Question

Assume that \(f\) and \(f^{\prime}\) are continuous, \(f, f^{\prime}, f^{\prime \prime} \in G(\mathbb{R})\), and \(x f(x)\) is also absolutely integrable. Assume that \(f\) satisfies the differential equation $$ f^{\prime \prime}(x)+2 x f^{\prime}(x)+2 f(x)=0 . $$ What differential equation does \(\mathcal{F}[f]\) satisfy?

Short answer

The Fourier Transform \(\mathcal{F}[f]\) satisfies \(-\omega^2 \mathcal{F}[f](\omega) + 2i\omega \frac{d}{d\omega} \mathcal{F}[f](\omega) + 2\mathcal{F}[f](\omega) = 0.\)

Step-by-step solution

Understanding the context

We are provided with a differential equation involving the function \(f\) and its derivatives, specifically \(f''(x) + 2x f'(x) + 2 f(x) = 0\). Our goal is to find what differential equation the Fourier Transform \(\mathcal{F}[f]\) satisfies.

Applying the Fourier Transform

The first step in our solution is to apply the Fourier Transform to each term in the original differential equation. Recall that the Fourier Transform of a derivative \(f'(x)\) is given by \(i \omega \mathcal{F}[f](\omega)\), and for a second derivative, it is \(-\omega^2 \mathcal{F}[f](\omega)\). Additionally, since \(x f(x)\) must be absolutely integrable, we use that the Fourier Transform of \(x \cdot f(x)\) is \(i \frac{d}{d\omega} \mathcal{F}[f](\omega)\).

Transforming each part

Transform each term:1. The Fourier Transform of \(f''(x)\) becomes \(-\omega^2 \mathcal{F}[f](\omega)\).2. The Fourier Transform of \(2x f'(x)\) is \(2i\omega \frac{d}{d\omega} \mathcal{F}[f](\omega)\).3. The Fourier Transform of \(2f(x)\) is simply \(2 \mathcal{F}[f](\omega)\).

Assembling the transformed equation

Combine these results using the original equation:\[-\omega^2 \mathcal{F}[f](\omega) + 2i\omega \frac{d}{d\omega} \mathcal{F}[f](\omega) + 2\mathcal{F}[f](\omega) = 0.\] This is the differential equation that \(\mathcal{F}[f]\) satisfies.

Key concepts

Differential Equations

Differential equations are mathematical equations that involve functions and their derivatives. They're essential in describing the change in different phenomena over time or space. In this exercise, we encounter a second-order differential equation:
\[ f''(x) + 2x f'(x) + 2 f(x) = 0. \]
This specific type of equation involves both the first and second derivatives of a function \( f \) with respect to \( x \).

• **Ordinary Differential Equations (ODEs):** These involve functions of a single variable and their derivatives. The equation provided is an ODE since \( f \) depends only on \( x \).
• **Linear vs. Non-linear:** The given equation is linear because it can be expressed as a linear combination of \( f(x) \), \( f'(x) \), and \( f''(x) \) without multiplying the derivatives together or involving functions of \( f \) itself.

Differential equations like this one model the dynamics of systems and processes in disciplines ranging from physics to engineering. Here, the problem assumes continuity and differentiability conditions for \( f \), ensuring proper application of the Fourier Transform.

Fourier Series

The Fourier series is a method used to express a periodic function as a sum of sine and cosine terms. While the original problem uses the Fourier Transform, it's beneficial to understand how these concepts are related.

• **Relation to Fourier Transform:** The Fourier series is akin to a discrete version of the Fourier Transform, limited to periodic functions. Essentially, the Fourier Transform generalizes the Fourier series for non-periodic functions.
• **Breaking Down Functions:** By decomposing a function into its frequency components, a Fourier series allows for an analysis similar to breaking light into its color spectrum.

Though not directly addressed in the problem, the foundation of Fourier analysis helps in understanding transformations applied to differential equations. This approach aids in simplifying and resolving complex functions into manageable wave components.

Integrable Functions

In the context of the problem, an integrable function is one whose integral over its domain is finite.

• **Absolutely Integrable:** For a function to be absolutely integrable, the integral of its absolute value must be finite. This condition ensures that the Fourier Transform can be properly applied.
• **Relevance in Fourier Transforms:** Functions must be absolutely integrable to apply the Fourier Transform, as this property influences convergence and the ability to analyze various phenomena.

When dealing with the exercise, ensuring that \( x f(x) \) is absolutely integrable guarantees that the Fourier Transform behaves correctly. This makes the function's transformation into the frequency domain feasible, which is crucial when solving differential equations using Fourier analysis.