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Mathematics LibreTexts

11.2E: Fourier Series I (Exercises)

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    18285
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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 \(\lim_{n\to\infty}a_n=\lim_{n\to\infty}b_n=0\).