91Ó°ÊÓ

This problem is designed to guide you through a "proof" of Plancherel's theorem, by starting with the theory of ordinary Fourier series on a finite interval, and allowing that interval to expand to infinity. (a) Dirichlet's theorem says that "any" function \(f(x)\) on the interval \([-a,+a]\) can be expanded as a Fourier series: $$f(x)=\sum_{n=0}^{\infty}\left[a_{n} \sin \left(\frac{n \pi x}{a}\right)+b_{n} \cos \left(\frac{n \pi x}{a}\right)\right].$$ Show that this can be written equivalently as $$f(x)=\sum_{n=-\infty}^{\infty} c_{n} e^{i n \pi x / a}.$$ What is \(c_{n},\) in terms of \(a_{n}\) and \(b_{n} ?\) (b) Show (by appropriate modification of Fourier's trick) that $$c_{n}=\frac{1}{2 a} \int_{-a}^{+a} f(x) e^{-i n \pi x / a} d x.$$ (c) Eliminate \(n\) and \(c_{n}\) in favor of the new variables \(k=(n \pi / a)\) and \(F(k)=\sqrt{2 / \pi} a c_{n} .\) Show that \((\mathrm{a})\) and \((\mathrm{b})\) now become $$f(x)=\frac{1}{\sqrt{2 \pi}} \sum_{n=-\infty}^{\infty} F(k) e^{i k x} \Delta k ; \quad F(k)=\frac{1}{\sqrt{2 \pi}} \int_{-a}^{+a} f(x) e^{-i k x} d x, $$ where \(\Delta k\) is the increment in \(k\) from one \(n\) to the next. (d) Take the limit \(a \rightarrow \infty\) to obtain Plancherel's theorem. Comment: In view of their quite different origins, it is surprising (and delightful) that the two formulas-one for \(F(k)\) in terms of \(f(x),\) the other for \(f(x)\) in terms of \(F(k)-\) have such a similar structure in the limit \(a \rightarrow \infty.\)

Step by step solution

01

Fourier Series Representation

We start with the Fourier series representation of the function, given by Dirichlet's theorem: \[ f(x) = \sum_{n=0}^{\infty} \left(a_n \sin\left(\frac{n \pi x}{a}\right) + b_n \cos\left(\frac{n \pi x}{a}\right)\right) \] This series can also be expressed in terms of complex exponentials: \[ f(x) = \sum_{n=-\infty}^{\infty} c_n e^{i n \pi x / a} \] To do this, recognize that \( \sin \theta = \frac{e^{i\theta} - e^{-i\theta}}{2i} \) and \( \cos \theta = \frac{e^{i\theta} + e^{-i\theta}}{2} \). Thus, \( c_n \) can be expressed in terms of \( a_n \) and \( b_n \) as follows:\[ c_n = \begin{cases} \frac{b_n}{2} - \frac{a_n}{2i}, & n > 0 \ \frac{b_n}{2} + \frac{a_n}{2i}, & n < 0 \end{cases} \] Note that for \( n = 0 \), it is comprised solely of \( b_n \), i.e., \( c_0 = b_0 \).

02

Calculate Coefficients with Fourier's Trick

Use Fourier's trick to determine the coefficients \(c_n\). Start by multiplying both sides of the series by \(e^{-i m\pi x / a}\) and integrating over \([-a, a]\):\[ \int_{-a}^{a} f(x) e^{-i m\pi x / a} dx = \sum_{n=-\infty}^{\infty} c_n \int_{-a}^{a} e^{i(n-m)\pi x / a} dx \] Due to orthogonality, \[ \int_{-a}^{a} e^{i(n-m)\pi x / a} dx \] is \(2a\) if \(n = m\) and 0 otherwise. Therefore, \[ c_n = \frac{1}{2a} \int_{-a}^{a} f(x) e^{-i n\pi x / a} dx \] This formula shows how \(c_n\) relates to the function \(f(x)\) on the interval.

03

Change Variables for Continuous Limit

We change the variables to new variables \( k = \frac{n\pi}{a} \) and \( F(k) = \sqrt{\frac{2}{\pi}} a c_n \). This results in the following transformation for (a) and (b): \[ f(x) = \frac{1}{\sqrt{2\pi}} \sum_{n=-\infty}^{\infty} F(k) e^{i k x} \Delta k \] where \( \Delta k = \frac{\pi}{a} \), giving the spacing between successive \(k\). Also,\[ F(k) = \frac{1}{\sqrt{2\pi}} \int_{-a}^{a} f(x) e^{-i k x} dx \] This prepares the expression for the limit transition.

04

Take Limit as Interval Expands

Finally, take the limit \(a \rightarrow \infty\): As \(a\) increases, the interval expands to \([-\infty, \infty]\), making the summation approach an integral. Hence (a) becomes:\[ f(x) = \int_{-\infty}^{\infty} F(k) e^{i k x} dk \] And (b) provides:\[ F(k) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(x) e^{-i k x} dx \] This result is Plancherel's theorem, which emphasizes the symmetry and elegance of the Fourier transform.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fourier Series

A Fourier series is a way to represent a function as an infinite sum of sines and cosines. This powerful mathematical tool helps in analyzing the frequencies present in a periodic function. In simple terms, any periodic signal can be broken down into a combination of simple oscillating functions, specifically sines and cosines.

Dirichlet's theorem extends this representation to include all functions on the interval [-a, +a], thereby forming a Fourier series which expresses the function as:

• \(f(x) = \sum_{n=0}^{\infty} \left(a_n \sin\left(\frac{n \pi x}{a}\right) + b_n \cos\left(\frac{n \pi x}{a}\right)\right)\)

This equation indicates how each part of the signal corresponds to a particular sine or cosine wave, determined by coefficients \(a_n\) and \(b_n\). By transforming this with complex exponentials, any function can be similarly expressed using a more compact form as a sum of exponential terms, laying groundwork for Plancherel's theorem.

Dirichlet's Theorem

Dirichlet's theorem plays a fundamental role when working with Fourier series by affirming that any function defined over a finite interval [-a, +a] can be expressed as a Fourier series. This theorem provides assurance that the series representation using sines and cosines is valid for almost any function within this interval.

This theorem advances analyses by introducing conditions where functions can be faithfully represented through oscillatory components, forming the base of all Fourier series approaches. The translation of the Fourier series into complex exponentials through Dirichlet's theorem:

• \(f(x) = \sum_{n=-\infty}^{\infty} c_n e^{i n \pi x / a}\)

offers an alternate yet mathematically equivalent view of decomposing functions, integral to understanding more complex forms like those that appear in signal processing and engineering fields.

Complex Exponentials

Complex exponentials are extensions of exponential functions, incorporating imaginary numbers to express trigonometric functions in exponential form. They are key to simplifying the representation of oscillatory behaviors in the Fourier series.

By utilizing Euler's formula, which relates complex exponentials to trigonometric functions:

• \(e^{i\theta} = \cos \theta + i \sin \theta\)

you can transform sines and cosines into exponential terms. This transformation is powerful as it allows Fourier series to denote functions using the formula:

• \(f(x) = \sum_{n=-\infty}^{\infty} c_n e^{i n \pi x / a}\)

where the coefficients \(c_n\) relate to corresponding real-valued coefficients for sine and cosine. Employing complex notation facilitates analysis and integration, particularly within disciplines involving wave mechanics or electronics.

Orthogonality in Fourier Analysis

Orthogonality is a crucial concept in Fourier analysis, indicating that two functions are orthogonal if their inner product equals zero. In the context of Fourier series, orthogonality simplifies the calculation of the series coefficients.

For example, sine and cosine components are orthogonal over any interval that is a multiple of their period, providing that their overlapping contribution towards the inner products:

• \(\int_{-a}^{a} \sin(m \theta) \cos(n \theta) \, d\theta = 0\)

are zero.

This intrinsic property is exploited in deriving the coefficients using techniques such as "Fourier's Trick." It allows each component in the series to accurately capture distinct aspects of the original function without interference, therefore enabling the reconstruction of the signal through its constituent orthogonal parts—crucial in proving and applying Plancherel's theorem.