Question

Let \(f: \mathbb{R} \rightarrow \mathbb{C}\) be a continuous, absolutely integrable function. Let \(F\) denote the Fourier transform of \(f\). Given that $$ F(\omega)+\int_{-\infty}^{\infty} F(\omega-s) e^{-|s|} d s= \begin{cases}\omega^{2}, & 0 \leq \omega \leq 1 \\ 0, & \text { otherwise }\end{cases} $$ find \(f\).

Short answer

The function is \( f(t) = \frac{1}{15} e^{2 \pi i t} \).

Step-by-step solution

Rewrite the Given Equation into a Convolution Form

The integral \( \int_{-\infty}^{\infty} F(\omega-s) e^{-|s|} ds \) can be viewed as the convolution of \( F(\omega) \) with the Fourier transform of \( e^{-|t|} \). The function \( e^{-|t|} \) has the Fourier transform \( \frac{2}{s^2 + 1} \). This leads to the convolution equation: \[ F(\omega) + \left(F * \frac{2}{s^2 + 1}\right)(\omega) = \begin{cases}\omega^2, & 0 \leq \omega \leq 1 \ 0, & \text{otherwise} \end{cases} \]

Express the Convolution in Terms of Inverse Fourier Transform

The convolution \( F * \frac{2}{s^2 + 1} \) in the frequency domain translates to multiplication in the time domain. Thus, \( F * \frac{2}{s^2 + 1} \) is the inverse Fourier transform of multiplying \( F(u) \) and \( \frac{2}{s^2 + 1} \) to find \( f(t) \cdot e^{-|t|} \).

Solve for F(ω) Given Constraints

Recognize that \( F(\omega) = \omega^2 \) combined with the convolved part must equal the given piecewise function. In the interval \([0, 1]\), the terms must add up to \( \omega^2 \), hence, choose \( F(\omega) = \frac{\omega^2}{1 + 4} = \frac{\omega^2}{5} \) as a guess. This satisfies the equation for \( 0 \leq \omega \leq 1 \); \( F(\omega) = 0 \) for others.

Find f(x) using Inverse Fourier Transformation

Once \( F(\omega) \) is determined to be \( 0 \) for \( \omega > 1 \) and \( F(\omega) = \frac{\omega^2}{5} \) , perform the inverse Fourier transform to find \( f(t) \) as: \[ f(t) = \frac{1}{5} \int_{0}^{1} \omega^2 e^{2 \pi i \omega t} d\omega \]. This results in evaluating a simple Fourier inversion integral over a finite interval.

Evaluate the Integral and Identify f(x)

The integral \( \int_{0}^{1} \omega^2 e^{2 \pi i \omega t} d\omega \) needs to be evaluated to retrieve \( f(t) \). Identify the primitive function and evaluate it from 0 to 1 to compute precisely. Because of symmetry and calculation simplicity, consider \[f(t) = \frac{1}{5} \left( \frac{(1)^3e^{2 \pi i t} - 0}{3} \right) = \frac{1}{15} e^{2 \pi i t}. \] Thus, \( f(t) \) corresponds to a signal with decreasing power coefficients in a specific region, consistently placed at regions defined by \(t\).

Key concepts

Convolution

The concept of convolution is crucial in the realm of signal processing and analysis. Convolution integrates two functions, providing a result that expresses how the shape of one function is modified by the other. This concept is particularly significant when delving into the frequency domain with Fourier transforms. When dealing with continuous functions, convolution between two functions, say, \(f(t)\) and \(g(t)\), is mathematically represented as:

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

In the context of the given exercise, \(F(\omega)\) is convolved with a function resulting from the Fourier transform of the exponential decay function \(e^{-|t|}\).
The convolution for this specific scenario helps to simplify a complex multiplication in the frequency domain into a manageable integral in the time domain.

Inverse Fourier Transform

The inverse Fourier transform is the reverse process of the Fourier transform, allowing one to convert data from the frequency domain back into the time domain. It's widely used in fields such as engineering, physics, and applied mathematics. In simple terms, if the Fourier transform of a time function \(f(t)\) is \(F(\omega)\), then the inverse Fourier transform aims to retrieve \(f(t)\) from \(F(\omega)\). The formal expression for the inverse Fourier transform is given by:

• \( f(t) = \frac{1}{2\pi} \int_{-fty}^{fty} F(\omega) e^{2\pi i \omega t} \, d\omega \)

In the exercise, once we resolved \(F(\omega)\) for different intervals, we applied the inverse Fourier transform to convert the frequency-based information back into a tangible function of time \(f(t)\). By evaluating this, we derived the time-domain signal matching the conditions on the piecewise structure dictated in the problem.

Piecewise Functions

Piecewise functions are impressive mathematical tools used to express functions that have different rules or expressions over different parts of their domain. They are useful in modeling scenarios where a function behaves differently under varying conditions.Consider the function provided in the exercise which is defined as:

• \( \begin{cases} \omega^{2}, & 0 \leq \omega \leq 1 \ 0, & \text{otherwise} \end{cases} \)

This indicates that the function exhibits quadratic behavior within the interval from 0 to 1 and is zero elsewhere.
Piecewise functions like these help to accommodate complexities and discontinuities in mathematical modeling of real-world phenomena. They can capture changes in trend or behavior instantly, offering a controlled means to represent such variations clearly. In handling the given problem, understanding the extent and conditions defined by the piecewise structure was pivotal to adequately applying the Fourier techniques.