チェザロ総和法（フーリエ部分級数和の平均）によるギブス現象の回避

矩形波に対してチェザロ総和法（N=100）を適用

clear; close all; clc;

T=2*pi;
f0=1/T;
dt=0.01;
t=-T/2:dt:T/2;
y=f(t);
plot(t,y,'r','DisplayName','original');
hold on;

a0=0;
a=@(n) 0;
b=@(n) 2*(1-(-1)^n)/(pi*n);

nmaxs=[100];

%Fourier series
for idx=1:length(nmaxs)
    nmax = nmaxs(idx);
    yt=fourier_series(t,f0,a0,a,b,nmax);
    plot(t,yt,'DisplayName',sprintf('N=%d',nmax));
end

%Cèsaro summation of Fourier series to suppress Gibbs' phenomena
for idx=1:length(nmaxs)
    nmax = nmaxs(idx);
    ytc=0;
    for n=1:nmax
      ytc=ytc+fourier_series(t,f0,a0,a,b,n);
    end
    ytc=ytc/nmax;
    plot(t,ytc,'DisplayName',sprintf('N=%d(Cesàro summation)',nmax));
end

legend();
big;


function res=f(t)
res = [];
for idx=1:length(t)
    ti=t(idx);
    if ti<0
        res(idx)=-1;
    else
        res(idx)=1;
    end
end
end

function ft=fourier_series(t,f0,a0,a,b,nmax)
ft = a0/2;
for n=1:nmax
    an = a(n);
    bn = b(n);
    ft = ft + an*cos(2*pi*f0*n*t) + bn*sin(2*pi*f0*n*t);
end

end