Question

Show that Parseval's theorem for two functions whose Fourier expansions have. cosine and sine coefficients \(a_{n}, b_{n}\) and \(\alpha_{n}, \beta_{n}\) takes the form $$ \frac{1}{L} \int_{0}^{L} f(x) g^{*}(x) d x=\frac{1}{4} a_{0} \alpha_{0}+\frac{1}{2} \sum_{n=1}^{\infty}\left(a_{n} \alpha_{n}+b_{n} \beta_{n}\right) $$ (a) Demonstrate that for \(g(x)=\sin m x\) or \(\cos m x\) this reduces to the definition of the Fourier coefficients. (b) Explicitly verify the above result for the case in which \(f(x)=x\) and \(g(x)\) is the square-wave function, both in the interval \(-1 \leq x \leq 1\)

Short answer

Parseval's theorem is verified by applying the orthogonality property, reducing to the definition of Fourier coefficients, and checking a specific case for given functions.

Step-by-step solution

- State Parseval's Theorem

Parseval's theorem states that for functions with Fourier expansions with cosine and sine coefficients, the integral of their product is equal to the series of products of their coefficients. In mathematical terms: \[ \frac{1}{L} \int_{0}^{L} f(x) g^{*}(x) dx = \frac{1}{4} a_{0} \alpha_{0} + \frac{1}{2} \sum_{n=1}^{\infty} (a_{n} \alpha_{n} + b_{n} \beta_{n}) \]

- Case 1: Set g(x) = sin(mx) or cos(mx)

Let us first consider the case where \(g(x)\) is either \(\sin(m x)\) or \(\cos(m x)\). Using the orthogonality property of sine and cosine functions, we know that: \[ \sin(nx) \text{ and } \cos(nx) \] are orthogonal to each other.

- Apply Orthogonality Property

Applying the orthogonality property: \[ \int_{0}^{L} \cos(nx) \sin(mx) dx = 0 \ \int_{0}^{L} \cos(nx) \cos(mx) dx = \frac{L}{2} \delta_{nm} \ \int_{0}^{L} \sin(nx) \sin(mx) dx = \frac{L}{2} \delta_{nm} \] Therefore, since \(a_n\) and \(b_n\) satisfy this property, the expression \(\frac{1}{L} \int_{0}^{L} f(x) g(x) dx\) simplifies to the definition of Fourier coefficients.

- Verify for Specific f(x) = x and g(x) as Square Wave

Consider \(f(x) = x\), and \(g(x)\) as a square wave in the interval \(-1 \leq x \leq 1\). The Fourier series of \(g(x)\) involves terms \( \alpha_n\) and \( \beta_n \) which can be calculated using: \[ \alpha_n = \int_{-1}^{1} g(x) \cos(nx) dx \ \beta_n = \int_{-1}^{1} g(x) \sin(nx) dx \] Substituting these values into Parseval’s theorem helps to verify the given relation.

- Conclusion

By calculating the integrals for the specific functions and using the orthogonality conditions, the simplified expression verifies Parseval’s theorem holds true for the given \(f(x) = x\) and the square wave function \(g(x)\).

Key concepts

Fourier Series

A Fourier series is a way to represent a function as a sum of sine and cosine terms. These series are particularly useful in analyzing periodic functions. Each function can be written as an infinite sum of sine and cosine waves, each multiplied by a different coefficient.
The general form of a Fourier series for a function \(f(x)\) defined on the interval \([0, L]\) can be written as:
\[ f(x) = a_0 + \sum_{n=1}^{\infty} \left(a_n \cos\left(\frac{2 \pi nx}{L}\right) + b_n \sin\left(\frac{2 \pi nx}{L}\right)\right) \]
Here, \(a_n\) and \(b_n\) are the Fourier coefficients that we determine for each n.
Fourier series are crucial in understanding signal representation, heat transfer, and other areas of applied mathematics.

Orthogonality Property

One essential aspect of Fourier series is the orthogonality property of sine and cosine functions. This property states that sine and cosine functions are orthogonal over any interval of length L, meaning their integral over one period is zero unless the functions are the same frequency.
Mathematically, this is represented by:
\[ \int_{0}^{L} \cos(nx) \sin(mx) dx = 0 \]
\[ \int_{0}^{L} \cos(nx) \cos(mx) dx = \frac{L}{2} \delta_{nm} \]
\[ \int_{0}^{L} \sin(nx) \sin(mx) dx = \frac{L}{2} \delta_{nm} \]
where \( \delta_{nm} \) is the Kronecker delta, which is 1 if n = m and 0 otherwise. This orthogonality makes it easier to calculate Fourier coefficients.

Fourier Coefficients

The Fourier coefficients \(a_n\) and \(b_n\) represent the amplitude of the cosine and sine terms, respectively. They are crucial for constructing the Fourier series of a function.
The coefficients can be calculated using integrals:
\[ a_n = \frac{2}{L} \int_{0}^{L} f(x) \cos\left(\frac{2 \pi nx}{L}\right) dx \]
\[ b_n = \frac{2}{L} \int_{0}^{L} f(x) \sin\left(\frac{2 \pi nx}{L}\right) dx \]
These formulas show how the function \(f(x)\) can be decomposed into a series of cosine and sine terms. By determining each \(a_n\) and \(b_n\), we can closely approximate any periodic function using its Fourier series.
Furthermore, to get the fundamental frequency part of the series, we often also compute the term \(a_0\), which is the average value of the function over one period.

Integral of Product of Functions

Parseval's theorem is a key concept linking the Fourier coefficients to the integral of the product of functions. It states that for functions with Fourier expansions, the integral of their product equals the sum of the product of their Fourier coefficients.
In mathematical terms, for two functions \(f(x)\) and \(g(x)\) with Fourier expansions, Parseval's theorem can be expressed as:
\[ \frac{1}{L} \int_{0}^{L} f(x) g^{*}(x) dx = \frac{1}{4} a_0 \alpha_0 + \frac{1}{2} \sum_{n=1}^{\infty} \left(a_n \alpha_n + b_n \beta_n\right) \]
This means that the integral of the product of two functions can be computed as a sum involving their Fourier coefficients. This simplification is particularly important in signal processing and other applications.