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Chapter 2: Problem 1

Let \(f(x)=x+\cos x\) and $$ f(x) \sim \frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n} \cos \frac{n x}{2}+b_{n} \sin \frac{n x}{2}\right] $$ be the Fourier series of \(f\) on \([0,4 \pi]\). (a) Determine the \(a_{n}\) and \(b_{n}\). (b) Let \(g(x)=\frac{A_{0}}{2}+\sum_{n=1}^{5} B_{n} \sin \frac{n x}{2}\). For what values of \(A_{0}\) and \(B_{n}\), \(1 \leq n \leq 5\), is the distance between \(f\) and \(g\) minimal?

Short Answer

Expert verified
(a) \(a_0 = 4\pi^2\), \(a_n = 0\), \(b_1 = -4\), \(b_n = 0\). (b) \(A_0 = 4\pi^2\), \(B_1 = -4\), all other \(B_n = 0\).

Step by step solution

01

Fourier Coefficients for Cosine

The Fourier coefficient \(a_n\) is derived using the formula \(a_n = \frac{2}{L} \int_0^{L} f(x) \cos\left(\frac{n \pi x}{L}\right) \, dx\). Here, the interval length \(L = 4\pi\). The function \(f(x)=x+\cos x\) results in terms that integrate separately. Calculations for \(a_0\) yield \(a_0 = \int_0^{4 \pi} (x+\cos x) \, dx = 4 \pi^2\). Analyzing symmetry reveals that \(a_n = 0\) for \(n\geq 1\), since all these cosine integrals result in zero due to symmetry properties.
02

Fourier Coefficients for Sine

The Fourier coefficient \(b_n\) is evaluated using \(b_n = \frac{2}{L} \int_0^{L} f(x) \sin\left(\frac{n \pi x}{L}\right) \, dx\). For \(f(x)=x+\cos x\), we have separate terms: \(b_n(f(x)=x)\) and \(b_n(f(x)=\cos x)\), which are derived separately. Utilizing symmetry, we find \(b_n(f(x)=\cos x)\to 0\). The first term integrates to zero except at non-zero odd harmonics of \(\sin\) when calculations are performed over the full interval, ultimately yielding \(b_1=-4\) and other \(b_n=0\) when \(n\) is even. Correspondingly, \(b_n\) lacks contribution in this problem context for even orders greater than 2.
03

Determine Coefficients Using Minimal Distance Condition

To minimize the distance between \(f(x)\) and \(g(x)\), it involves matching terms where possible: \(A_0 = a_0 = 4 \pi^2\), for sine terms, taking lower orders minimizes deviations, thus: \(B_1 = b_1 = -4\), all other \(B_n = 0\) for \(1 \leq n \leq 5\) except when providing additional constraints in the higher-order terms.

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Key Concepts

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

Fourier Coefficients
Fourier coefficients, represented by \(a_n\) and \(b_n\), are crucial to developing the Fourier series for a function like \(f(x)=x+\cos x\). These coefficients tell us how much of each trigonometric base function should be included in the series representation.
For cosine terms, the coefficient \(a_n\) is determined using the formula: \[a_n = \frac{2}{L} \int_0^{L} f(x) \cos\left(\frac{n \pi x}{L}\right) \, dx\]Here, \(L\) is the length of the interval over which the function is defined. In our case, since we're considering \([0,4 \pi]\), \(L=4\pi\). By calculating it, we find that all the cosine harmonics above the constant term (\(n \geq 1\)) integrate to zero due to symmetry.
Similarly, the sine coefficients \(b_n\) use the formula:\[b_n = \frac{2}{L} \int_0^{L} f(x) \sin\left(\frac{n \pi x}{L}\right) \, dx\]The function contributions for sine terms get canceled for specific values when considering the full function and its components separately. Notably, the symmetry of sine integrate to zero unless dealing with specific harmonics as highlighted in the solution.
Trigonometric Polynomials
Trigonometric polynomials are expressions consisting of sums of sine and cosine functions. In the context of Fourier series, these polynomials help approximate complex periodic functions.
Our function \(f(x)\) is expressed as a Fourier series:\[f(x) \sim \frac{a_0}{2} + \sum_{n=1}^{\infty} \left(a_n \cos\frac{n x}{2} + b_n \sin\frac{n x}{2}\right)\]where each term represents a harmonic of the fundamental frequencies involved. When \(a_n\) and \(b_n\) are identified, the trigonometric polynomial formed provides an approximation to \(f(x)\) over the interval. This is essentially a synthesis of sine and cosine waves to recreate or closely mimic the original function.
The Fourier coefficients adjust the amplitudes of these sine and cosine elements, allowing a perfect approximation over the period, should enough terms be used, highlighting a core concept of harmonic analysis.
Harmonic Analysis
Harmonic analysis involves breaking down functions or signals into their fundamental frequency components. In the case of our function \(f(x)=x+\cos x\), the use of Fourier series is a prime example of how a function can be expressed as a sum of basic trigonometric components.
In practical terms:
  • Each sine and cosine component represents a harmonic that contributes to the overall function.
  • By analyzing these harmonics, you can understand the structure and characteristics of the function at different fundamental frequencies.
This is fundamental in fields such as signal processing, where complex waveforms are analyzed for their frequency content, a technique made feasible by the principles of harmonic analysis.
For our exercise, to make the distance between the two functions, \(f(x)\) and \(g(x)\), minimal, the process involves selecting coefficients \(A_0\) and \(B_n\) such that their trigonometric expressions align as closely as possible. This results in a minimal deviation between the two functions, demonstrating the precision offered by harmonic analysis in approximating functions.

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Most popular questions from this chapter

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 \(f\) be a \(2 \pi\)-periodic piecewise continuous function satisfying $$ \int_{-\pi}^{\pi} f(x) d x=0 . $$ \(\operatorname{Set} g(x)=\int_{0}^{x} f(t) d t\). (a) Prove that \(g\) is \(2 \pi\)-periodic. (b) Let \(\sum_{n=-\infty}^{\infty} c_{n} e^{i n x}\) be the complex Fourier series of the function \(f\) and \(\sum_{n=-\infty}^{\infty} d_{n} e^{i n x}\) the complex Fourier series of the function \(g\). Prove that for all real \(x\) we have the equality $$ g(x)=\sum_{n=-\infty}^{\infty} d_{n} e^{i n x} $$ where \(d_{n}=\frac{c_{n}}{i n}\) for every integer \(n \neq 0\).

For each real \(p,-\pi \leq p \leq \pi\), find the Fourier series of the function $$ f_{p}(x)= \begin{cases}0, & -\pi \leq x \leq p, \\ 1, & p

Find the Fourier series of $$ f(x)= \begin{cases}x-[x], & x \text { is not an integer, } \\ \frac{1}{2}, & x \text { is an integer. }\end{cases} $$ To what values does the Fourier series converge at the points \(x=5, x=3\), and \(x=1.5\) ?

Set \(f(x)=1-x^{2}\) in the interval \([-\pi, \pi]\) and let $$ f(x) \sim \frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n} \cos n x+b_{n} \sin n x\right] $$ be the Fourier series of \(f\). (a) Calculate the \(a_{n}\) and \(b_{n}\). (b) To what values does the Fourier series of \(f\) converge at the points \(x=5 \pi\) and \(x=6 \pi\) ? Explain.
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