ S_Nf(x) = \sum_{n=-N}^N \hat{f}(n) e^{2\pi inx} \]

Theorem 2. Suppose that \(f\) is continuous on \([0,1]\) and \(\hat{f}(n)\) is absolutely summable. Then \(S_Nf(x)\rightarrow f(x)\) uniformly on \([0,1]\).

Theorem 3. Suppose \(f\) is Riemann integrable on \([0, 1]\) and \(\hat{f}(n) = O(1/|n|)\).

1. If \(f\) is continuous at \(x_0\in[0,1]\) then \(S_Nf(x_0)\rightarrow f(x_0)\)
2. If \(f\) is continuous on \([0,1]\) then \(S_Nf(x)\rightarrow f(x)\) uniformly on \([0,1]\).

The following two results are useful corollaries of the last theorem:

Corollary 4. Suppose that \(f\) is Riemann integrable on \([0, 1]\) and \(\hat{f}(n) = O(1/|n|)\). Then whenever \(f(x-)\) and \(f(x+)\) exist, \[ S_Nf(x_0)\rightarrow \frac{f(x+) + f(x-)}{2} \]

Corollary 5. Suppose that:

1. \(f\) is continuously differentiable except at finitely many points in \([0,1]\),
2. \(f'(x)\) is bounded on the set of points at which it exists.

Then, \(f(x+)\) and \(f(x-)\) exist for every \(x\in[0,1]\) and \[ S_Nf(x) \rightarrow \frac{f(x+) + f(x-)}{2} \] for all \(x\).

Theorem 6. (Carleson) Let \(f\in L^p(0, 1)\) for \(1 < p \le \infty\). Then \(S_Nf(x)\) converges pointwise almost everywhere to \(f(x)\).

Theorem 7. (Fejér) Define the Fejér sum of \(f\) to be \[ \sigma_Nf(x) = \frac{1}{N+1}\sum_{n=0} ^ N S_nf(x) = \sum_{n=-N}^N \frac{N + 1 - |n|}{N+1} \hat{f}(n) e^{2\pi i nt} \] Then,

1. If \(f\) is continuous at \(x_0\) then \(\sigma_Nf(x_0) \rightarrow f(x_0)\), and
2. If \(f\) is continuous on \([0,1]\) then \(\sigma_Nf(x) \rightarrow f(x)\) uniformly on \([0,1]\).

Theorem 8. (Parseval's Theorem) Let \(f \in L^2(0,1)\). Then its Fourier coefficients are in \(\ell^2(\ZZ)\) and \[ \int_0^1 |f(x)|^2 dx = \sum_{n=-\infty}^\infty |\hat{f}(n)|^2 \] Thus the map \(f \in L^2(0,1) \rightarrow {f}\in\ell^2(\ZZ)\) is unitary.

Fourier Transforms

Definition 9. Let \(f\in L^1(\RR)\) and define the Fourier transform \[ (\F f)(\xi) = \int_{-\infty}^\infty f(x) e^{-2\pi i x\xi} \,dx \] Likewise for integrable \(g(\xi)\) define the inverse Fourier transform \[ (\F^{-1}g)(x) = \int_{-\infty}^\infty g(\xi) e^{2\pi i x\xi} \, d\xi \]

Theorem 10. (Fourier Inversion Theorem) If \(f \in L^1(\RR)\) and also \(\F f \in L^1(\RR)\) then \((\F^{-1}\F f)(x) = f(x)\) for almost every \(x\in\RR\).

Corollary 11. If, in addition, \(f\) is continuous, then \((\F^{-1}\F f)(x) = f(x)\) for all \(x\in\RR\).

Theorem 12. (Plancherel Theorem) If \(f\in L^1(\RR)\cap L^2(\RR)\) then \(\F f\in L^2(\RR)\) and \[ \int_{-\infty}^\infty |(\F f)(\xi)|^2\,d\xi = \int_{-\infty}^\infty |f(x)|^2 \,dx \]

Thus the map \(f\rightarrow\F f\) is an isometry from \(f\in L^1(\RR)\cap L^2(\RR)\) into \(L^2(\RR)\), which extends to a unitary \(L^2(\RR)\rightarrow L^2(\RR)\). This unitary is called the Plancherel transform.

The fact that \(\F\) is unitary means that \(\ip{\F f}{\F g} = \ip{f}{g}\) and so for \(f, g \in L^2(\RR)\rightarrow L^2(\RR)\), \[ \begin{align} \ip{\F f}{\F g} &= \int \overline{ f(x)e^{-2\pi i x \xi}} g(x') e^{-2\pi i x' \xi} dx dx' d\xi \\ & = \int \overline{f(x)}g(x') e^{2\pi i (x-x')\xi} dx dx' d\xi \\ &= \ip{f}{g} \\ &= \int \overline{f(x)}g(x) dx \end{align} \] This justifies use of the shortcut formula \[ \int e^{2\pi i (x-x')\xi} d\xi = \delta(x-x') \]