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Chapter 11: Problem 7

Suppose \(f \in L^{1}(\partial \mathbf{D}) .\) Show that \(f \in L^{2}(\partial \mathbf{D})\) if and only if \(\sum_{n=-\infty}|\hat{f}(n)|^{2}<\infty .\)

Short Answer

Expert verified
In conclusion, a function \(f\) belongs to the \(L^2(\partial \mathbf{D})\) space if and only if \(\sum_{n=-\infty}^{\infty} |\hat{f}(n)|^2 < \infty \). This statement was proven by first assuming the convergence of the series and showing that \(f\) belongs to the \(L^2\) space, and then by assuming that \(f\) is in the \(L^2\) space and showing that the series converges. Both directions of the proof relied on Parseval's identity, a crucial result in Fourier analysis.

Step by step solution

01

Definitions and Properties

To begin, let us recall some definitions and results related to \( L^1 \) and \( L^2 \) spaces and Fourier coefficients: 1. A function \( f \) belongs to the \( L^1(\partial \mathbf{D}) \) space if its integral on the boundary of the unit disk is finite, i.e., \( \int_{\partial \mathbf{D}} |f(\theta)|d\theta < \infty \). 2. A function \( f \) belongs to the \( L^2(\partial \mathbf{D}) \) space if its squared integral on the boundary of the unit disk is finite, i.e., \( \int_{\partial \mathbf{D}} |f(\theta)|^2 d\theta < \infty \). 3. The Fourier coefficients of the function \(f\), \( \hat{f}(n) \), are defined as follows: \[ \hat{f}(n) = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(\theta) e^{-in\theta} d\theta \] We can now proceed with the proof.
02

Proof: If part

Suppose that the series \( \sum_{n=-\infty} |\hat{f}(n)|^2 \) converges. We will show that \( f \in L^2(\partial \mathbf{D}) \). Start by using Parseval's identity, which states: \[ \frac{1}{2\pi} \int_{-\pi}^{\pi} |f(\theta)|^2 d\theta = \sum_{n=-\infty}^{\infty} |\hat{f}(n)|^2 \] Since we are given that the series converges, we can deduce that the integral of \( |f(\theta)|^2 \) over the boundary of the unit disk is finite: \[ \int_{\partial \mathbf{D}} |f(\theta)|^2 d\theta < \infty \] By definition, this implies that \( f \) belongs to the \( L^2(\partial \mathbf{D}) \) space.
03

Proof: Only if part

Now, suppose that \( f \in L^2(\partial \mathbf{D}) \). We will show that the series \( \sum_{n=-\infty} |\hat{f}(n)|^2 \) converges. Let us use Parseval's identity again: \[ \frac{1}{2\pi} \int_{-\pi}^{\pi} |f(\theta)|^2 d\theta = \sum_{n=-\infty}^{\infty} |\hat{f}(n)|^2 \] Since \( f \) belongs to the \( L^2(\partial \mathbf{D}) \) space, the integral of \( |f(\theta)|^2 \) over the boundary of the unit disk is finite: \[ \int_{\partial \mathbf{D}} |f(\theta)|^2 d\theta < \infty \] Thus, the sum of the Fourier coefficients must also be finite: \[ \sum_{n=-\infty}^{\infty} |\hat{f}(n)|^2 < \infty \] In conclusion, we have shown that \( f \) belongs to the \( L^2(\partial \mathbf{D}) \) space if and only if \( \sum_{n=-\infty}^{\infty} |\hat{f}(n)|^2 < \infty \).

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Key Concepts

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

Parseval鈥檚 Identity
In the context of Fourier analysis, Parseval鈥檚 identity is fundamental in understanding the relationship between a function and its Fourier coefficients. In essence, this identity provides a statement of equality between the sum of the squares of the function鈥檚 Fourier coefficients and the integral of the square of the absolute value of the function itself.

Mathematically, for a function f that is square-integrable over 袄谤摆鈭抃谤蟺, \r蟺\r], the identity is expressed as:\r
\r
袄谤袄摆袄谤袄蹿谤补肠调1皑调2袄谤蟺皑袄谤袄颈苍迟冲调鈭抃谤蟺\r}^{\r蟺}\r |f(\r胃\r)|^2 d\r胃\r = \r\sum_{n=鈭抃r鈭瀩^{\r鈭瀩\r |\r\hat{f}(n)\r|^2\r\]\r

The identity not only serves as a bridge between the time domain and frequency domain representations but is also a powerful tool in signal processing and data analysis. It suggests that the energy of the signal鈥攚hich can be thought as the integral of its squared modulus鈥攊s equal to the sum of the squares of its Fourier coefficients. This principle is particularly useful in proving whether a function that belongs to the L1 space also belongs to L2 space, as outlined in the exercise.
Fourier Coefficients
The Fourier coefficients of a function provide a mechanism to represent the function as a sum of sinusoids, effectively decomposing it into its constituent frequencies. Each Fourier coefficient corresponds to a specific frequency component of the function.

The Fourier coefficient \(\r\hat{f}(n)\r\) is defined using the integral:
\r\[\r\hat{f}(n) = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(\theta) e^{-in\theta} d\theta\r\]

Where n is an integer representing the frequency component, and e^{-in\theta} is the complex exponential function which forms the basis for the Fourier series. The determination of these coefficients is essential in many fields, like electrical engineering, quantum physics, and number theory because it provides insight into the amplitude and phase of each frequency present in the original function. It's through this transformation that we begin to understand the behavior of f in the frequency domain as opposed to the time or spatial domain.
L1 Space
The concept of an L1 space 鈥� formally known as a Lebesgue integrable space 鈥� pertains to functions whose absolute value鈥檚 integral is finite. Specifically, a function f belongs to the L1 space on a domain if:
\r\[\r\int_{\text{domain}} |f(\theta)|d\theta < \infty\r\]

This space is significant when we discuss functions in terms of their integrability and is a crucial concept in areas such as real analysis and probability theory. Importantly, it serves as the starting point in our original exercise; the given function f is assumed to be in the L1 space. This assumption allows us to progress further in considering its Fourier coefficients, and by leveraging Parseval鈥檚 identity, investigating its membership in L2 space. In practical applications, being in the L1 space is often related to a function鈥檚 total 鈥渟ize鈥� or 鈥渟ignal strength鈥� in a given context.

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Most popular questions from this chapter

Use techniques similar to Example 11.32 to evaluate \(\sum_{n=1} \frac{1}{n^{4}}\). [If you feel industrious, you may also want to evaluate \(\sum_{n=1}^{\infty} 1 / n^{6} .\) Similar techniques work to evaluate \(\sum_{n=1}^{\infty} 1 / n^{k}\) for each positive even integer \(k .\) You can become famous if you figure out how to evaluate \(\sum_{n=1}^{\infty} 1 / n^{3},\) which currently is an open question.

Suppose \(\zeta \in \partial \mathbf{D}\). Show that the function $$ w \mapsto \frac{1-|w|^{2}}{|1-\bar{\zeta} w|^{2}} $$ is harmonic on \(\mathbf{C} \backslash\\{\zeta\\}\) by finding an analytic function on \(\mathbf{C} \backslash\\{\zeta\\}\) whose real part is the function above.

Suppose \(f(x)=4 \pi x^{2} e^{-\pi x^{2}}-e^{-\pi x^{2}}\) for all \(x \in \mathbf{R}\). Show that \(\hat{f}=-f\).

Suppose \(p \in[1, \infty]\) and \(f \in L^{p}(\mathbf{R})\). Prove that \(\mathcal{P}_{y}\left(\mathcal{P}_{y^{\prime}} f\right)=\mathcal{P}_{y+y^{\prime}} f\) for all \(y, y^{\prime}>0\)

Suppose \(n \in \mathbf{Z}^{+}\) and \(f \in L^{1}(\mathbf{R})\) is \(n\) times continuously differentiable and \(f^{(n)} \in L^{1}(\mathbf{R}) .\) Prove that if \(t \in \mathbf{R},\) then $$ \left(f^{(n)}\right)^{\wedge}(t)=(2 \pi i t)^{n} \hat{f}(t) $$
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