Question

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

Short answer

The average value of \( f(x)g(x) \) is \( \frac{1}{4} a_0 a_0' + \frac{1}{2} \sum_{n=1}^{\infty} a_n a_n ' + \frac{1}{2} \sum_{n=1}^{\infty} b_n b_n ' \)

Step-by-step solution

- Represent the product

Express the product of the functions using their Fourier series.Given:\[ f(x) = \frac{1}{2} a_0 + \sum_{n=1}^{\infty} a_n \cos(nx) + \sum_{n=1}^{\infty} b_n \sin(nx) \]\[ g(x) = \frac{1}{2} a_0' + \sum_{n=1}^{\infty} a_n' \cos(nx) + \sum_{n=1}^{\infty} b_n' \sin(nx) \]We want to find the average value of their product, \( f(x)g(x) \).

- Expand the product

Multiply the two Fourier series together:\[ h(x) = f(x)g(x) \]\[ h(x) = \left( \frac{1}{2} a_0 + \sum_{n=1}^{\infty} a_n \cos(nx) + \sum_{n=1}^{\infty} b_n \sin(nx) \right) \left( \frac{1}{2} a_0' + \sum_{n=1}^{\infty} a_n' \cos(nx) + \sum_{n=1}^{\infty} b_n' \sin(nx) \right) \]

- Distribute and collect terms

Expand the expression by distributing all terms:\[ h(x) = \frac{1}{4} a_0 a_0' + \frac{1}{2} a_0 \sum_{n=1}^{\infty} a_n' \cos(nx) + \frac{1}{2} a_0 \sum_{n=1}^{\infty} b_n' \sin(nx) + \frac{1}{2} a_0' \sum_{n=1}^{\infty} a_n \cos(nx) + \frac{1}{2} a_0' \sum_{n=1}^{\infty} b_n \sin(nx) + \sum_{n=1}^{\infty} a_n \cos(nx) \sum_{m=1}^{\infty} a_m' \cos(mx) + \sum_{n=1}^{\infty} a_n \cos(nx) \sum_{m=1}^{\infty} b_m' \sin(mx) + \sum_{n=1}^{\infty} b_n \sin(nx) \sum_{m=1}^{\infty} a_m' \cos(mx) + \sum_{n=1}^{\infty} b_n \sin(nx) \sum_{m=1}^{\infty} b_m' \sin(mx) \]

- Integrate over one period

To find the average value of \( f(x)g(x) \), integrate \( h(x) \) over one period and divide by the period length:\[ \text{Average value of } h(x) = \frac{1}{2\pi} \int_{0}^{2\pi} h(x) \, dx \]When integrating, note that integrals of products of different trigonometric functions over one period are zero. Therefore, only terms with the same arguments survive.

- Simplify using orthogonality

Use the orthogonality relations of sine and cosine functions:\[ \int_{0}^{2\pi} \cos(nx) \cos(mx) \, dx = \begin{cases}2\pi & \text{if } n = m \ 0 & \text{if } n eq m \end{cases} \]\[ \int_{0}^{2\pi} \sin(nx) \sin(mx) \, dx = \begin{cases}2\pi & \text{if } n = m \ 0 & \text{if } n eq m \end{cases} \]\[ \int_{0}^{2\pi} \cos(nx) \sin(mx) \, dx = 0 \]

- Combine surviving terms

The terms that survive are:\[ \frac{1}{4} a_0 a_0' \]\[ \frac{1}{2} \sum_{n=1}^{\infty} a_n a_n ' \]\[ \frac{1}{2} \sum_{n=1}^{\infty} b_n b_n ' \]Thus, the average value of \( f(x)g(x) \) is:\[ \frac{1}{4} a_0 a_0' + \frac{1}{2} \sum_{n=1}^{\infty} a_n a_n ' + \frac{1}{2} \sum_{n=1}^{\infty} b_n b_n ' \]

Key concepts

Fourier Series

Fourier Series are a powerful tool for analyzing periodic functions. By expressing a function as a sum of sine and cosine terms, you can break down complex waveforms into simpler components. A typical Fourier series for a function \( f(x) \) is given by:
\( f(x) = \frac{1}{2} a_0 + \sum_{n=1}^{\infty} a_n \cos(nx) + \sum_{n=1}^{\infty} b_n \sin(nx) \).
Each term in this series corresponds to a different harmonic frequency, helping to capture the essence of the original function over one period. When used in various fields like signal processing, engineering, and physics, they provide insight into the frequency domain representation of time-domain functions.

Orthogonality

In mathematics, orthogonality is a property that signifies perpendicularity in multidimensional space. For Fourier series, orthogonality of sine and cosine functions means that the integral of the product of different sine or cosine functions over one period is zero:
\( \int_{0}^{2\pi} \cos(nx) \cos(mx) \, dx = 0 \) for \( n eq m \).
This property simplifies calculations greatly, allowing us to focus on terms where the frequencies match. Orthogonal functions help isolate individual terms in a series, making it easier to analyze and interpret complex functions.

Trigonometric Integrals

Trigonometric integrals involve integrating functions that include trigonometric terms like sine and cosine. These integrals are crucial when working with Fourier series. For example, when you integrate the product of sine and cosine terms over one period, only terms with matching frequencies contribute non-zero values:
\( \int_{0}^{2\pi} \cos(nx) \cos(mx) \, dx = \begin{cases} 2\pi & \text{if } n = m \ 0 & \text{if } n eq m \end{cases} \).
This property is pivotal in simplifying the process of finding the average value of the product of two functions. Calculating these integrals often helps in isolating significant terms.

Average Value of Functions

The average value of a function over a period is a measure of its central tendency. For periodic functions expressed as Fourier series, the average value can be found by integrating the function over one period and dividing by the period length. For example, the average value of \( h(x) = f(x)g(x) \) over one period, such as \( [0, 2\pi] \), is determined by:
\( \text{Average value of } h(x) = \frac{1}{2\pi} \int_{0}^{2\pi} h(x) \, dx \).
In the context of Parseval's theorem, this average value integrates relevant terms from the product of two Fourier series, focusing solely on the coefficients where frequencies match.

Mathematical Proofs

Mathematical proofs provide a logical sequence of statements demonstrating the truth of a given proposition. In proving Parseval's theorem for Fourier series, one follows systematic steps:
1. Represent the product of functions using their Fourier series.
2. Expand the product to include all cross terms.
3. Integrate over one period to find the average value.
4. Utilize the orthogonality of sine and cosine functions to simplify the result.
This structured approach helps confirm that the average value of the product of two functions equals the sum of terms involving their corresponding Fourier coefficients. Proofs ensure that the concepts and results stand on a solid theoretical foundation.