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

$$ f(x) = T_m(x) + T_{m+1}(x) + \cdots + T_n(x), $$

where $T_k$ is the Chebyshev polynomial of degree $k$, 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,'interval',[.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,1.2)
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 2.744878 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,1)
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,1)
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 1.239600 seconds.