Question

For each \(a>0\), let \(f_{a}(x)=e^{-a|x|}\), and \(g_{a}(x)=\frac{2 a}{x^{2}+a^{2}}\). (a) Find the Fourier transform of \(f_{a}\). (b) Find the Fourier transform of \(g_{a}\). (c) Does there exist a function \(\varphi \in G(\mathbb{R})\) such that \(\int_{-\infty}^{\infty} \frac{\varphi(t)}{(x-t)^{2}+16} d t=\) \(\frac{1}{x^{2}+49}\) ? If yes, find it. If no, explain why it cannot exist. (d) Does there exist a function \(\varphi \in G(\mathbb{R})\) such that \(\int_{-\infty}^{\infty} \frac{\varphi(t)}{(x-t)^{2}+49} d t=\) \(\frac{1}{x^{2}+16}\) ? If yes, find it. If no, explain why it cannot exist.

Short answer

(a) \( F(f_a)(\nu) = \frac{2a}{a^2 + (2\pi \nu)^2} \). (b) \( F(g_a)(\nu) = 2\pi e^{-2\pi a |\nu|} \). (c) No such \( \varphi \). (d) No such \( \varphi \).

Step-by-step solution

Set up the Fourier Transform for f_a(x)

The Fourier transform of a function \( f(x) \) is given by \( F(f)(u) = \int_{-\infty}^{\infty} f(x)e^{-i2\pi u x} \,dx \). For \( f_a(x) = e^{-a|x|} \), this translates to \( F(f_a)(u) = \int_{-\infty}^{\infty} e^{-a|x|}e^{-i2\pi u x} \,dx \). Note that we will have to consider the integral for both positive and negative \( x \) separately.

Evaluate the Integral for f_a(x)

Split the integral into two parts, \( x \geq 0 \) and \( x < 0 \). For \( x \geq 0 \), the integral becomes \( \int_{0}^{\infty} e^{-ax} e^{-i2\pi u x} \,dx = \int_{0}^{\infty} e^{-(a + i2\pi u) x} \,dx \). For \( x < 0 \), it becomes \( \int_{-\infty}^{0} e^{ax} e^{-i2\pi u x} \,dx = \int_{-\infty}^{0} e^{(a - i2\pi u) x} \,dx \). Each integral is evaluated using the formula \( \int e^{cx} \,dx = \frac{e^{cx}}{c} \).

Result of the Fourier Transform of f_a(x)

After evaluating, you get \( F(f_a)(u) = \frac{1}{a + i2\pi u} + \frac{1}{a - i2\pi u} \), simplifying to \( \frac{2a}{a^2 + (2\pi u)^2} \).

Set up the Fourier Transform for g_a(x)

The function \( g_a(x) = \frac{2a}{x^2 + a^2} \) resembles the form of a Lorentzian distribution. Its Fourier transform is known to be \( F(g_a)(u) = 2\pi e^{-2\pi a |u|} \).

Confirm Fourier Transform of g_a(x)

The Fourier transform \( F(g_a)(u) = 2\pi e^{-2\pi a |u|} \) results because \( g_a(x) \) essentially describes a damped distribution, and its Fourier transform reduсes to an exponential function.

Analyze Function Existence for Part (c)

For the equation \( \int_{-\infty}^{\infty} \frac{\varphi(t)}{(x-t)^2 + 16} \,dt = \frac{1}{x^2 + 49} \), consider the transform properties of convolution and inverses. This represents a convolution in the form \( F^{-1}(\frac{1}{x^2 + 16}) \ast \varphi = \frac{1}{x^2 + 49} \). By comparison, such \( \varphi(x) \) cannot exist since these forms describe fundamentally different transformation properties.

Analyze Function Existence for Part (d)

For the equation \( \int_{-\infty}^{\infty} \frac{\varphi(t)}{(x-t)^2 + 49} \,dt = \frac{1}{x^2 + 16} \), analogous reasoning applies as in Part (c). The setup, convolution, and existing transforms again contradict the possibility of \( \varphi(x) \)'s existence, given the differing magnitude in damping terms.

Key concepts

Integral Transforms

Integral transforms are mathematical operations that convert functions into a different domain, often to simplify analysis or solve differential equations more easily. The Fourier Transform is a critical example of an integral transform. This transform takes a time-domain function and expresses it in terms of frequencies. By examining how different frequencies combine, we gain insights into the original function's behavior.

The general form for a Fourier Transform of a function \( f(x) \) is given by:

• \( F(f)(u) = \int_{-\infty}^{\infty} f(x) e^{-i2\pi ux} \,dx \)

This equation tells us how much oscillation at a given frequency \( u \) is present in the original function.

When applying the Fourier Transform to a specific function, we generally deal with two parts of the calculation: understanding the impact of complex numbers (through \( e^{-i2\pi ux} \)) and solving the integral over the entire real number line. These transforms often break down complex problems into manageable frequency components.

Convolution

Convolution is a mathematical operation that combines two functions to produce a third function. It reflects how the shape of one function is "smeared" by another. For two functions \( f(t) \) and \( g(t) \), their convolution \( f * g \) is defined as:

• \( (f * g)(t) = \int_{-\infty}^{\infty} f(\tau) g(t - \tau) \,d\tau \)

This operation is pivotal in systems analysis and signal processing as it represents the effect of a linear time-invariant system on a signal.

The convolution also plays a significant role in solving differential equations and understanding the product of their Fourier Transforms. In the context of the given problem, convolution allows one to interpret expressions like \( \int_{-\infty}^{\infty} \frac{\varphi(t)}{(x-t)^{2}+16} \,dt \) as finding how one function modifies another over an entire domain.

When the right-hand side of such expressions differs significantly from what is attained through convolution, it indicates the non-existence of the function \( \varphi \) that would satisfy the equation.

Lorentzian Distribution

The Lorentzian Distribution, also known as the Cauchy distribution, is a type of probability distribution. It is characterized by its peak at the center and long tails, which describe the likelihood of observing various values. This distribution is often used in physics and signal processing due to its shape.

Mathematically, the Lorentzian function is given by the expression:

• \( g_a(x) = \frac{2a}{x^2 + a^2} \)

This form represents a typical Lorentzian shape, peaked at \( x=0 \) with "a" regulating width. Its Fourier Transform is known to simplify into an exponential function, often reducing computational complexity.

In terms of integral transforms, its Fourier Transform reveals that the smooth peak correlates with a rapidly decaying exponential function in the frequency domain, providing insights into how quickly the amplitudes of different frequencies reduce.

Exponential Function

The exponential function is one of the most significant mathematical functions, represented by \( e^{x} \). In our context, exponentials appear prominently both in the setup and solution of Fourier Transforms.

The function \( f_{a}(x)=e^{-a|x|} \) is an exponential that decays as \( x \) moves away from zero, with the decay rate controlled by \( a \). This decaying exponential is crucial in damping oscillations and is frequently used in signal processing and differential equations.

The exponential function also manifests in the solution of transforms, such as \( 2\pi e^{-2\pi a |u|} \). Here, it illustrates how particular patterns and frequencies diminish leading to simplifications.

• This behavior reflects how the exponential functions handle decay and growth phenomena in both spatial and temporal contexts.
• They also play a role in converting linear decreases into exponential reductions through integral transformations.

Exponential functions appear ubiquitously in both natural sciences and engineering applications, due to their fundamental properties related to growth and decay dynamics.