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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 39 "Fourier Series of Piecewi
se Polynomials" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 39 "  Define some functions on -Pi< x < Pi." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 61 "f1:=x->piecewise(x<-3*Pi/4,0,x<-Pi/4,-(x+Pi/4)*(x+3*P
i/4),0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "f2:=x->piecewis
e(x<0,0,x<Pi/4,x/4,x<Pi/2,(Pi/2-x)/4,0);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "  construct the 1D \"imag
e\" & plot the bumps" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f :
=x-> f1(x)+f2(x);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot(\{f1,f2\}
,-Pi..Pi,numpoints=200);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 35 " calculate zero cosine coefficient " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "(1/Pi)*Int(f(x),x=-Pi..Pi);
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "a0:=evalf(%);" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "  calculate cos
ine coefficients, a(n), which needs to be a function of n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "a:=n-> value((1/Pi)*Int(f(x)*cos(n*
x),x=-Pi..Pi));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 75 "  calculate sine coefficients, b(n), which also need
s to be a function of n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "
b:=n-> value((1/Pi)*Int(f(x)*sin(n*x),x=-Pi..Pi)) ;" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "  check out the
 results" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "a(n); b(n);" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 " \+
 sum a truncated series" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "
S:=x-> value((a0)/2+Sum(a(n)*cos(n*x)+b(n)*sin(n*x),n=1..5));" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "  \+
plot the original function & truncated series" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 30 "plot(\{f(x), S(x)\}, x=-Pi..Pi);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "  plot th
e error of the truncated series" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 32 "plot(abs(f(x)-S(x)), x=-Pi..Pi);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "  a low-tech way of findi
ng the max error is to focus in on the plot" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 39 "plot(abs(f(x)-S(x)), x=-.786...-0.784);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "23" 0 }
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