Unlock the Power of Convolution: The Surprising Math Behind Fourier Transforms

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so that \(f*g=g*f\). It is easy to see that the integral converges absolutely if one of the functions \(f\) or \(g\) is bounded. The most important property of the Fourier transform is that it converts convolution to ordinary pointwise multiplication:

\[\widehat{(f*g)}(x)=\widehat{f}(x)\widehat{g}(x).\]

For a proof, one can change the order of integration by using Fubini’s Theorem (for Fubini’s theorem see, e.g., [50], pp. 75-80) to write

\[\widehat{(f*g)}(x) =\int_{-\infty}^{\infty}e^{-ixt}\int_{-\infty}^{\infty}f(s)g(t-s )ds\,dt\] \[=\int_{-\infty}^{\infty}e^{-ixs}f(s)\int_{-\infty}^{\infty}e^{- ix(t-s)}\,g(t-s)dt\,ds\] \[=\widehat{f}(x)\widehat{g}(x).\]

This relationship is important when one tries to solve partial differential equations. For example, it could be the case that one has the product of two Fourier transforms; by the above relationship we can recognize it as a Fourier transform of a single function formed by the convolution.

### Fourier Transform and the Laplacian

For applications to the theory of partial differential equations, one of the important Fourier transforms is the one defined on Euclidean space \(\mathbb{R}^{n}\). Suppose \(f:\mathbb{R}^{n}\to\mathbb{C}\) is the _plane wave function_ defined as:

\[f(x)=c_{\xi}e^{2\pi ix\cdot\xi}.\]

Here both \(x,\xi\in\mathbb{R}^{n}\), where \(x\) is the position, \(\xi\) is the frequency of the plane wave, \(x\cdot\xi\) is the dot product, and \(c_{\xi}\) is a complex number whose magnitude is the _amplitude_. The reason why this function is named the plane wave function stems from the fact that if one considers the hyperplane \(H_{\alpha}=\{x:\ x\cdot\xi=\alpha\}\), then \(f(x)\) is constant on \(H_{\alpha}\) and the value taken by \(f\) on \(H_{\alpha}\) is equal to the value taken on \(H_{\alpha+2\pi}\). It turns out that if \(f\) is sufficiently smooth, then it can be represented as the superposition of plane waves. The following formulas show how to determine a Fourier transformed function from the original function and vice versa. Namely, we have:

\[f(x)=\int_{\mathbb{R}^{n}}\widehat{f}(\xi)e^{2\pi ix\cdot\xi}d\xi,\]

and the Fourier transform is

\[\widehat{f}(\xi)=\int_{\mathbb{R}^{n}}f(x)e^{-2\pi ix\cdot\xi}dx.\]

Again, it turns out that if \(f\) is sufficiently smooth, one can easily verify convergence of these integrals. Now given a function \(f:\mathbb{R}^{n}\to\mathbb{C}\), its _Laplacian_\(\triangle f\) is defined by the formula

\[\triangle\,f(x)=\sum_{i=1}^{n}\frac{\partial^{2}f}{\partial x_{i}^{2}},\]

where \(f\) is a function of \(n\) real variables. For simplicity, consider only functions \(f\) where the above formula is well defined. Now suppose \(f\) is a plane wave function \(f(x)=e^{2\pi ix\cdot\xi}\), then its Laplacian is

\[\triangle\,e^{2\pi ix\cdot\xi}=-4\pi^{2}|\xi|^{2}e^{2\pi ix\cdot\xi}.\]

Thus, the effect of the Laplacian on the plane wave function is to multiply by the scalar \(-4\pi^{2}|\xi|^{2}\). This means that plane wave function is an eigenfunction for the Laplacian \(\triangle\). Even though in general there is no obvious relationship between a function \(f\) and its Laplacian \(\triangle f\), we have a very different situation for the wave function. If we study the Laplacian via the Fourier transform, then we can write an arbitrary function as a superposition of plane waves and the Laplacian has a striking effect on each plane wave. We can even obtain a formula for the Laplacian of the general function. More precisely, if we have a nice enough function \(f\) so that we can interchange the integral \(\int\) and the Laplacian \(\triangle\),

\[\triangle\,f(x)=\triangle\int_{\mathbb{R}^{n}}\hat{f}(\xi)e^{2\pi ix\cdot\xi} d\xi=\int_{\mathbb{R}^{n}}\hat{f}(\xi)\,\triangle\,e^{2\pi ix\cdot\xi}d\xi.\]Then the Laplacian

\[\triangle\,f(x)=\int_{\mathbb{R}^{n}}(-4\pi^{2}|\xi|^{2})\hat{f}(\xi)e^{2\pi ix \cdot\xi}d\xi.\]

This formula represents \(\triangle f(x)\) as a superposition of plane waves. Moreover, such a representation is unique. Recalling the Fourier inversion formula

\[\triangle\,f(x)=\int_{\mathbb{R}^{n}}\widehat{\triangle f}(\xi)e^{2\pi ix\cdot \xi}d\xi,\]

or using integration by parts formula in the definition of Fourier transform, we obtain

\[\widehat{\triangle f}(\xi)=(-4\pi^{2}|\xi|^{2})\widehat{f}(\xi).\]

This identity shows that the Fourier transform diagonalizes the Laplacian operator, i.e., the operation of taking the Laplacian is multiplication of a function \(F(\xi)\) by the multiplier \((-4\pi^{2}|\xi|^{2})\).

**Remark 59**.: The classical text in Fourier Series has been Zygmund’s _Trigonometric Series_[56] which was published first in 1930. Stein and Sakarachi’s book _Fourier Analysis an Introduction_[49], Folland’s _Fourier Series and its Applications_[21], and Korner’s _Fourier Analysis_[33] are books that all provide a solid introduction to the topic. Since Fourier analysis has applications to a wide range of areas, there is an ample supply of books on applications. For example, B. Hubbard’s [27]_The World According to Wavelets_, _Fourier Series and Numerical Methods for Partial Differential Equations_ by R. Bernatz [10], _Fourier Analysis in Probability Theory_ by T. Kawata [31], and _Fourier Analysis of Time Series_ by Bloomfield [12] are good sources.

### Exercises

1. Let \(f:\mathbb{R}\to\mathbb{R}\) be a \(2\pi\)-periodic and Riemann integrable function on \([-\pi,\pi]\). If \(f\) is even (respectively odd) show that the Fourier series can be written using only cosine (respectively sine) terms.

2. Let \(f:\mathbb{R}\to\mathbb{R}\) be a unique function such that \(f(x)=x\) if \(\pi\leq x<\pi\) and \(f(x+2n\pi)=f(x)\) for all \(n\in\mathbb{Z}\).

1. Show that Fourier series for \(f\) is \(\sum_{n=1}^{\infty}\frac{(-1)^{n+1}2\mathrm{sin}nx}{n}\).
2. Prove that the series above does not converge uniformly.

3. Define \(f(x)=\pi-x\) for \(0

4. On the vector space

\[L^{2}[-\pi,\pi]=\left\{f:[-\pi,\pi]\to\mathbb{C}:\quad\int_{-\pi}^{\pi}|f|^{2}< \infty\right\},\]

show that

\[\langle f,g\rangle=\int_{-\pi}^{\pi}f(x)\,\overline{g(x)}\,dx\]

is an inner product.

5. Let \(f(x)=|x|\) for \(x\in[-\pi,\pi]\) be extended periodically to the whole line by \(f(x+2\pi)=f(x)\). Find Fourier coefficients and the Fourier series for \(f\).

6. If \(f\) and \(g\) are Riemann integrable on \([-\pi,\pi]\), then show that \(s_{n}(\alpha f+\beta g)=\alpha s_{n}(f)+\beta s_{n}(g)\) for every \(n\). In other words, show that the map \(f\mapsto s_{n}(f)\) is linear.

7. Prove the trigonometric form of the Weierstrass approximation theorem given in Corollary 19.

8. Prove that \(||f-\sigma_{n}(f)(x)||_{2}\to 0\) as \(n\to\infty\), where \(\sigma_{n}(f)(x)=\frac{1}{n+1}\sum_{k=0}^{n}s_{k}(x)\).

9. Let \(f(t)=e^{-c|t|}\) for some constant \(c>0\). Show that its Fourier transform is \(\widehat{f}(x)=\frac{2c}{c^{2}+x^{2}}\).

10. Let \(a\neq 0\) and \(b\) be real constants and \(g(t)=f(at+b)\). Prove that

\[\widehat{g}(x)=\frac{1}{|a|}e^{\frac{ibx}{a}}\widehat{f}\left(\frac{x}{a} \right).\]

11. Show that the operation of convolution behaves much like multiplication of numbers: it is commutative, associative, and distributive over addition:

\[f*g=g*f,\qquad\quad f*(g*h)=(f*g)*h,\qquad\quad f*(g+h)=f*g+f*h.\]

12. Consider the convolution

\[(f*g)(x)=\int_{\mathbb{R}}f(x-y)g(y)dy.\]

1. Show that \(f*g\) is uniformly continuous when \(f\) is integrable and \(g\) is bounded.
2. If in addition \(g\) is integrable, show that \((f*g)(x)\to 0\) as \(|x|\to\infty\).

13. Let \(f\) be twice differentiable real-valued function on \([0,2\pi]\) with

\[\int_{0}^{2\pi}f(x)\,dx=0=f(2\pi)-f(0).\]

Show that

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