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Chapter 2: Problem 11

Let \(f\) be a \(2 \pi\)-periodic piecewise continuous function and $$ f(x) \sim \sum_{n=1}^{\infty}\left[a_{n} \cos n x+b_{n} \sin n x\right] $$ its Fourier series on \([-\pi, \pi]\). Set $$ g(x)=\int_{-\pi}^{x}[f(t)+f(\pi-t)] d t $$ and let $$ g(x) \sim \frac{A_{0}}{2}+\sum_{n=1}^{\infty}\left[A_{n} \cos n x+B_{n} \sin n x\right] $$ be the Fourier series of \(g\) on \([-\pi, \pi]\). Express the \(A_{n}\) and \(B_{n}\) in terms of \(a_{n}\) and \(b_{n}\).

Short Answer

Expert verified
\( A_n = \frac{2}{n}(a_n \sin(\pi n) + b_n(1 - \cos(\pi n))) \), \( B_n = 0 \).

Step by step solution

01

Understand the Fourier Series of f(x)

The given periodic function \( f(x) \) has a Fourier series on the interval \([-, ]\). This series is expressed as \( f(x) \sim \sum_{n=1}^{\infty}[a_{n} \cos n x + b_{n} \sin n x] \). This represents the function in terms of its sine and cosine components.
02

Define the Function g(x)

The function \( g(x) \) is defined as the integral \( g(x) = \int_{-}^{x} [f(t) + f(-t)] dt \). As \( t \) varies from \( - \) to \( x \), this integral accumulates the values of \( f(t) + f(-t) \), a modification of \( f(t) \) with symmetric consideration about \(  \).
03

Analyze the Symmetry of f(t) + f(-t)

The function \( f(t) + f(-t) \) introduces elements of symmetry. Since the original function \( f(x) \) is periodic, this alteration aids in considering both the original form and its reflection across \(  \), which is important in determining the Fourier coefficients of \( g(x) \).
04

Integrate to Find g(x)

To express \( g(x) \) in terms of its Fourier series, substitute \( f(x) \) in the integral: \[ g(x) = \int_{-}^{x} [a_0 + \sum_{n=1}^{\infty} (a_{n} \cos n t + b_{n} \sin n t) + \sum_{n=1}^{\infty} (a_{n} \cos n(-t) + b_{n} \sin n(-t))] dt \].
05

Simplify the Integral Expression

Simplify \( g(x) \) by breaking it into simpler integrals for each component of the series. Consider the trigonometric identities and symmetries involved when applying to \( f(t) + f(-t) \). This step groups terms and adjusts the bounds to better accommodate integration.
06

Derive Fourier Coefficients A_n and B_n for g(x)

Using the properties of definite integrals and the periodic nature of trigonometric functions, derive expressions for \( A_n \) and \( B_n \) involving \( a_n \) and \( b_n \). Solving these integrals will result in the series expansion terms for \( g(x) \).
07

Express A_n and B_n

From the integration and symmetry analysis, express: \[ A_n = \frac{2}{n}(a_n \sin( n) + b_n(1 - \cos( n))) \] and \[ B_n = 0 \] because the symmetry cancels out the sine terms due to evenness of the integrated expression.

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Key Concepts

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

Piecewise Continuous Function
When dealing with Fourier series, a key aspect is understanding the type of function involved. A **piecewise continuous function** is a function that is continuous for the most part, except at a finite number of points where it may have holes, jump discontinuities, or infinite discontinuities. This type of function is essential because Fourier series are particularly good at representing piecewise continuous functions. The function is defined this way to ensure that there are well-defined limits on both sides of any point of discontinuity, which is crucial for convergence issues within the Fourier series.Why choose piecewise continuous functions? Well, they are flexible and can model many real-world signals and phenomena. For instance, rectangular waves and other signals that naturally arise in engineering and physics. Additionally, focusing on functions within the \( [ -\pi, \pi ]\) interval is a standard practice because it simplifies the computation of Fourier coefficients and aligns with the periodic nature of trigonometric functions utilized in Fourier analysis.
Fourier Coefficients
To reconstruct a function using its Fourier series, knowing about **Fourier coefficients** is fundamental. These coefficients, denoted as \(a_n\) and \(b_n\), reflect the amplitude of the cosine and sine components at each harmonic frequency.
  • The \(a_n\) coefficients correspond to the even, or cosine, parts of the function.
  • The \(b_n\) coefficients correspond to the odd, or sine, parts of the function.
These coefficients represent the weights of various frequency components in the Fourier series. By determining them, we can understand how much of each frequency is present in the signal, a process which is vital in signal processing, audio compression, and other areas such as image analysis.In exercises where a function is modified 鈥� like in the given problem where \(f(t) + f(\pi - t)\) is used 鈥� the task is to express new Fourier coefficients \(A_n\) and \(B_n\) in terms of the original ones. This involves integrating the original functions over the period and applying trigonometric identities to simplify and solve for these coefficients.
Symmetry
The concept of **symmetry** plays a crucial role in simplifying and analyzing functions within Fourier series. Symmetry helps determine which Fourier coefficients will be non-zero, therefore it can significantly simplify computations.There are different types of symmetry relevant to Fourier series:
  • **Even symmetry** where \(f(-x) = f(x)\). Functions with even symmetry have only cosine terms in their series since cosine is even.
  • **Odd symmetry** where \(f(-x) = -f(x)\). These functions contain only sine terms due to the odd nature of sine.
  • **Half-wave symmetry**, which affects coefficients based on periodicity adjustments.
In this problem, symmetry is considered through the expression \(f(t) + f(\pi - t)\), introducing a reflective property about \(\pi\). This reflection implies an enhancement of certain symmetries within the function, helping us deduce that the \(B_n\) coefficients vanish, simplifying the Fourier series of \(g(x)\). This elimination stems from the even symmetry of the expression over the interval.
Integral Transforms
Integral transforms like the **Fourier transform** facilitate converting functions from one domain into another, generally making analysis more manageable, especially for periodic functions. In the given exercise, integrating the function \(f(x)\) involves breaking it down into simpler integral components to identify how each part contributes to the transformed function \(g(x)\).
  • The integral of a sum of functions can be split into the sum of integrals, allowing for handling individual components one at a time.
  • Often, transformations take place over a specific range \( [-\pi, \pi] \), which ties into the periodicity of the functions.
For the function \(g(x)\), the integration step examines the transformed components from the original basis, forming a new Fourier series. This requires applying trigonometric identities and understanding properties of definite integrals. By doing so, you derive the expressions for coefficients like \(A_n\) and \(B_n\) based on the integral properties of sine and cosine functions over symmetrically defined intervals.

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Most popular questions from this chapter

Let \(f \in E\) and $$ f(x) \sim \sum_{n=-\infty}^{\infty} c_{n} e^{i n x} $$ be the complex Fourier series of \(f\). Determine the complex Fourier series of \(f(\bar{x}), \overline{f(x)}\), and \(f(-x)\).

Set \(f(x)=|\sin x|\) and let $$ f(x) \sim \frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n} \cos n x+b_{n} \sin n x\right] $$ denote the Fourier series of \(f\) on \([-\pi, \pi]\). (a) Calculate the \(a_{n}\) and \(b_{n}\). (b) Set \(g(x)=\sum_{n=1}^{\infty}\left[-n a_{n} \sin n x+n b_{n} \cos n x\right]\). Determine \(g\) and sketch the graph of \(g\) on \([-\pi, \pi]\). (c) Calculate the sums $$ \sum_{n=1}^{\infty} \frac{1}{4 n^{2}-1}, \quad \sum_{n=1}^{\infty} \frac{(-1)^{n}}{4 n^{2}-1}, \quad \sum_{n=1}^{\infty} \frac{1}{\left(4 n^{2}-1\right)^{2}}, \quad \sum_{n=1}^{\infty} \frac{n^{2}}{\left(4 n^{2}-1\right)^{2}} . $$

Find the complex Fourier series of \(f(t)=\frac{1}{1-\frac{1}{2} e^{-i t}}\) on the interval \([-\pi, \pi]\).

Let \(C[-\pi, \pi]^{n}\) denote the space of complex-valued continuous functions defined on the cube \([-\pi, \pi]^{n}\). On this linear space we define the inner product $$ \langle f, g\rangle=\frac{1}{(2 \pi)^{n}} \underbrace{\int_{-\pi}^{\pi} \ldots \int_{-\pi}^{\pi}}_{n \text { times }} f\left(x_{1}, \ldots, x_{n}\right) \overline{g\left(x_{1}, \ldots, x_{n}\right)} d x_{1} \cdots d x_{n} . $$ Prove that the family of functions $$ \left\\{e^{i\left(m_{1} x_{1}+\cdots+m_{n} x_{n}\right)}\right\\}_{m_{1}, \ldots, m_{n} \in \mathbb{Z}} $$ is an orthonormal system with respect to the above inner product. If we define $$ c_{m_{1}, \ldots, m_{n}}=\frac{1}{(2 \pi)^{n}} \underbrace{\int_{-\pi}^{\pi} \cdots \int_{-\pi}^{\pi} f\left(x_{1}, \ldots, x_{n}\right) e^{-i\left(m_{1} x_{1}+\cdots+m_{n} x_{n}\right)} d x_{1} \cdots d x_{n}} $$ then the series $$ \sum_{m_{1}, \ldots, m_{n}=-\infty}^{\infty} c_{m_{1}, \ldots, m_{n}} e^{i\left(m_{1} x_{1}+\cdots+m_{n} x_{n}\right)} $$ is called the multivariate complex Fourier series of \(f\). Prove that if \(f_{1}, f_{2}\), \(\ldots, f_{n}\) are functions in \(C[-\pi, \pi]\) and $$ f\left(x_{1}, x_{2}, \ldots, x_{n}\right)=f_{1}\left(x_{1}\right) f_{2}\left(x_{2}\right) \cdots f_{n}\left(x_{n}\right) $$ then $$ c_{m_{1}, \ldots, m_{n}}=a_{m_{1}}^{1} a_{m_{2}}^{2} \cdots a_{m_{n}}^{n} $$ where the \(a_{m}^{k}\) are the coefficients of the complex Fourier series of \(f_{k}\). That is, \(f_{k}(x) \sim \sum_{m=-\infty}^{\infty} a_{m}^{k} e^{i m x}\).

Let \(f \in E\) and assume $$ \frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n} \cos n x+b_{n} \sin n x\right] $$ is the Fourier series of \(f\). Show that if there exist constants \(c\) and \(d\) such that $$ \left|a_{n}\right| \leq \frac{c}{n^{2}}, \quad\left|b_{n}\right| \leq \frac{d}{n^{2}} $$ for all \(n\), then \(f\) may be considered to be continuous on \([-\pi, \pi]\), satisfying \(f(-\pi)=f(\pi)\), and the Fourier series of \(f\) converges uniformly to \(f\) on \([-\pi, \pi]\).
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