Question

22\. Since \(F\) commutes with rotations, the Fourier transform of a radial function is radial; that is, if \(F \in L^{1}\left(\mathbb{R}^{n}\right)\) and \(F(x)=f(|x|)\), then \(\widehat{F}(\xi)=g(|\xi|)\), where \(f\) and \(g\) are related as follows. a. Let \(J(\xi)=\int_{S} e^{i x \xi} d \sigma(x)\) where \(\sigma\) is surface measure on the unit sphere \(S\) in \(\mathbb{R}^{n}\) (Theorem 2.49). Then \(J\) is radial - say, \(J(\xi)=j(|\xi|)\) - and \(g(\rho)=\int_{0}^{\infty} j(2 \pi r \rho) f(r) r^{n-1} d r .\) b. \(J\) satisfies \(\sum_{1}^{n} \partial_{k}^{2} J+J=0\). c. \(j\) satisfies \(\rho j^{\prime \prime}(\rho)+(n-1) j^{\prime}(\rho)+\rho j(\rho)=0\). (This equation is a variant of Bessel's equation. The function \(j\) is completely determined by the fact that it is a solution of this equation, is smooth at \(\rho=0\), and satisfies \(j(0)=\sigma(S)=\) \(2 \pi^{n / 2} / \Gamma(n / 2)\). In fact, \(j(\rho)=(2 \pi)^{n / 2} \rho^{(2-n) / 2} J_{(n-2) / 2}(\rho)\) where \(J_{\alpha}\) is the Bessel function of the first kind of order \(\alpha .)\) d. If \(n=3, j(\rho)=4 \pi \rho^{-1} \sin \rho\). (Set \(f(\rho)=\rho j(\rho)\) and use (c) to show that \(f^{\prime \prime}+f=0\). Alternatively, use spherical coordinates to compute the integral defining \(J(0,0, \rho)\) directly.) (use induction on \(k\) ), and in particular, $$ h_{k}(x)=\frac{(-1)^{k}}{\left[\pi^{1 / 2} 2^{k} k !\right]^{1 / 2}} e^{x^{2} / 2}\left(\frac{d}{d x}\right)^{k} e^{-x^{2}} . $$ f. Let \(H_{k}(x)=e^{x^{2} / 2} h_{k}(x)\). Then \(H_{k}\) is a polynomial of degree \(k\), called the \(k\) th normalized Hermite polynomial. The linear span of \(H_{0}, \ldots, H_{m}\) is the set of all polynomials of degree \(\leq m\). (The kth Hermite polynomial as usually defined is \(\left[\pi^{1 / 2} 2^{k} k !\right]^{1 / 2} H_{k}\).) g. \(\left\\{h_{k}\right\\}_{0}^{\infty}\) is an orthonormal basis for \(L^{2}(\mathbb{R})\). (Suppose \(f \perp h_{k}\) for all \(k\), and let \(g(x)=f(x) e^{-x^{2} / 2}\). Show that \(\widehat{g}=0\) by expanding \(e^{-2 \pi i \xi \cdot x}\) in its Maclaurin series and using (f).) h. Define \(A: L^{2} \rightarrow L^{2}\) by \(A f(x)=(2 \pi)^{1 / 4} f(x \sqrt{2 \pi})\), and define \(\tilde{f}=\) \(A^{-1} \mathcal{F} A f\) for \(f \in L^{2}\). Then \(A\) is unitary and \(\widetilde{f}(\xi)=(2 \pi)^{-1 / 2} \int f(x) e^{-i \xi x} d x\). Moreover, \(\widetilde{T f}=-i T(\widetilde{f})\) for \(f \in \mathcal{S}\), and \(\widetilde{h}_{0}=h_{0}\); hence \(\widetilde{h}_{k}=(-i)^{k} h_{k}\). Therefore, if \(\phi_{k}=A h_{k} .\left\\{\phi_{k}\right\\}_{0}^{\infty}\) is an orthonormal basis for \(L^{2}\) consisting of eigenfunctions for \(\mathcal{F}\); namely, \(\widehat{\phi}_{k}=(-i)^{k} \phi_{k}\).

Short answer

The Fourier transform of a radial function is also radial, and for \( n=3 \), \( j(\rho) = 4 \pi \rho^{-1} \sin \rho \).

Step-by-step solution

Understanding the Fourier Transform of Radial Functions

A radial function is one that depends only on the distance from the origin, i.e., it is invariant under rotations. The Fourier transform of such a function in a given space is also radial. The original radial function, denoted as \( F(x) = f(|x|) \), depends only on the magnitude of \( x \). After taking the Fourier transform, \( \widehat{F}(\xi) = g(|\xi|) \) holds, meaning the transform depends solely on the magnitude of \( \xi \).

Determining the Integral \( J(\xi) \)

We are given that the integral \( J(\xi) = \int_{S} e^{i x \xi} d \sigma(x) \) is radial and can be expressed as \( J(\xi) = j(|\xi|) \). This implies that \( j \) is a function only of \( |\xi| \), and \( g(\rho) = \int_{0}^{\infty} j(2 \pi r \rho) f(r) r^{n-1} d r \).

Derivation and Solution of Bessel's Equation

\( J(\xi) \) satisfies a PDE related to Bessel's equation: \( \sum_{1}^{n} \partial_{k}^{2} J + J = 0 \). The function \( j(\rho) \) is treated within the framework of Bessel functions since it satisfies \( \rho j''(\rho) + (n-1) j'(\rho) + \rho j(\rho) = 0 \). A solution for \( j \) considering boundary conditions is \( j(\rho) = (2\pi)^{n/2} \rho^{(2-n)/2} J_{(n-2)/2}(\rho) \).

Special Case for \( n=3 \)

For \( n = 3 \), the specific form of \( j(\rho) \) given is \( j(\rho) = 4\pi \rho^{-1} \sin \rho \). This is derived from simplifying Bessel's equation and its boundary conditions. Additionally, this example uses spherical coordinates or sets an auxiliary function \( f(\rho) \), i.e., \( f(\rho) = \rho j(\rho) \), to show \( f'' + f = 0 \), confirming the form with trigonometric functions.

Key concepts

Radial Functions

Radial functions are fascinating mathematical constructs. When we say a function is radial, it means the function's value at a point depends solely on the distance of that point from the origin, not the direction. Imagine dropping a pebble in a pond and the ripples move outwards uniformly. Each ripple can be visualized as a radial function in two dimensions. Mathematically, if a function is radial, it can be represented as \( F(x) = f(|x|) \), where \(|x|\) is the Euclidean distance from the origin to the point \( x \).
Given this nature, the Fourier transform of a radial function is also radial. This means that if a function depends only on its distance from the origin, so does its transform. This property is incredibly useful in physics and engineering, where symmetrical problems often lead to radial functions. In these cases, dealing with the Fourier transform can simplify computations because you can handle the function based on radius alone, reducing the complexity of multidimensional problems.
In Fourier terminology, if \( F(x) = f(|x|) \), then \( \widehat{F}(\xi) = g(|\xi|) \). This implies symmetry in the frequency domain as well.

Bessel Functions

Bessel functions are a family of solutions to a particular type of differential equation known as Bessel's equation, which appears in a wide range of physical problems, especially those involving radial symmetry. These functions are used to model situations like heat conduction in cylindrical objects or wave propagation in circular objects.
There's a standard form of Bessel's equation:\[\rho j''(\rho) + (n-1) j'(\rho) + \rho j(\rho) = 0,\]where \( j \) is the function in question. Specific solutions to this equation, known as Bessel functions of the first kind, are denoted \( J_\alpha(\rho) \), where \( \alpha \) is the order of the Bessel function. These functions exhibit oscillatory behavior similar to sine and cosine functions but are damped or amplified based on distance from the origin.
In the exercise, it's given that the function \( j \) is described in terms of a Bessel function to satisfy specific boundary conditions and smoothness criteria. This relation is essential in analyzing radial wave equations and many other applications where cylindrical or spherical symmetry is present.

Hermite Polynomials

Hermite polynomials are a sequence of orthogonal polynomials that arise in probability theory, combinatorics, and particularly in quantum mechanics as solutions to the quantum harmonic oscillator. They are especially useful because of their relationship with Gaussian distributions and their orthogonality property.
The Hermite polynomial of degree \( k \) is given by:\[H_k(x) = e^{x^2 / 2} h_k(x),\]where \( h_k(x) \) is obtained using derivatives of an exponential function as shown in the exercise text. These functions form a complete orthonormal basis in the space \( L^2(\mathbb{R}) \), meaning any function in this space can be expressed as a series of Hermite polynomials.
The orthogonality property of Hermite polynomials means that their integrals over all space, weighted by a Gaussian function, are zero unless they are the same polynomial. This property is incredibly beneficial for simplifying integral calculations in physics and engineering applications.
Their involvement in quantum mechanics further underscores their importance, providing a convenient mathematical framework for modeling wavefunctions in potential fields.