Let \( F_N (x) \) be the finite Fourier sum for the periodic function f(x) with N+1 terms:

where the coefficients 𝑎_n and b_n were defined previously. The Gibbs phenomenon is the peculiar manner in which the Fourier series of a piecewise continuously differentiable periodic function behaves at a jump of discontinuity. The partial Fourier sums ripple near every point of discontinuity in an amount proportional to the finite jump. So finite Fourier sums (as well as other eigenfunction series) provide an overshoot/undershoot (or "ringing") occurring at discontinuities of finite length. Mathematically, it reflects the difference between uniform convergence and pointwise convergence. The Nth partial sum of the Fourier series has large oscillations near the jump, which increases/decreases the maximum/minimum of the partial sum of the function itself. The overshoot does not die out as N increases, but instead approaches a finite limit. If a function f(x) has a discontinuity at the point \( x_0 \) of amount \( c = f(x_0 +0) - f(x_0 -0) , \) then finite Fourier sums experience overshoot and undershoot in a neighborhood of the amount \( c\, w , \) where w = 0.0894899… is the Wilbraham constant. These undershoots or overshoots cannot be eliminated by increasing the number of terms in the finite Fourier partial sums; actually these finite sums swap the range when \( N\mapsto \infty . \) Here Si is the sine integral a special function that can be evaluated by Mathematica with a one line build-in command Division by π/2 yields So the overshoot/undershoot of a finite Fourier series for a function having a unit jump of discontinuity will exceed the actual value of the function at that point by the Wilbraham constant which is evaluated with 20 decimale places by Mathematica as

If the function is continuous (actually, its periodic continuation on all real axis), we do not observe the Gibbs phenomenon. The considered examples show that if a function is a piecewise continuous with finite jumps, its Fourier coefficients decay at a rate of 1/k; if a continuous function has corners (not differentiable), then its Fourier coefficients decay at a rate of 1/k². The more derivatives exist, the faster Fourier coefficients decay. The general case provides the following lemma.

 syms x
t=linspace(-2,2,2000); %time 
N=[10,50,100]; %different n terms
for n=1:3
an=[];
for  m=1:N(n)
    a=int(cos(m*pi*x/2), x, 0, 2);
    b= int(cos(m*pi*x/2), x, -2, -1);
    an=[an,(b/2)+a];  %Intergal
    %an=[an,-sin(m*pi/2)/(m*pi)];
end
bn=[];
for  m=1:N(n)
    a1=int(sin(m*pi*x/2), x, 0, 2);
    b1= int(sin(m*pi*x/2), x, -2, -1);
    %bn=[bn,((-1)^m-2+cos(m*pi/2))/(m*pi)];
    bn=[bn,(b1/2)+a1]; %Intergal
end
fN=5/4;
for m=1:N(n)
    fN=fN+an(m)*cos(m*pi*t/2)+bn(m)*sin(m*pi*t/2); %Fourier approximation equation 
end
nq=int2str(N(n));
subplot(2,2,n)
syms x
y = piecewise( -2 < x < -1, 1, -1 < x < 0, 0, 0 < x < 2, 2); %piecewise function
fplot(y,'r','LineWidth',2);hold on;
ylim([-2.1 2.1])
plot(t,fN,'LineWidth',2); hold off; axis([-2 2 -0.5 2.5]);grid; %plot of fourier approximation
xlabel('Time'), ylabel('y_N(t)');title(['N= ',nq]);hold on;
yticks([-0.1789797444721675 0.91051 1.0894899 2.1789797444721675])
end

syms t
syms x
t=linspace(-2,2,2000); %time 
N=[1,3,6,8,9,10]; %different n terms
for n=1:6;
an=[];
for  m=1:N(n)
    a=int(cos(m*pi*x/2), x, 0, 2);
    b= int(cos(m*pi*x/2), x, -2, -1);
    an=[an,(b/2)+a];  %Intergal
    %an=[an,-sin(m*pi/2)/(m*pi)];
end;
bn=[];
for  m=1:N(n)
    a1=int(sin(m*pi*x/2), x, 0, 2);
    b1= int(sin(m*pi*x/2), x, -2, -1);
    %bn=[bn,((-1)^m-2+cos(m*pi/2))/(m*pi)];
    bn=[bn,(b1/2)+a1]; %Intergal
end;
fN=5/4;
for m=1:N(n)
    fN=fN+an(m)*cos(m*pi*t/2)+bn(m)*sin(m*pi*t/2); %Fourier approximation equation 
end; 
nq=int2str(N(n));
subplot(3,2,n)
syms x
y = piecewise( -2 < x < -1, 1, -1 < x < 0, 1, 0 < x < 2, 2); %piecewise function
fplot(y,'r','LineWidth',2);hold on;
ylim([-2.1 2.1])
plot(t,fN,'LineWidth',2); hold off; axis([-2 2 -0.5 2.5]);grid; %plot of fourier approximation
xlabel('Time'), ylabel('y_N(t)');title(['N= ',nq]);
end;

syms x
t=linspace(-2.2,2.5,100); %time space
%  u=linspace(-2,2,2000);
% sq=[zeros(1,500),2*ones(1,1000),zeros(1,500)];
N=[10,20,100]; % different n terms 
for n=1:3;
an=[];
for  m=1:N(n)
    an=[an,(8*m*pi*((-1)^m)+4*m*pi*cos(m*pi/2)-(8+m^2*pi^2)*sin(m*pi/2))/(m^3*pi^3)]; %Fourier approximation equation 
end;
bn=[];
for  m=1:N(n)
    bn=[bn,(2*((-1)^m)*(-4+(m*pi)^2)+(8+((m*pi)^2)*cos(m*pi/2))+4*m*pi*sin(m*pi/2))/(m^3*pi^3)]; %Fourier approximation equation 
end;
fN=1/4;
for m=1:N(n)
    fN=fN+an(m)*cos(m*pi*t/2)-bn(m)*sin(m*pi*t/2); %Combined Fourier approximation equation 
end;
nq=int2str(N(n));
subplot(3,1,n)
y = piecewise(-2Copy
        

f = chebfun(’sign(x)’,10);subplot(1,2,1), plot(f,’.-’), grid onf=chebfun(’sign(x)’,20);subplot(1,2,2), plot(f,’.-’), grid on