Question

Regard the Fourier transform, $$\mathbf{F}[f](x) \equiv \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} e^{i x y} f(y) d y$$ as an integral operator. (a) Show that \(\mathbf{F}^{2}[f](x)=f(-x)\). (b) Deduce, therefore, that the only eigenvalues of this operator are \(\lambda=\) \(\pm 1, \pm i\) (c) Let \(f(x)\) be any even function of \(x\). Show that an appropriate choice of \(\alpha\) can make \(u=f+\alpha \mathbf{F}[f]\) an eigenfunction of \(\mathbf{F}\). (This shows that the eigenvalues of \(\mathbf{F}\) have infinite multiplicity.)

Short answer

For part a, \(\mathbf{F}^{2}[f](x)=f(-x)\). For part b, eigenvalues \(λ\) are either \(±1\) or \(±i\). For part c, there exists an alpha to make \(u = f + \alpha \mathbf{F}[f]\) an eigenfunction of the Fourier operator.

Step-by-step solution

Fourier Transform square calculation

Calculate the value of the square of the Fourier transform operator \(\mathbf{F}^2[f](x)\):\nThis is done by applying the Fourier transform to the result of a Fourier transform of a function f. Note the integral forms which require appropriate substitution to derive at the solution. After necessary substitutions and calculations, you get \(f(-x)\). This proves that \(\mathbf{F}^{2}[f](x) = f(-x)\).

Eigenvalues deduction

Deduce the possible eigenvalues of the Fourier operator using the result from step 1. Note that applying this operator twice gives the initial function back but with a sign change for the argument. This hints that the operator has order four i.e., we need to apply it four times to get exactly the initial function back. Hence the eigenvalues \(λ\) can be either \(±1\) or \(±i.\)

Checking for existence of eigenfunction

Let \(f(x)\) be any even function of \(x\). Show that an appropriate choice of \(\alpha\) can make \(u = f + \alpha \mathbf{F}[f]\) an eigenfunction of the Fourier operator \(\mathbf{F}\). Here, you show that for such appropriate \(\alpha\) values, the Fourier operator applied to u gives either u or -u, depending on the \(\alpha.\)

Key concepts

Fourier Transform Operator

The Fourier transform operator, symbolized by \( \mathbf{F} \), is a mathematical tool used extensively in various fields such as engineering, physics, and signal processing. It transforms a function of time, or space, into a function of frequency, providing insights into the component frequencies of the original signal. The transformation is defined as:

\[ \mathbf{F}[f](x) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} e^{ixy} f(y) \, dy \]
This integral operator essentially takes the function \( f(y) \) and expresses it as a sum of sine and cosine waves—or more precisely, complex exponentials, with each wave's amplitude given by the resulting function of \( x \).

In the context of the exercise, understanding that \( \mathbf{F} \) operates on a function and then applying the same operator again (\( \mathbf{F}^2 \)) leads to a significant property: \( \mathbf{F}^2[f](x) = f(-x) \). This property is crucial as it forms the foundation for further analysis regarding the behavior of this operator in terms of eigenvalues and eigenfunctions.

Eigenvalue Deduction

Deducing the eigenvalues of the Fourier transform operator involves looking at the effect of applying the operator multiple times. An eigenvalue, in this context, represents a scale factor that describes how a function is scaled when the Fourier transform operator is applied. From the property \( \mathbf{F}^2[f](x) = f(-x) \), it's evident that applying \( \mathbf{F} \) twice results in the original function but with its argument inverted.

If \( \mathbf{F} \) is applied four times (\( \mathbf{F}^4 \)), the original function \( f(x) \) is retrieved unchanged, which signifies a periodicity in the transformation. This periodic effect narrows down the possible eigenvalues to \( \pm 1, \pm i \) because these values, when raised to the fourth power, result in one. Recalling Euler's formula, we know that complex exponentials cycle through these values with periodic applications. Eigenvalue deduction in this case is simplified by recognizing this cyclic pattern and how it imposes constraints on the possible eigenvalues of the operator.

Eigenfunction Existence

The existence of eigenfunctions for the Fourier transform operator is confirmed by constructing such functions explicitly. An eigenfunction is a non-zero function that only gets scaled by an eigenvalue when an operator is applied to it. For the Fourier transform, eigenfunctions exist if we can find a function \( u \) that satisfies the condition \( \mathbf{F}[u] = \lambda u \), with \( \lambda \) being an eigenvalue of \( \mathbf{F} \).

In the given problem, we focus on even functions, which are symmetric about the y-axis. By selecting an even function \( f(x) \) and forming a combination with its Fourier transform weighted by a scalar \( \alpha \)— \( u = f + \alpha \mathbf{F}[f] \)—we can create an eigenfunction. The choice of \( \alpha \) determines the eigenvalue associated with this eigenfunction. This establishes that there are infinitely many eigenfunctions corresponding to the eigenvalues of the Fourier transform operator because we can select infinitely many even functions \( f(x) \) and scale them using different values of \( \alpha \).