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Chapter 1: Problem 1

Check that Gibbs' phenomenon must present itself in the vicinity of any jump of a piecewise smooth function. At such a jump \(\lim _{n+\infty} S_{n}(x)=\frac{1}{2} f(x-)+\frac{1}{2} f(x+) .\) Hint: The problem may be localized by Theorem 1.5.3. The other ingredient is an easy extension of Theorem 1.4.2 to jump-free piecewise smooth functions.

Short Answer

Expert verified
Gibbs' phenomenon manifests as oscillations around discontinuities in piecewise smooth functions.

Step by step solution

01

Understanding the Gibbs' Phenomenon

Gibbs' phenomenon refers to the overshoot (or 'ringing') that occurs when approximating a discontinuous function using its Fourier series. Near a jump discontinuity, the Fourier series does not uniformly converge to the function value but instead creates oscillations above and below the jump. Recognizing this helps us identify the expectation that this overshoot appears in piecewise smooth functions with jumps.
02

Expression for Partial Sums of Fourier Series

The partial sum of the Fourier series of a function at a point is given by:\[ S_n(x) = \sum_{k=-n}^{n} c_k e^{ikx} \] where \(c_k\) are the Fourier coefficients. At a discontinuity in a piecewise smooth function, we evaluate the convergence behavior of these partial sums.
03

Examining the Convergence Near a Jump

At a jump, the Fourier series attempts to converge to a value that is the average of the left and right limits of the function at that point. Thus, we have:\[ \lim_{n \to \infty} S_n(x) = \frac{1}{2} f(x-) + \frac{1}{2} f(x+) \] This convergence formula indicates the Gibbs' overshoot as the series converges to this average, resulting in oscillations near the jump.
04

Application of Theorems

The problem suggests using Theorem 1.5.3 and an extension of Theorem 1.4.2. Theorem 1.5.3 likely deals with localization or convergence of Fourier series in the presence of discontinuities, reinforcing that Gibbs' phenomenon is a local behavior near jumps. Theorem 1.4.2, applied to jump-free functions, asserts smoother convergence, highlighting the contrast with jump behavior.
05

Conclusion on Gibbs' Phenomenon

From these observations, we conclude that the partial sum overshoots will indeed manifest around discontinuities of a piecewise smooth function. The approximate convergence to the average value of the jump's endpoints results in noticeable oscillations, consistent with Gibbs' phenomenon.

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Key Concepts

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

Fourier Series
The Fourier series is a way to represent a function as a sum of sinusoidal components. Essentially, you're breaking down a complex waveform into simpler sine and cosine waves. This is extremely useful in analyzing functions that repeat periodically. By doing so, you can approximate complex functions with potentially just a few components if the function is smooth. A Fourier series of a function contains terms calculated through Fourier coefficients, which are determined from the function itself. Once calculated, you can reconstruct the function using the series. This process is powerful, but when dealing with functions having discontinuities, the series approximates "average points," around these breaks, contributing to unique behaviors like Gibbs' phenomenon.
Piecewise Smooth Function
A piecewise smooth function is made up of sections, where each section is smooth but jumps in value at certain points. This characteristic makes them quite common in real-world applications, like signal processing where signals have abrupt changes. For such functions, you can still use a Fourier series for approximation but need to be extra cautious around the points of discontinuity. At these points, a smooth transition is not possible, and this is where the function's nature introduces complexity when it's being broken into Fourier series components.
Jump Discontinuity
Jump discontinuity occurs at points where a function suddenly "jumps" to a different value, rather than transitioning smoothly. Imagine a step function that jumps from one level to another without a slope in between. In the context of Fourier series, jump discontinuities present a challenge. The Fourier series can approximate the function either side of the jump well, however, it cannot perfectly replicate the sudden change at the jump point. Instead, the series tends to average the two sides, often producing oscillations as it strives to approximate the function in the vicinity of the jump, which connects closely with Gibbs' phenomenon.
Convergence Behavior
Convergence behavior refers to how a series approaches a certain function as more terms are added. For Fourier series, convergence at a smooth part of the function is generally well-behaved. However, at a jump discontinuity in a piecewise smooth function, the convergence behavior becomes unique. The Fourier series tends towards the average of the limits on either side of the discontinuity, leading to non-uniform convergence characterized by overshoots and oscillations, emphasizing the impact of Gibbs' phenomenon. This means that instead of smoothing out to meet the function precisely, the series creates an overshoot when trying to align with the jump, illustrating a behavioral pattern that is a hallmark of Fourier series near discontinuities.

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Most popular questions from this chapter

Check that the Lebesgue integral \(\int_{0}^{1} f\) of the indicator function \(f\) of the rational numbers exists \((=0)\) but that \(f\) is not integrable in the Riemann sense.

Dini's test states that if \(f \in \mathrm{L}^{1}\left(S^{1}\right)\) and if for fixed \(|x| \leqslant \frac{1}{2}\), the function \(y^{-1}[f(x+y)-f(x)]\) is summable, then \(\lim _{n} \uparrow \infty S_{n}(x)=f(x)\). Prove it. Hint: Use the Dirichlet kernel and the Riemann-Lebesgue lemma, as earlier.

The infinite-dimensional complex number space \(C^{\infty}\) is the class of all sequences \(c=\left(c_{1}, c_{2}, \ldots\right)\) of complex numbers with $$ \|c\|=\left(\sum\left|c_{n}\right|^{2}\right)^{1 / 2}<\infty . $$ Check that \(C^{\infty}\) is a Hilbert space under the inner product $$ (a, b)=\sum a_{n} b_{n}^{*} . $$ The proof is the same as for \(L^{2}(Q)\) only easier. Another name for \(C^{\infty}\) is \(\mathrm{L}^{2}\left(Z^{+}\right), Z^{+}\)being the integers \(n \geqslant 1\); this brings out the analogy with \(\mathrm{L}^{2}(Q)\). The space \(L^{2}\left(Z^{1}\right)\) of complex functions on the integers \(n=0, \pm 1, \pm 2, \ldots\) \(\left(=Z^{1}\right)\) is defined similarly.

Check that if \(f_{n}\) converges to \(f\) pointwise a.e. and if \(\sup _{n} \geqslant 1\left|f_{n}\right|^{2}\) is summable, then \(f_{n}\) also converges to \(f\) in \(\mathrm{L}^{2}(Q)\). Hint: Use dominated convergence. Step 3 (concluded): The proof of completeness can now be made. The problem is to check that if \(\left\|f_{n}-f_{m}\right\|\) tends to 0 as \(n\) and \(m \uparrow \infty\), then there is an actual function \(f \in \mathrm{L}^{2}(Q)\) such that $$ \lim _{n \dagger \infty}\left\|f_{n}-f\right\|=0 . $$ PROOF. The proof is a little more elaborate than the preceding one. The first take is to produce a plausible candidate for \(f\). Pick \(n_{1}

Define the angle between \(\alpha \neq 0\) and \(\beta \neq 0\) by the rule \(\cos \theta=\operatorname{Re}(\alpha, \beta) /(\|\alpha\|\|\beta\|) .\) Check the "law of cosines": $$ \|\alpha-\beta\|^{2}=\|\alpha\|^{2}-2\|\alpha\|\|\beta\| \cos \theta+\|\beta\|^{2} . $$ Step 2: For any \(f\) and \(g\) from \(L^{2}(Q)\), the elementary bound $$ 2\left|f g^{*}\right|=2|f||g| \leqslant|f|^{2}+|g|^{2} $$ shows that \(f g^{*}\) is a summable function, so that the inner product $$ (f, g)=\int_{Q} f g^{*} $$ makes sense. The reader will have no difficulty in checking that all the customary rules for inner products hold; after all, from an algebraic standpoint, \(\int f g^{*}\) is just the same as the inner product for \(C^{n}(n<\infty)\), only now you have an integral in place of a finite sum. An automatic consequence is the Schwarz inequality: $$ |(f, g)|^{2}=\left|\int_{Q} f g^{*}\right|^{2} \leqslant\|f\|^{2}\|g\|^{2}=\int_{Q}|f|^{2} \int_{Q}|g|^{2} . $$ \(\alpha\) and \(\beta\) are declared to be perpendicular if \((\alpha, \beta)=0\). In this circumstance, the law of cosines becomes the "Pythagorean rule": $$ \|\alpha+\beta\|^{2}=\|\alpha\|^{2}+\|\beta\|^{2} . $$ Check the extended Pythagorean rule: $$ \left\|\sum_{k=1}^{n} \alpha_{k}\right\|^{2}=\sum_{k=1}^{n}\left\|\alpha_{k}\right\|^{2} \quad \text { if } \quad\left(\alpha_{i}, \alpha_{j}\right)=0 \text { for } i \neq j . $$ So much for elementary geometry. The proof that \(L^{2}(Q)\) is a Hilbert space will now be presented in a series of simple steps with explanatory asides. Step 1: Check that \(L^{2}(Q)\) is closed under multiplication by complex numbers and under addition. The first is self-evident, while the second follows from the elementary bound $$ |f+g|^{2} \leqslant 2|f|^{2}+2|g|^{2} $$ upon integrating over \(Q\).
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