FejerJackson.m

%% Fejer-Jackson inequality
% Nick Trefethen, July 2015

%%
% (Chebfun example fourier/FejerJackson.m)
% [Tags: #Fejer-Jackson, Gibbs phenomenon]

%%
% The Fejer-jackson inequality concerns the sums
% $$ f_n(t) = \sum_{k=1}^n {\sin(k t)\over k}, \quad
% x \in (0,\pi) . $$
% In Chebfun, we can compute $f_n$ like this: 
fnx = @(n,x) sum(sin((n:-1:1)*x)./(n:-1:1));
fn = @(n) chebfun(@(x) fnx(n,x),[0 pi],'vectorize');

%%
% Here for example is the case $n=32$:
XT = 'xtick'; XL = 'xticklabel'; YT = 'ytick';
YL = 'yticklabel'; FS = 'fontsize'; LW = 'linewidth';
MS = 'markersize'; ax = [-.1 3.3 0 2];
f32 = fn(32); plot(f32,LW,1.6), axis(ax), grid on
set(gca,XT,pi*(0:.5:1),XL,{'0','\pi/2','\pi'})
set(gca,YT,pi*(0:.25:.5),YL,{'0','\pi/4','\pi/2'})
ss = 'Min and max of f%d:  %9.6f, %9.6f';
title(sprintf(ss,32,minandmax(f32)),FS,12)

%%
% Here is $n=128$.
f128 = fn(128); plot(f128,LW,1), axis(ax), grid on
set(gca,XT,pi*(0:.5:1),XL,{'0','\pi/2','\pi'})
set(gca,YT,pi*(0:.25:.5),YL,{'0','\pi/4','\pi/2'})
title(sprintf(ss,128,minandmax(f128)),FS,12)

%%
% And here is $n=512$, now with the plot showing
% just a closeup near $x=0$.
f512 = fn(512); plot(f512{0,0.2},LW,1), axis([0 .2 0 2]), grid on
set(gca,YT,pi*(0:.25:.5),YL,{'0','\pi/4','\pi/2'})
title(sprintf(ss,512,minandmax(f512)),FS,12)

%%
% The oscillations don't go away: this is the Gibbs phenomenon.

%%
% The Fejer-Jackson inequality asserts that for
% every $n\ge 1$, $f_n(x) > 0$ for all $x\in (0,\pi)$.
% This was conjectured by Fejer in 1910 and subsequently proved
% by various people including Jackson, Gronwall, Landau, Turan,
% and Fejer himself.  I learned of the matter this past April
% from Dick Askey, who has been interested in sharpenings of
% the inequality.

%%
% It is interesting to look at this function from the
% Chebfun point of view.  The functions above construct
% $f_n$ as an algebraic polynomial, nonperiodic, represented
% by Chebyshev coefficients.  Here is a plot of length as
% a function of $n$, showing a linear relationship with
% a proportionality constant of $\pi/2$.
nn = 10:10:500; ln = [];
for n = nn
  ln = [ln length(fn(n))];
end
plot(nn,ln,'.',MS,14)
title('Length of chebfuns',FS,12)
xlabel('n',FS,10), ylabel('length(fn(n))',FS,10)

%%
% Since these are trigonometric functions, 
% an alternative would be to construct them as trig chebfuns,
% though now the domain would have to be $[0, 2\pi]$.
% This is slightly less efficient as
% now the proportionality constant is $2$:
fntrig = @(n) chebfun(@(x) fnx(n,x),[0 2*pi],'trig','vectorize');
lntrig = [];
for n = nn
  lntrig = [lntrig length(fntrig(n))];
end
hold on, plot(nn,lntrig,'or',MS,6), hold off
title('Length of chebfuns, both cheb and trig',FS,12)
xlabel('n',FS,10), ylabel('length(fn(n))',FS,10)
legend('cheb','trig','location','northwest')

%%
% Of course, algebraically the trigonometric approximations are
% exactly the partial sums in the definition of $f_n$, and
% they could be constructed that way explicitly rather
% than using the general trigfun constructor as above.