Question

A general form of Parseval's theorem says that if two functions are expanded in Fourie? series $$ \begin{aligned} &f(x)=\frac{1}{2} a_{0}+\sum_{1}^{x} a_{n} \cos n x+\sum_{1}^{\infty} b_{n} \sin n x \\ &g(x)=\frac{1}{2} a_{0}^{\prime}+\sum_{1}^{\infty} a_{n}^{\prime} \cos n x+\sum_{i}^{\infty} b_{n}^{\prime} \sin n x \end{aligned} $$ then the average value of \(f(x) g(x)\) is \(\frac{1}{4} a_{0} a_{0}^{\prime}+\frac{1}{2} \sum_{1}^{\infty} a_{n} a_{n}^{\prime}+\frac{1}{2} \sum_{1}^{\infty} b_{n} b_{n}^{\prime}\). Prove this.

Short answer

Average value of \(f(x)g(x)\) is \(\frac{1}{4}a_0a_0' + \frac{1}{2}\sum_{1}^{\infty}a_n a_n' + \frac{1}{2}\sum_{1}^{\infty}b_n b_n'\).

Step-by-step solution

Understand the Fourier Series Expansion

Given the Fourier series of two functions: \[\begin{aligned}&f(x) = \frac{1}{2}a_0 + \sum_{n=1}^{\infty} a_n \cos(nx) + \sum_{n=1}^{\infty} b_n \sin(nx), \&g(x) = \frac{1}{2}a_0' + \sum_{n=1}^{\infty} a_n' \cos(nx) + \sum_{n=1}^{\infty} b_n' \sin(nx)\end{aligned}\]Here, the coefficients \(a_n, b_n, a_n', b_n'\) are the Fourier coefficients for \(f(x)\) and \(g(x)\) respectively.

Compute the Product \(f(x) g(x)\)

To find the average value of the product \(f(x)g(x)\), first compute the product:\[\begin{aligned}f(x)g(x) &= \left( \frac{1}{2}a_0 + \sum_{n=1}^{\infty} a_n \cos(nx) + \sum_{n=1}^{\infty} b_n \sin(nx) \right) \left( \frac{1}{2}a_0' + \sum_{n=1}^{\infty} a_n' \cos(nx) + \sum_{n=1}^{\infty} b_n' \sin(nx) \right)\end{aligned}\]

Expand the Product

Expand the product \(f(x)g(x)\) to separate terms involving \(a_0, a_0', a_n, a_n', b_n, b_n'\):\[\begin{aligned}f(x)g(x) &= \frac{1}{4}a_0a_0' + \frac{1}{2}a_0 \sum_{n=1}^{\infty} (a_n' \cos(nx) + b_n' \sin(nx)) + \frac{1}{2}a_0' \sum_{n=1}^{\infty} (a_n \cos(nx) + b_n \sin(nx)) \&\quad + \left( \sum_{n=1}^{\infty} a_n \cos(nx) + \sum_{n=1}^{\infty} b_n \sin(nx) \right) \left( \sum_{m=1}^{\infty} a_m' \cos(mx) + \sum_{m=1}^{\infty} b_m' \sin(mx) \right)\end{aligned}\]

Simplify Using Orthogonality

Simplify the expanded product using the orthogonality of sine and cosine functions. Orthogonal properties state that the average value over one period \( \frac{1}{2\pi} \int_{-\pi}^{\pi} \cos(nx) \cos(mx) \, dx = \frac{1}{2} \delta_{mn} \) and similarly for sine terms. Simplifying, we get:\[\frac{1}{4}a_0a_0' + \frac{1}{2} \sum_{n=1}^{\infty} a_n a_n' + \frac{1}{2} \sum_{n=1}^{\infty} b_n b_n'\]

Conclude the Derivation

By applying the orthogonality conditions, the average value of \(f(x) g(x)\) over one period is:\[\frac{1}{2\pi} \int_{-\pi}^{\pi} f(x)g(x) dx = \frac{1}{4}a_0a_0' + \frac{1}{2} \sum_{n=1}^{\infty} a_n a_n' + \frac{1}{2} \sum_{n=1}^{\infty} b_n b_n'\]We have hence proven the given expression using Parseval's theorem.

Key concepts

Fourier Series

Understanding Fourier series is essential for analyzing periodic functions. A Fourier series is a way to represent a function as a sum of sine and cosine terms. Specifically, any periodic function can be expressed as a combination of sines and cosines with varying frequencies and amplitudes. By breaking down a function into its constituent sine and cosine waves, we can analyze and manipulate it more effectively. This is particularly useful in signal processing and for solving differential equations. The standard form of a Fourier series for a function \(f(x)\) can be written as:
\[f(x) = \frac{1}{2} a_0 + \sum_{n=1}^{\infty} a_n \cos(nx) + \sum_{n=1}^{\infty} b_n \sin(nx)\]
Where \(a_n\) and \(b_n\) are Fourier coefficients determined by integrating the function over one period. These coefficients give the sine and cosine terms their amplitudes, making the Fourier series a powerful tool for studying periodic phenomena.

Orthogonality of Sine and Cosine Functions

The orthogonality of sine and cosine functions is a fundamental concept in Fourier series and other areas of mathematical analysis. Orthogonality means that, over one period, the integral of the product of different sine or cosine terms is zero. This property is critical for simplifying the computation of Fourier coefficients. Specifically, for two functions \(\cos(nx)\) and \(\cos(mx)\)
\[\int_{-\pi}^{\pi} \cos(nx) \cos(mx) \, dx =\, 0\, \text{\,(if}\, n \eq m)\, \text{and}\, \pi\, \text{\,(if}\, n = m)\]
Similarly, the integral of the product of two sine functions is zero, and the integral of the product of a sine and a cosine function is also zero. This orthogonality enables us to isolate individual terms in a Fourier series when computing the average value of the product of two functions. By leveraging orthogonality, we can greatly simplify complex expressions and obtain meaningful results.

Average Value of Function Products

The average value of the product of two functions over one period is a useful metric in many fields, including physics and engineering. When analyzing two periodic functions, \(f(x)\) and \(g(x)\), given by their Fourier series expansions, the average value of their product \(f(x)g(x)\) can be computed using Parseval's theorem. According to Parseval's theorem, the average value involves the sum of the products of their respective Fourier coefficients. For functions expanded in Fourier series:
\[f(x) = \frac{1}{2} a_0 + \sum_{n=1}^{\infty} a_n \cos(nx) + \sum_{n=1}^{\infty} b_n \sin(nx)\]
\[g(x) = \frac{1}{2} a_0^{'} + \sum_{n=1}^{\infty} a_n^{'} \cos(nx) + \sum_{n=1}^{\infty} b_n^{'} \sin(nx)\]
The average value of the product \(f(x)g(x)\) over one period is:
\[\frac{1}{2\pi} \int_{-\pi}^{\pi} f(x)g(x)dx = \frac{1}{4}a_0a_0^{'} + \frac{1}{2} \sum_{n=1}^{\infty} a_n a_n^{'} + \frac{1}{2} \sum_{n=1}^{\infty} b_n b_n^{'}\]
This result is derived by expanding the product using Fourier series and applying the orthogonality of sine and cosine functions.