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11.2E: Fourier Series I (Exercises)

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[exer:11.2.1] Prove Theorem [thmtype:11.1.5}.

[exer:11.2.2] $$L=1$$; $$f(x)=2-x$$

[exer:11.2.3] $$L=\pi$$; $$f(x)=2x-3x^2$$

[exer:11.2.4] $$L=1$$; $$f(x)=1-3x^2$$

[exer:11.2.5] $$L=\pi$$; $$f(x)=|\sin x|$$

[exer:11.2.6] $$L=\pi$$; $$f(x)=x\cos x$$

[exer:11.2.7] $$L=\pi$$; $$f(x)=|x|\cos x$$

[exer:11.2.8] $$L=\pi$$; $$f(x)=x\sin x$$

[exer:11.2.9] $$L=\pi$$; $$f(x)=|x|\sin x$$

[exer:11.2.10] $$L=1$$; $$f(x)= \left\{\begin{array}{cl} 0,&-1<x<{1\over2},\ $5pt] \cos\pi x,&-{1\over2}<x<{1\over2},\ \[5pt] 0,&\phantom{-}{1\over2}<x<1 \end{array}\right.$$ [exer:11.2.11] $$L=1$$; $$f(x)= \left\{\begin{array}{cl} 0,&-1<x<{1\over2},\ \[5pt] x\cos\pi x,&-{1\over2}<x<{1\over2},\ \[5pt] 0,&\phantom{-}{1\over2}<x<1 \end{array}\right.$$ [exer:11.2.12] $$L=1$$; $$f(x)= \left\{\begin{array}{cl} 0,&-1<x<{1\over2},\ \[5pt] \sin\pi x,&-{1\over2}<x<{1\over2},\ \[5pt] 0,&\phantom{-}{1\over2}<x<1 \end{array}\right.$$ [exer:11.2.13] $$L=1$$; $$f(x)= \left\{\begin{array}{cl} 0,&-1<x<{1\over2},\ \[5pt] |\sin\pi x|,&-{1\over2}<x<{1\over2},\ \[5pt] 0,&\phantom{-}{1\over2}<x<1 \end{array}\right.$$ [exer:11.2.14] $$L=1$$; $$f(x)= \left\{\begin{array}{cl} 0,&-1<x<{1\over2},\ \[5pt] x\sin\pi x,&-{1\over2}<x<{1\over2},\ \[5pt] 0,&\phantom{-}{1\over2}<x<1 \end{array}\right.$$ [exer:11.2.15] $$L=4$$; $$f(x)= \left\{\begin{array}{cl} 0,&-4<x<0,\\x,&\phantom{-}0<x<4 \end{array}\right.$$ [exer:11.2.16] $$L=1$$; $$f(x)= \left\{\begin{array}{cl} x^2,&-1< x<0, \\1-x^2,&\phantom{-}0<x<1 \end{array}\right.$$ [exer:11.2.17] Verify the Gibbs phenomenon for $$f(x)= \left\{\begin{array}{rl} 2,&-2< x< -1,\\1,&-1<x<1,\\-1,&\phantom{-}1< x<2. \end{array}\right.$$ [exer:11.2.18] Verify the Gibbs phenomenon for $$f(x)= \left\{\begin{array}{rl} 2,&-3< x< -2,\\3,&-2<x<2,\\1,&\phantom{-}2< x<3. \end{array}\right.$$ [exer:11.2.19] Deduce from Example [example:11.2.5} that \[\sum_{n=0}^\infty{1\over(2n+1)^2}={\pi^2\over 8}.$

[exer:11.2.20]

Find the Fourier series of $$f(x)=e^x$$ on $$[-\pi,\pi]$$.

Deduce from

a

that

$\sum_{n=0}^\infty{1\over n^2+1}={\pi\coth\pi-1\over2}.$

[exer:11.2.21] Find the Fourier series of $$f(x)=(x-\pi)\cos x$$ on $$[-\pi,\pi]$$.

[exer:11.2.22] Find the Fourier series of $$f(x)=(x-\pi)\sin x$$ on $$[-\pi,\pi]$$.

[exer:11.2.23] Find the Fourier series of $$f(x)=\sin kx$$ ($$k\ne$$ integer) on $$[-\pi,\pi]$$.

[exer:11.2.24] Find the Fourier series of $$f(x)=\cos kx$$ ($$k\ne$$ integer) on $$[-\pi,\pi]$$.

[exer:11.2.25]

Suppose $$g'$$ is continuous on $$[a,b]$$ and $$\omega\ne0$$. Use integration by parts to show that there’s a constant $$M$$ such that

$\left|\int_a^bg(x)\cos\omega x\,dx\right|\le{M\over\omega} \mbox{\quad and \quad} \left|\int_a^bg(x)\sin\omega x\,dx\right|\le{M\over\omega},\quad \omega>0.$

Show that the conclusion of

a

also holds if $$g$$ is piecewise smooth on $$[a,b]$$. (This is a special case of Riemann’s Lemma.

We say that a sequence $$\{\alpha_n\}_{n=1}^\infty$$ is of order $$n^{-k}$$ and write $$\alpha_n=O(1/n^k)$$ if there’s a constant $$M$$ such that

$|\alpha_n|<{M\over n^k},\quad n=1,2,3,\dots.$

Let $$\{a_n\}_{n=1}^\infty$$ and $$\{b_n\}_{n=1}^\infty$$ be the Fourier coefficients of a piecewise smooth function. Conclude from

b

that $$a_n=O(1/n)$$ and $$b_n=O(1/n)$$.

[exer:11.2.26]

Suppose $$f(-L)=f(L)$$, $$f'(-L)=f'(L)$$, $$f'$$ is continuous, and $$f''$$ is piecewise continuous on $$[-L,L]$$. Use Theorem [thmtype:11.2.4} and integration by parts to show that

$f(x)=a_0+\sum_{n=1}^\infty\left(a_n\cos{n\pi x\over L}+b_n\sin{n\pi x\over L}\right),\quad -L\le x\le L,$

with

$a_0={1\over 2L}\int_{-L}^L f(x)\,dx,$

$a_n= -{L\over n^2\pi^2}\int_{-L}^L f''(x)\cos{n\pi x\over L}\,dx,\mbox{\quad and \quad} b_n=-{L\over n^2\pi^2}\int_{-L}^L f''(x)\sin{n\pi x\over L}\,dx,\,n\ge1.$

Show that if, in addition to the assumptions in

a

, $$f''$$ is continuous and $$f'''$$ is piecewise continuous on $$[-L,L]$$, then

$a_n={L^2\over n^3\pi^3}\int_{-L}^Lf'''(x)\sin{n\pi x\over L}\,dx.$

[exer:11.2.27] Show that if $$f$$ is integrable on $$[-L,L]$$ and

$f(x+L)=f(x),\quad -L<x<0$

(Figure [figure:11.2.8}), then the Fourier series of $$f$$ on $$[-L,L]$$ has the form

$A_0+\sum_{n=1}^\infty\left(A_n\cos{2n\pi\over L}+B_n\sin{2n\pi\over L}\right)$

where

$A_0={1\over L}\int_0^Lf(x)\,dx,$

and

$A_n={2\over L}\int_0^Lf(x)\cos{2n\pi x\over L}\,dx, \quad B_n={2\over L}\int_0^Lf(x)\sin{2n\pi x\over L}\,dx,\quad n=1,2,3,\dots.$

[exer:11.2.28] Show that if $$f$$ is integrable on $$[-L,L]$$ and

$f(x+L)=-f(x),\quad -L<x<0$

(Figure [figure:11.2.9}), then the Fourier series of $$f$$ on $$[-L,L]$$ has the form

$\sum_{n=1}^\infty\left(A_n\cos{(2n-1)\pi x\over L}+B_n\sin{(2n-1)\pi x\over L}\right),$

where

$A_n={2\over L}\int_0^Lf(x)\cos{(2n-1)\pi x\over L}\,dx \text{\quad and \quad } B_n={2\over L}\int_0^Lf(x)\sin{(2n-1)\pi x\over L}\,dx,\quad n=1,2,3,\dots.$

[exer:11.2.29] Suppose $$\phi_1$$, $$\phi_2$$, …, $$\phi_m$$ are orthogonal on $$[a,b]$$ and

$\int_a^b\phi_n^2(x)\,dx\ne0,\quad n=1,2,\dots,m.$

If $$a_1$$, $$a_2$$, …, $$a_m$$ are arbitrary real numbers, define

$P_m=a_1\phi_1+a_2\phi_2+\cdots+a_m\phi_m.$

Let

$F_m=c_1\phi_1+c_2\phi_2+\cdots+c_m\phi_m,$

where

$c_n={\int_a^bf(x)\phi_n(x)\,dx\over\int_a^b\phi_n^2(x)\,dx};$

that is, $$c_1$$, $$c_2$$, …, $$c_m$$ are Fourier coefficients of $$f$$.

Show that

$\int_a^b(f(x)-F_m(x))\phi_n(x)\,dx=0,\quad n=1,2,\dots,m.$

Show that

$\int_a^b(f(x)-F_m(x))^2\,dx\le \int_a^b(f(x)-P_m(x))^2\,dx,$

with equality if and only if $$a_n=c_n$$, $$n=1,2,\dots, m$$.

Show that

$\int_a^b(f(x)-F_m(x))^2\,dx=\int_a^bf^2(x)\,dx-\sum_{n=1}^mc_n^2\int_a^b \phi_n^2\,dx.$

Conclude from

c

that

$\sum_{n=1}^m c_n^2\int_a^b\phi_n^2(x)\,dx\le \int_a^bf^2(x)\,dx.$

[exer:11.2.30] If $$A_0$$, $$A_1$$, …, $$A_m$$ and $$B_1$$, $$B_2$$, …, $$B_m$$ are arbitrary constants we say that

$P_m(x)=A_0+\sum_{n=1}^m\left(A_n\cos{n\pi x\over L}+B_n\sin{n\pi x\over L}\right)$

is a trigonometric polynomial of degree $$\le m$$.

Now let

$a_0+\sum_{n=1}^\infty\left(a_n\cos{n\pi x\over L}+b_n\sin{n\pi x\over L}\right)$

be the Fourier series of an integrable function $$f$$ on $$[-L,L]$$, and let

$F_m(x)= a_0+\sum_{n=1}^m\left(a_n\cos{n\pi x\over L}+b_n\sin{n\pi x\over L}\right).$

Conclude from Exercise [exer:11.2.29}

b

that

$\int_{-L}^L(f(x)-F_m(x))^2\,dx\le \int_{-L}^L(f(x)-P_m(x))^2\,dx,$

with equality if and only if $$A_n=a_n$$, $$n=0$$, $$1$$, …, $$m$$, and $$B_n=b_n$$, $$n=1$$, $$2$$, …, $$m$$.

Conclude from Exercise [exer:11.2.29}

d

that

$2a_0^2+\sum_{n=1}^m(a_n^2+b_n^2)\le{1\over L}\int_{-L}^Lf^2(x)\,dx$

for every $$m\ge0$$.

Conclude from

b

that $$\lim_{n\to\infty}a_n=\lim_{n\to\infty}b_n=0$$.