使用切比雪夫多项式应用在逼近理论中。

% Using Chebyshev Polynomials
clear all; close all; clc;

% Define:
f = @(x) x.^4;
%f = @(x) 4*(x.^2-x.^4).*exp(-x./2);

x = (-1:0.1:1)'; % for -1 <= x <= 1

function chebP = chebyshevP(x,k)
% compute the value of the chebyshev polinomial of degree 'k' for every
% column value 'x'

%number of points
p = k+1;

switch k
    case {0} % when k = 0
        chebP = ones(size(x));
    case {1} % when k = 1
        chebP = x;
    otherwise % k >= 2 
        chebP0 = ones(size(x));
        chebP1 = x;
        for m = 1:k-1
            chebPX = (2*x.*chebP1 - chebP0);
            chebP0 = chebP1;
            chebP1 = chebPX;
        end
        chebP = chebPX;
end

% Using dcf and dst 
clear all; close all; clc;

f = @(x) x.^2/pi^2;

N = 16; j = 0:N;
x = (2*pi/N)*j;

y = f(x)';

yct = dct(y);
yst = dst(y);

figure(1)
hold on; grid on;
plot(abs(yst),'-or');
plot(abs(yct),'--sg');
legend('dst','dct')
hold off;

% Using FFT 
clear all; close all; clc;

f = @(x) cos(3*x);
%f = @(x) 1*(x>=0 & x<pi) + (-1)*(x>=pi & x<2*pi);

N = 16; j = 0:N-1;

x = (2*pi/N)*j;

y = f(x)';

yt = fft(y);

% Plot coeficients
figure(1)
grid on; plot(abs(yt),'or'); legend('fft')

% Using FFT 
clear all; close all; clc;

f = @(x) cos(3*x);
%f = @(x) 1*(x>=0 & x<pi) + (-1)*(x>=pi & x<2*pi);

N = 16; j = 0:N-1;

x = (2*pi/N)*j;

y = f(x)';

yt = fft(y);

% Plot coeficients
figure(1)
grid on; plot(abs(yt),'or'); legend('fft')

% Using FFT 
clear all; close all; clc;

f = @(x) cos(3*x);
%f = @(x) 1*(x>=0 & x<pi) + (-1)*(x>=pi & x<2*pi);

N = 16; j = 0:N-1;

x = (2*pi/N)*j;

y = f(x)';

yt = fft(y);

% Plot coeficients
figure(1)
grid on; plot(abs(yt),'or'); legend('fft')