Solution Assignment 2 (MATH464, Spring 2020)

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1(a) Sketch graphs of f(t) and f(1/3-t), graphs of f(t) and f(2/3-t)

(The vertical lines are not really part of the graphs.)

format compact; format short g
frac = @(x) x-floor(x);      % fractional part
f = @(x) frac(x)<.5;         % define function f

t = -1:.001:1;
plot(t,f(t),'g',t,f(1/3-t),'k:','LineWidth',3); axis equal
grid on; xticks(-1:.5:1); yticks(-1:.5:1);
legend('f(t)','f(1/3-t)')

snapnow

plot(t,f(t),'g',t,f(2/3-t),'k:','LineWidth',3); axis equal;
grid on; xticks(-1:.5:1); yticks(-1:.5:1);
legend('f(t)','f(2/3-t)')

1(a)(iii) Sketch graph of q(x) on [-1,1]

We found: q(x) is 1-periodic function with

$$q(x) = \cases{x & for $x\in[0,\frac12)$ \cr -x & for $x\in[-\frac12,0)$}=|x| \mbox{ for $x\in [-\frac12,\frac12)$}$$

x = linspace(-1,1,1000);
q = abs(x-round(x));
plot(x,q,'LineWidth',3);
xticks(-1:.25:1); yticks(0:.25:.5); axis equal;  axis([-1 1 0 .5]); grid on

1(b) Plot $q_{(10)}$ for convolution q=f*f

Note that $\hat q_{-k}=\hat q_k$.

nv = 1:10;                   % vector of frequencies
qhpos = -1./(pi^2*nv.^2);    % vector qhpos = [qhat_1,...,qhat_{10}]
qhpos(2:2:end) = 0;          % set to zero for even k

x = linspace(-1,1,1000);     % 1000 equidistant points from -1 to 1 for plotting
plot(x,foursum(x,.25,qhpos)); grid on; axis equal

Warning: Imaginary parts of complex X and/or Y arguments ignored

2(b) Plot $f*p$ for h=1/8

Note $\hat f_{-k}=-\hat f_k$, $\hat p_{-k}=\hat p_k$.

h = 1/8;
nv = 1:30;
fh0 = .5;
fhpos = -1i./(pi*nv);        % vector fhpos = [qhat_1,...,qhat_{10}]
fhpos(2:2:end) = 0;          % set to zero for even k

ph0 = 1;
phpos = sinc(nv*h);          % NOTE: sinc_Matlab(x) = sinc(pi*x)

x = linspace(-1,1,1000);     % 1000 equidistant points from -1 to 1 for plotting
plot(x,foursum(x,fh0*ph0,fhpos.*phpos));
grid on; title('f{\ast}p')

Warning: Imaginary parts of complex X and/or Y arguments ignored

2(b) Plot $q*p$

Note $\hat q_{-k}=\hat q_k$, $\hat p_{-k}=\hat p_k$.

qh0 = fh0*fh0;
qhpos = fhpos.*fhpos;
plot(x,foursum(x,qh0*ph0,qhpos.*phpos));
grid on; title('q{\ast}p')

Warning: Imaginary parts of complex X and/or Y arguments ignored

4(a)

Consider 1-periodic functions $f,F$ with $f(x)=e^x$ and $F(x)=f(x)+(e-1)*(1-x)$ for $x\in[0,1]$.

x = (0:999)/1000;
f = exp(x);
F = f + (exp(1)-1)*(1-x);
plot([x-1,x],[f,f],[x-1,x],[F,F])
legend('f','F','Location','SouthEast')

4(c),(d) Plot $f_{(10)}=f*D_{10}$ together with $f*\sigma_{10}$

We have

$$\hat f_k = \frac{e-1}{1-2\pi i k}$$

N = 10;
Nv = 1:N;
c = exp(1) - 1;
fh0 = c;
fhpos = c./(1-2i*pi*Nv);
shpos = (1-Nv/(N+1));     % f*sigma10: fhat_k*(1-|k|/(N+1))
qhpos = sinc(Nv/(N+.5));  % f*A10:     fhat_k*sinc_Matlab(k/(N+.5))

x = 0:.001:1;
plot(x,foursum(x,fh0,fhpos),x,foursum(x,fh0,fhpos.*shpos),...
     x,foursum(x,fh0,fhpos.*qhpos),x,exp(x),'k:')
legend('f_{(10)}','f*\sigma_{10}','f*A_{10}','f','Location','NorthWest')

snapnow
x = 0:.001:.5;
f = exp(x);
plot(x,foursum(x,fh0,fhpos)-f,x,foursum(x,fh0,fhpos.*shpos)-f,...
     x,foursum(x,fh0,fhpos.*qhpos)-f)
grid on; axis tight; title('errors')
legend('f_{(10)}-f','f*\sigma_{10}-f','f*A_{10}-f')

Warning: Imaginary parts of complex X and/or Y arguments ignored 

Warning: Imaginary parts of complex X and/or Y arguments ignored

4(d) Plot Dirichlet kernel $D_{10}$, kernel $A_{10}$

For $g=A_N$ we have the Fourier coefficients $\hat g_k=\mathrm{sinc}_{Matlab}(\frac{k}{N+1/2})$ for $k=-N,\ldots,N$, $\hat g_k=0$ for $|k|>N$.

Note that for $x\in[.25,.75]$ we have $D_N(x)\approx c\cdot \sin(2\pi M x)$ with $M=N+\frac12$. Therefore $D_N*p\approx 0$ in [.25,.75], see problem 3(i).

x = -.5:.001:.5;
plot(x,foursum(x,1,Nv.^0),x,foursum(x,1,qhpos));
axis tight; grid on; grid minor
legend('Dirichlet kernel D_N','averaged kernel A_N')

Warning: Imaginary parts of complex X and/or Y arguments ignored


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