Question

Let \(f, g \in E\) be \(2 \pi\)-periodic functions, and $$ f(x) \sim \sum_{n=-\infty}^{\infty} a_{n} e^{i n x}, \quad g(x) \sim \sum_{n=-\infty}^{\infty} b_{n} e^{i n x} $$ be the complex Fourier series of \(f\) and \(g .\) For each \(x \in \mathbb{R}\) we define $$ h(x)=\frac{1}{2 \pi} \int_{-\pi}^{\pi} f(x-t) g(t) d t . $$ (a) Prove that \(h\) is piecewise continuous and \(2 \pi\)-periodic. (b) Let \(h(x) \sim \sum_{n=-\infty}^{\infty} c_{n} e^{i n x}\) be the complex Fourier series of \(h\). Prove that \(c_{n}=a_{n} b_{n}\) for all \(n \in \mathbb{Z}\).

Short answer

(a) h(x) is 2π-periodic and piecewise continuous. (b) c_n = a_n b_n for all n.

Step-by-step solution

Prove Piecewise Continuity and Periodicity

First, show that the function \( h(x) \) defined by the convolution \( h(x) = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x-t)g(t) \, dt \) is \(2\pi\)-periodic and piecewise continuous. 1. **Periodicity**: Since both \( f(x) \) and \( g(x) \) are \(2\pi\)-periodic, \( f(x-t) \) and \( g(t) \) remain \(2\pi\)-periodic when integrated over a full period. Therefore, \( h(x + 2\pi) = \frac{1}{2\pi} \int_{-\pi}^{\pi} f((x+2\pi)-t)g(t) \, dt = h(x) \).2. **Piecewise Continuity**: Both \( f(x) \) and \( g(x) \) are assumed piecewise continuous. Convolution of two piecewise continuous functions is also piecewise continuous.

Express Convolution as Fourier Series

Next, express \( h(x) \) in terms of the complex Fourier series coefficients and show that the series for the convolution is a multiplication of two series.The convolution property for Fourier series states that the Fourier coefficient \( c_n \) of the convolution \( h(x) = f * g \) is given by the product of the Fourier coefficients of \( f(x) \) and \( g(x) \). Hence,\[ h(x) \sim \sum_{n=-\infty}^{\infty} c_n e^{inx} \] where \( c_n = a_n b_n \). We derive \( c_n \) by first substituting \( f(x-t) \) and \( g(t) \) with their Fourier expansions:\[ f(x-t) = \sum_{m=-\infty}^{\infty} a_m e^{im(x-t)} \]\[ g(t) = \sum_{k=-\infty}^{\infty} b_k e^{ikt} \]Now substitute into the convolution:\(h(x) = \frac{1}{2\pi} \int_{-\pi}^{\pi}\left( \sum_{m=-\infty}^{\infty} a_m e^{im(x-t)} \right) \left( \sum_{k=-\infty}^{\infty} b_k e^{ikt} \right) \ dt\).

Key concepts

Complex Fourier Series

The complex Fourier series is a fascinating approach to representing periodic functions using exponential terms. For a given function, the series consists of terms

• \( e^{i n x} \), where \( n \) is an integer
• coefficients \( a_n \) that determine the contribution of each harmonic

This series transforms complex signals or functions into a sum of sinusoidal components. Each term in the series correlates to a specific frequency, with the coefficient indicating its amplitude and phase.

A function \( f(x) \) can thus be expressed as: \[f(x) hicksim sim \sum_{n=-\infty }^{\infty } a_n e^{inx}.\]This representation is immensely powerful in engineering and physics, as it simplifies the analysis of periodic signals. The coefficients \( a_n \) are crucial as they encapsulate all necessary information about the original function's frequency content.

Periodic Functions

Periodic functions have a characteristic property of repeating their values at regular intervals. A function \( f(x) \) is called periodic if there exists a positive number \( T \) such that:

• \( f(x + T) = f(x) \) for all \( x \)
• The smallest such \( T \) is known as the period

In our exercise, \( f \) and \( g \) are \( 2\pi\)-periodic functions, meaning they repeat every \( 2\pi \) units. This periodicity is fundamental in forming Fourier series as it allows the functions to be described over one period and extended indefinitely.

Periodic functions can be simple sines and cosines, or more complex forms, but their periodic nature means they are highly predictable. This predictability aids in the study of complex phenomena, as very elaborate waveforms can be analyzed by their periodic components.

Convolution in Analysis

Convolution is a mathematical operation that combines two functions into a third. It is particularly useful in signal processing and analysis.

• In our context, the convolution of two functions,\( f(x) \) and \( g(x) \), results in a new function \( h(x) \)
• Defined by: \[h(x) = \frac{1}{2\pi}\int_{-\pi}^{\pi} f(x-t)g(t) \, dt\]

The integral evaluates the area under the product \( f(x-t) \) and \( g(t) \), essentially blending their properties over each shift.

In Fourier analysis, convolution of periodic functions is linked to multiplication of their Fourier series coefficients. This gives a beautiful symmetry: instead of convolving time-domain signals, their spectra are multiplied, simplifying many problems in engineering. This principle, captured by the statement that \( c_n = a_n b_n \), highlights how convolution in the time domain correlates to simple multiplication in the frequency domain.