A wiggly function and its best approximations

Ricardo Pachon and Nick Trefethen, November 2010

Ken Lord, whose doctoral supervisor was the Chebyshev technology wizard Charles Clenshaw, has explored functions of the form

f(x) = T_m(x) + T_m+1(x) + ... + T_n(x),

where T_k is the kth Chebyshev polynomial, as challenging functions for minimax approximation by polynomials of lower order. We can construct such functions in a single Chebfun command:

fmn = @(m,n) sum(chebpoly(m:n),2);

For example, here we plot f(30,40) and its best approximation of degree 29:

LW = 'linewidth'; FS = 'fontsize'; fs = 14;
tic, m = 30; n = 40;
f = fmn(m,n);
subplot(2,2,1), plot(f,LW,1)
grid on, title('f(30,40)',FS,fs)
subplot(2,2,2), plot(f{.8,1},LW,1.6)
grid on, title('closeup',FS,fs)
p = remez(f,m-1); err = f-p;
subplot(2,2,3), plot(err,'r',LW,1)
grid on, title('f - p',FS,fs)
subplot(2,2,4), plot(err{.8,1},'r',LW,1.6)
grid on, title('closeup',FS,fs), toc

Elapsed time is 2.079370 seconds.

Here are f(200,220) and its best approximation of degree 199:

tic, m = 200; n = 220;
f = fmn(m,n);
subplot(2,2,1), plot(f,LW,.8)
grid on, title('f(200,220)',FS,fs)
subplot(2,2,2), plot(f{.995,1},LW,1.6)
grid on, title('closeup',FS,fs), xlim([.995 1])
p = remez(f,m-1); err = f-p;
subplot(2,2,3), plot(err,'r',LW,.8)
grid on, title('f - p',FS,fs)
subplot(2,2,4), plot(err{.995,1},'r',LW,1.6)
grid on, title('closeup',FS,fs), xlim([.995 1]), toc

Elapsed time is 1.461200 seconds.