Question

For each \(a>0\) we define the functions $$ f_{a}(x)=\left\\{\begin{array}{ll} 1, & |x|

Short answer

\((f_a * f_a)(x) = 2a \mathbb{1}_{|x| < 2a}\) and \((g_a * g_a)(x)\) is a triangular function with width \(2a\).

Step-by-step solution

Understanding Convolution

To find the convolution of two functions, we use the formula: \\[ (f * g)(x) = \int_{-\infty}^{\infty} f(t) g(x-t) \, dt. \] \We will apply this definition to both \( f_a(x) \) and \( g_a(x) \).

Calculating Convolution for \( f_a * f_a \)

Start by applying the definition of convolution: \\[ (f_a * f_a)(x) = \int_{-\infty}^{\infty} f_a(t) f_a(x-t) \, dt. \] \Given that \( f_a(x) = 1 \) for \(|x|<a\) and \( f_a(x) = 0 \) otherwise, the interval of integration is from \([-a, a]\) where both \(|t| < a\) and \(|x-t| < a\). This gives the condition: \\[ |t| < a \quad \text{and} \quad |x-t| < a. \] \The resulting condition simplifies to \(|x| < 2a\) and thus, \\[ (f_a * f_a)(x) = 2a \cdot \mathbb{1}_{|x| < 2a}, \] \where \( \mathbb{1} \) denotes the indicator function.

Setting Up Convolution for \( g_a * g_a \)

Now consider \( g_a(x) = 1 - \frac{|x|}{a} \) for \(|x|<a\) and \( g_a(x) = 0 \) otherwise. Apply the convolution definition: \\[ (g_a * g_a)(x) = \int_{-\infty}^{\infty} g_a(t) g_a(x-t) \, dt. \] \Integrate over the interval \([-a, a]\) under the condition \(|t| < a\) and \(|x-t| < a\).

Integrating for \( g_a * g_a \)

To handle the integral, note: \\[ g_a(t) g_a(x-t) = \left(1 - \frac{|t|}{a}\right)\left(1 - \frac{|x-t|}{a}\right). \] \The integral simplifies to: \\[ (g_a * g_a)(x) = \int_{-a}^{a} \left(1 - \frac{|t|}{a}\right)\left(1 - \frac{|x-t|}{a}\right) dt. \] \After evaluating this piecewise integral across the intervals \(-a < x < a\), \(-a < x - t < a\), and \(t < a\), the result is a triangular function that peaks at \(x=0\) and width \(2a\).

Conclusion

Thus, upon evaluating the integrals, we find that: \\[ (f_a * f_a)(x) = 2a \cdot \mathbb{1}_{|x| < 2a} \] \and \\[ (g_a * g_a)(x) \text{ is a triangular function with maximum }\frac{a}{2} \text{ at } x = 0\text{ and width } 2a. \]

Key concepts

Fourier series

Fourier series are a fascinating tool in mathematical analysis. They allow us to express periodic functions as an infinite sum of sine and cosine functions.
This concept is especially significant when analyzing signals or any form of wave behavior, such as sound or light.
Imagine you're trying to recreate a complex wave. Instead of trying to replicate every single point, Fourier series lets you construct it using basic waves—sines and cosines. These basic waves are like the building blocks of many types of functions.

The main advantage of a Fourier series is its ability to handle a variety of functions, including those that are not necessarily smooth or continuous.

• The Fourier series can approximate any function if the function is periodic.
• It breaks down complicated functions into simple oscillatory functions.
• This is incredibly useful in engineering and physics.

When applying Fourier series to piecewise functions—like we have in our original exercise—this tool becomes a perfect fit. This is because piecewise functions often exhibit abrupt changes. Fourier series smooth these out by capturing the essence of the function’s periodic components.

piecewise functions

Piecewise functions are a type of function defined by multiple sub-functions, each applied to a specific interval. They are quite common in mathematics and applicable in many real-world situations.
Each piece of the function can be a different expression. It allows for flexibility when modeling scenarios with varying conditions.

In our exercise, piecewise functions model behaviors that change based on the interval of x-values:

• For example, the function \( f_{a}(x) \) equals 1 for \(|x| < a\) and 0 otherwise.
• This kind of function is useful for modeling systems where outcomes depend on a conditional range of inputs.

Working with piecewise functions often involves ensuring continuity where different pieces meet. However, not all piecewise functions are continuous, which can create fascinating points of analysis.

Practically, they are used in control systems or signal processing, where systems react differently under varying conditions.

integration techniques

Integration techniques are crucial when working with convolutions, like in our original problem.
The fascinating aspect of convolution is how it blends two functions to produce a third function. It involves integrating the product of one function with a shifted version of the other.

Let's look at some integration techniques that stand out when dealing with convolutions:

• **Substitution Method**: Useful when modifying the integration variables to simplify the expression.
• **By Parts**: Applicable when dealing with products of functions, breaking the problem into more manageable parts.
• **Piecewise Integration**: Especially relevant in our case, where functions differ across intervals.

In the exercise, integration is performed over intervals determined by the definitions of \( f_{a}(x) \) and \( g_{a}(x) \). This piecemeal approach enables handling the function’s discontinuities.
Handling integration in cases like these often involves identifying the boundaries carefully, ensuring each piece is evaluated accurately. This maintains the integrity of piecewise intents and produces results true to both function limits.