Term-wise Integration and Differentiation of Fourier Series

Section 4.3 Term-wise Integration and Differentiation of Fourier Series

Suppose that

\begin{equation*} g(x)\sim a_0 +\sum_{n=1}^{\infty}\left[ a_n \cos (n x) + b_n \sin (nx) \right]. \end{equation*}

is the Fourier series of $g\in \cR[-\pi, \pi]\text{.}$ We address whether we can integrate this relation term-wise and, when $g'$ exists and is in $\cR[-\pi, \pi]\text{,}$ whether we can differentiate this relation term-wise.

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Theorem 4.3.1. Term-wise Integration of Fourier Series.

Suppose that

\begin{equation*} g(x)\sim a_0 +\sum_{n=1}^{\infty}\left[ a_n \cos (n x) + b_n \sin (nx) \right] \end{equation*}

is the Fourier series of $g\in \cR[-\pi, \pi]\text{.}$ Then for any $a, b \in (-\pi, \pi)\text{,}$

\begin{equation*} \int_{a}^{b} g(x) = a_{0}(b-a) + \sum_{n=1}^{\infty}\int_{a}^{b} \left[ a_n \cos (n x) + b_n \sin (nx) \right]\, dx. \end{equation*}

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Proof.

The assertion is equivalent to

\begin{equation} \int_{a}^{b} \left[ g(x)-S_{N}[g](x)\right]\, dx \to 0 \text{ as $N\to \infty$.}\tag{4.3.1} \end{equation}

But

\begin{align*} \vert \int_{a}^{b} \left[ g(x)-S_{N}[g](x)\right]\, dx\vert \amp \le \int_{a}^{b} \vert g(x)-S_{N}[g](x)\vert \, dx\\ \amp \le \left(2\pi\right)^{1/2}\left( \int_{-\pi}^{\pi} \vert g(x)-S_{N}[g](x)\vert^{2} \, dx\right)^{1/2}, \end{align*}

and using $\int_{-\pi}^{\pi} \vert g(x)-S_{N}[g](x)\vert^{2} \, dx\to 0\text{,}$ we conclude (4.3.1).

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Recall that a function is called piecewise continuous on $[a, b]$ if there is a finite partition $a=a_{0}\lt a_{1}\lt \cdots \lt a_{m}=b$ of $[a, b]$ such that its restriction on any $(a_{k}, a_{k+1})$ is continuous and has a continuous extension to $[a_{k}, a_{k+1}]\text{.}$ This implies that the function is continuous at every point of $[a, b]$ with possibly the exception at the $a_{k}$’s and that it has both left and right limits at each of these $a_{k}$’s.

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Theorem 4.3.2. Term-wise Differentiation of Fourier Series.

Suppose that $g(x)$ is continuous on $[-\pi, \pi]\text{,}$ and $g(-\pi)=g(\pi)\text{,}$ and that $g'(x)$ exists except at a finite number of points and is piecewise continuous. Denote the Fourier series expansion of $g'(x)$ on $[-\pi, \pi]$ by

\begin{equation*} g'(x)\sim a_0' +\sum_{n=1}^{\infty}\left[ a_n' \cos (nx)+ b_n' \sin (nx) \right]. \end{equation*}

Then

\begin{equation*} a_0'=0, \text{ and } a_n'=n b_{n}, b_n'=- n a_{n}. \end{equation*}

In other words, under our assumptions here, the Fourier series expansion of $g'(x)$ on $[-\pi, \pi]$ can be obtained by term-wise differentiation of the Fourier series expansion of $g(x)$ on $[-\pi, \pi]\text{.}$

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Proof.

Note that, for $n=0\text{,}$

\begin{equation*} 2\pi a_{0}'=\int_{-\pi}^{\pi} g'(x)\, dx=g(\pi)-g(-\pi)=0, \end{equation*}

and for $n\ge 1\text{,}$ integration by parts gives

\begin{align*} \pi a_{n}' = \amp \int_{-\pi}^{\pi } g'(x) \cos (nx) \, dx \\ = \amp g(x) \cos (nx) \Big|_{x=-\pi}^{x=\pi}+ n \int_{-\pi}^{\pi} g(x) \sin (nx) \, dx \text{ (using $g \in C[-\pi, \pi] $)}\\ = \amp n \int_{-\pi}^{\pi} g(x) \sin (nx) \, dx \text{ (using $ g(-\pi)=g(\pi)$) }\\ = \amp \pi n b_{n};\\ \pi b_{n}'= \amp \int_{-\pi}^{\pi} g'(x) \sin (nx) \, dx \\ = \amp g(x) \sin (nx) \Big|_{x=-\pi}^{x=\pi}- n \int_{-\pi}^{\pi} g(x) \cos (nx) \, dx \text{ (using $ g \in C[-\pi, \pi]$)} \\ = \amp - n \int_{-\pi}^{\pi} g(x) \cos (nx) \, dx \\ = \amp -\pi n a_{n}. \end{align*}

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Remark 4.3.3.

Note that the continuity and periodicity of $g$ can’t be dropped, as can be seen by the Fourier expansion of $g(x)=x$ on $(-\pi, \pi)\text{.}$

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Exercise 4.3.4. Relation Between the Fourier Series of a Function and its Derivative.

Compute the Fourier series of $g(x)=x$ and $g'(x)=1$ on $(-\pi, \pi)\text{,}$ then study the relation between the two series.

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Remark 4.3.5.

The orthogonal family of functions $\{\sin (nx)\}_{n=1}^{\infty}$ on $[0, \pi]$ happens to be the restriction to $[0, \pi]$ of the odd functions $\sin(nx)\text{.}$ For any continuous (or Riemann integrable) $g$ on $[0, \pi]\text{,}$ let $g_{\text{odd}}$ be the odd extension of $g$ to $[-\pi, \pi]\text{.}$ Then the Fourier series of $g_{\text{odd}}$ on $[-\pi, \pi]$ would only consist of the $\sin(nx)$ terms, and would converge to $g_{\text{odd}}$ in the mean square on $[-\pi, \pi]\text{.}$ In particular, the restriction of the Fourier series would converge to $g$ in the mean square on $[0, \pi]\text{.}$

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Note that if $b_{n}$ denotes the Fourier coefficient of $g_{\text{odd}}$ with respect to $\sin(nx)$ on $[-\pi, \pi]\text{,}$ then

\begin{equation*} b_{n}=\frac 1\pi \int_{-\pi}^{\pi} g_{\text{odd}}(x)\sin(nx)\, dx=\frac 2\pi \int_{0}^{\pi} g(x)\sin (nx)\, dx. \end{equation*}

The series $\sum_{n=1}^{\infty}b_{n}\sin(nx)$ is typically called the Fourier sine series of $g$ on $[0, \pi]\text{.}$

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Likewise, the family of functions $\{\cos (nx)\}_{n=0}^{\infty}$ on $[0, \pi]$ is orthogonal on $[0,\pi]$ and happens to be the restriction to $[0, \pi]$ of the even functions $\cos(nx)\text{.}$ For any continuous (or Riemann integrable) $g$ on $[0, \pi]\text{,}$ let $g_{\text{even}}$ be the even extension of $g$ to $[-\pi, \pi]\text{.}$ Then the Fourier series of $g_{\text{even}}$ on $[-\pi, \pi]$ would only consist of the $\cos(nx)$ terms, and would converge to $g_{\text{even}}$ in the mean square on $[-\pi, \pi]\text{.}$ In particular, the restriction of the Fourier series would converge to $g$ in the mean square on $[0, \pi]\text{.}$ This series is called the Fourier cosine series of $g$ on $[0, \pi]\text{.}$ Note that for $n\ge 1$

\begin{equation*} a_{n}=\frac 1\pi \int_{-\pi}^{\pi} g_{\text{even}}(x)\cos(nx)\, dx=\frac 2\pi \int_{0}^{\pi} g(x)\cos (nx)\, dx, \end{equation*}

while

\begin{equation*} a_{0}=\frac {1}{2\pi} \int_{-\pi}^{\pi} g_{\text{even}}(x)\, dx= \frac 1\pi \int_{0}^{\pi} g(x)\, dx. \end{equation*}

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Exercise 4.3.6. Fourier Sine Series.

Compute the Fourier sine series of $g(x)=1$ on $(0, \pi)\text{,}$ then study the sense in which this Fourier sine series approximates $g(x)\text{.}$

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Exercise 4.3.7. Fourier Cosine Series.

Compute the Fourier cosine series of $g(x)=x$ on $(0, \pi)\text{,}$ then study the sense in which this Fourier cosine series approximates $g(x)\text{.}$

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Exercise 4.3.8.

Suppose that $g(x)$ is continuous on $[-l, l]\text{,}$ $g(-l)=g(l)\text{,}$ and that $g'(x)$ exists except at a finite number of points and is piecewise continuous. Suppose that

\begin{equation*} g \sim a_0 +\sum_{n=1}^{\infty}\left[ a_n \cos \left(\frac{n \pi}{l} x \right) + b_n \sin \left(\frac{n \pi}{l} x \right) \right] \end{equation*}

is the Fourier series of $g$ on $(-l, l)\text{.}$ Prove that

\begin{equation*} \int_{-l}^{l} |g'(x)|^{2} \, dx = \sum_{n=1}^{\infty} \left(\frac{n \pi}{l} \right)^{2}l \left[ |a_n|^{2} + |b_n|^{2}\right]. \end{equation*}

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Exercise 4.3.9. Wirtinger’s inequality.

Suppose that $g(x)$ is continuous on $[0, L]\text{,}$ $g(0)=g(L)\text{,}$ and that $g'(x)$ exists except at a finite number of points and is piecewise continuous. Prove that

\begin{equation} \int_{0}^{L} |g(x)-\bar g|^{2} \, dx \le \left(\frac{L}{2\pi}\right)^{2} \int_{0}^{L} |g'(x)|^{2} \, dx\tag{4.3.2} \end{equation}

with equality iff $g=\bar g + a_{1}\cos \left(\frac{2\pi}{L} x \right) +b_{1} \sin \left(\frac{2\pi}{L} x \right)$ for some constants $a_{1}, b_{1}\text{.}$ Here $\bar g =L^{-1} \int_{0}^{L} g(x)\, dx$ is the average of $g$ over $(0, L)\text{.}$

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Exercise 4.3.10. Another Wirtinger’s inequality.

Suppose that $g(x)$ is continuous on $[0, L]$ and that $g'(x)$ exists except at a finite number of points and is piecewise continuous. Prove that

\begin{equation} \int_{0}^{L} |g(x)-\bar g|^{2} \, dx \le \left(\frac{L}{\pi}\right)^{2} \int_{0}^{L} |g'(x)|^{2} \, dx\tag{4.3.3} \end{equation}

with equality iff $g=\bar g + a_{1}\cos \left(\frac{\pi}{L} x \right)$ for some constants $a_{1}\text{.}$ Here $\bar g =L^{-1} \int_{0}^{L} g(x)\, dx$ is the average of $g$ over $(0, L)\text{.}$

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Hint.

$\{ \cos \left(\frac{n\pi}{L} x \right) \}_{n=0}^{\infty}$ is a complete system of orthogonal functions on $(0, L)$ and so is $\{ \sin \left(\frac{n\pi}{L} x \right) \}_{n=1}^{\infty}\text{.}$ Expand $g$ in the former and $g'$ in the latter.

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Exercise 4.3.11.

Prove that a sequence of functions converges in $L^{2}(-l, l)$ iff the sequence of their Fourier coefficients converges in $l^{2}\text{.}$

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Exercise 4.3.12.

Let $M > 0$ be a finite number. Consider the set $S_{M}$ of continuous $g(x)$ on $[-l, l]$ such that $g'(x)$ exists except at a finite number of points and is piecewise continuous and $\left| \int_{-l}^{l} g(x)\, dx\right|, \int_{-l}^{l} |g'(x)|^{2} \, dx \le M\text{.}$ Prove that the closure of $S_{M}$ in $L^{2}(-l, l)$ is compact.

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Hint.

Use Exercise 2.8.1 and Exercise 4.3.8 to prove that any sequence in $S_{M}$ has a subsequence converging in $L^{2}(-l, l)\text{.}$

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