% Approximation of a functions by the scaling function and its translates. h = daub(6); %h = [-1 0 9 16 9 0 -1 0]'/16; N = length(h); [x1,phi] = phivals(h,4); nx = 2^4; L = 12; n = L * nx; x = (0:n)/n*6; sf = [phi; zeros((L-N+1)*2^4,1)]; for num = 1:5 y = zeros(size(sf)); tmp = ((-N+2)*nx:(-N+2+32)*nx-1)'/n; v = x'/6; if num == 1 ck = [zeros(N-2,1); h; zeros(L,1)]; elseif num == 2 f = ones(n+1,1); elseif num == 3 f = v; ck = scalecoeffs(tmp,32,h,0,4); elseif num == 4 f = 4*v.*v-4*v+1; ck = scalecoeffs(4*tmp.*tmp-4*tmp+1,32,h,0,4); elseif num == 5 f = -6*v.*v.*v+9*v.*v-3*v; ck = scalecoeffs(-6*tmp.*tmp.*tmp+9*tmp.*tmp-3*tmp,32,h,0,4); end clf minval = 0; maxval = 0; for k = -N+2:L-1 if num == 2 g = eoshift(sf,k*nx); else g = ck(k+N-1)*eoshift(sf,k*nx); end hold on plot(x,g,':') hold off y = y + g; minval = min(minval,min(g)); maxval = max(maxval,max(g)); end hold on plot(x,y) hold off s = maxval - minval; minval = minval - 0.2 * s; maxval = maxval + 0.2 * s; axis([min(x) max(x) minval maxval]) xlabel('x') ylabel('f(x)') if num == 1 title('Representation of a scaling function by its translates') v = axis; v(2) = 6; axis(v); elseif num == 2 title('Representation of a constant function by translates of a scaling function') elseif num == 3 title('Representation of a linear function by translates of a scaling function') elseif num == 4 title('Representation of a quadratic by translates of a scaling function') elseif num == 5 title('Representation of a cubic by translates of a scaling function') end if num > 1 pause plot(x(1:n),f(1:n)-y(1:n)) xlabel('x') ylabel('f(x)-f_{approx}(x)') title('Approximation error') end pause end