Chapter 7: Fourier Series and Transforms

Verify Parseval’s theorem (12.24) for the special cases in Problems 31 to 33.

32. $\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}$and $\mathit{g}\mathbf{\left(}\mathit{\alpha }\mathbf{\right)}$as in problem 21.

Verify Parseval’s theorem (12.24) for the special cases in Problems 31 to 33.

32. $\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}$and $\mathit{g}\mathbf{\left(}\mathit{\alpha }\mathbf{\right)}$as in problem 24a.

Show that if (12.2) is written with the factor $\frac{\mathbf{1}}{\sqrt{\mathbf{2}\mathbf{\pi }}}$multiplying each integral, then the corresponding form of Parseval’s (12.24) theorem is $\underset{\mathbf{-}\mathbf{\infty }}{\overset{\mathbf{\infty }}{\mathbf{\int }}}{\mathbf{\left|}\mathbf{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{\right|}}^{\mathbf{2}}\mathit{d}\mathit{x}\mathbf{=}\underset{-\infty }{\overset{\infty }{\int }}{\left|g\left(\alpha \right)\right|}^2\mathit{d}\mathit{\alpha }$.

Starting with the symmetrized integrals as in Problem 34, make the substitutions $\mathit{\alpha }\mathbf{=}\frac{\mathbf{2}\mathbf{\pi }\mathbf{p}}{\mathbf{h}}$(where pis the new variable, his a constant), $\mathit{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\mathit{\psi }\mathbf{\left(}\mathbf{x}\mathbf{\right)}$, localid="1664270725133" $\mathit{g}\mathbf{\left(}\mathbf{\alpha }\mathbf{\right)}\mathbf{=}\sqrt{\frac{\mathbf{h}}{\mathbf{2}\mathbf{\pi }}}\mathit{\varphi }\mathbf{\left(}\mathbf{p}\mathbf{\right)}$; show that then

$\begin{array}{c}\mathbf{\psi }\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\frac{\mathbf{1}}{\sqrt{\mathbf{h}}}{\int }_{-\infty }^{\infty }\varphi \left(p\right)e^{\frac{2\pi ipx}{h}}dp\\ \mathbf{\varphi }\mathbf{\left(}\mathbf{p}\mathbf{\right)}\mathbf{=}\frac{\mathbf{1}}{\sqrt{\mathbf{h}}}{\int }_{-\infty }^{\infty }\psi \left(x\right)e^{-\frac{2\pi ipx}{h}}dx\\ {\int }_{-\infty }^{\infty }{\left|\psi \left(x\right)\right|}^{2}dx={\int }_{-\infty }^{\infty }{\left|\varphi \left(p\right)\right|}^{2}dp\end{array}$

This notation is often used in quantum mechanics.

Normalize $\mathit{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$in Problem 21; that is find the factor Nso that ${\mathbf{\int }}_{\mathbf{-}\mathbf{\infty }}^{\mathbf{\infty }}{\mathbf{\left|}\mathbf{N}\mathbf{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{\right|}}^{\mathbf{2}}\mathbf{=}\mathbf{1}$.Let $\mathit{\psi }\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\mathit{N}\mathit{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$, and find $\mathit{\varphi }\mathbf{\left(}\mathbf{p}\mathbf{\right)}$as given in Problem 35. Verify Parseval’s theorem, that is, show that${\mathbf{\int }}_{\mathbf{-}\mathbf{\infty }}^{\mathbf{\infty }}{\mathbf{\left|}\mathbf{\varphi }\mathbf{\left(}\mathbf{p}\mathbf{\right)}\mathbf{\right|}}^{\mathbf{2}}\mathit{d}\mathit{p}\mathbf{=}\mathbf{1}$.

We have said that Fourier series can represent discontinuous functions although power series cannot. It might occur to you to wonder why we could not substitute the power series for $\mathbf{sin}\mathbf{nx}$and $\mathbf{cos}\mathbf{nx}$(which converge for all x) into a Fourier series and collect terms to obtain a power series for a discontinuous function. As an example of what happens if we try this, consider the series in Problem 9.5. Show that the coefficients of x, if collected, form a divergent series; similarly, the coefficients of ${\mathbf{x}}^3$form a divergent series, and so on.

If f(x)is complex, we usually want the average of the square of the absolute value of f(x). Recall that${\left|f\left(x\right)\right|}^{\mathbf{2}}\mathbf{=}\mathbf{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{·}\overline{\mathbf{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}}$where$\overline{\mathbf{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}}$means the complex conjugate of f(x). Show that if a complex$\mathbf{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}{\mathbf{\sum }}_{\mathbf{-}\mathbf{\infty }}^{\mathbf{\infty }}{\mathbf{c}}_{\mathbf{n}}{\mathbf{e}}^{\mathbf{i}\mathbf{n}\mathbf{\pi }\mathbf{x}\mathbf{/}\mathbf{l}}$, then (11.5)holds

Find the exponential Fourier transform of the given f(x) and write f(x)as a Fourier integral [that is, find $\mathbf{\text{g}}\mathbf{\left(}\mathbf{\text{α}}\mathbf{\right)}$in equation (12.2) and substitute your result into the first integral in equation (12.2)].

role="math" localid="1664339656101" $\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{\{}\begin{array}{l}\begin{array}{rr}\mathbf{-}\mathbf{1}\mathbf{,}& \mathbf{-}\mathbf{\pi }\mathbf{<}\mathbf{x}\mathbf{<}\mathbf{0}\end{array}\\ \begin{array}{rr}\mathbf{1}\mathbf{,}& \mathbf{0}\mathbf{<}\mathbf{x}\mathbf{<}\mathbf{\pi }\end{array}\\ \begin{array}{rr}\mathbf{0}\mathbf{,}& \mathbf{\left|}\mathbf{x}\mathbf{\right|}\mathbf{>}\mathbf{\pi }\end{array}\end{array}$

Sketch several periods of the corresponding periodic function of period . Expand the periodic$\mathbf{2}\mathit{\pi }$ function in a sine-cosine Fourier series.

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{\{}\begin{array}{llllll}\mathbf{0}\mathbf{,}\mathbf{}\mathbf{}& \mathbf{-}\mathbf{\pi }& \mathbf{<}& \mathbf{x}& \mathbf{<}& \frac{\mathbf{\pi }}{\mathbf{2}}\\ \mathbf{1}\mathbf{,}& \frac{\mathbf{\pi }}{\mathbf{2}}& \mathbf{<}& \mathbf{x}& \mathbf{<}& \mathit{\pi }\mathbf{,}\end{array}$

The functions in Problems 1 to 3 are neither even nor odd. Write each of them as the sum of an even function and an odd function.

(a) ${\mathit{x}}^{\mathbf{5}}\mathbf{-}{\mathit{x}}^{\mathbf{4}}\mathbf{+}{\mathit{x}}^{\mathbf{3}}\mathbf{-}\mathbf{1}$

(b)$\mathbf{1}\mathbf{+}{\mathit{e}}^{\mathbf{x}}$