n = 18;

t = linspace(-1,1,n+1)';

y = [zeros(9,1);1;zeros(n-9,1)]; % 10th cardinal function

plot(t,y,'.'), hold on

x = linspace(-1,1,400)';

plot(x,interp1(t,y,x,'spline'))

title('Piecewise cubic cardinal function') % ignore this line

xlabel('x'), ylabel('p(x)') % ignore this line

The piecewise cubic cardinal function is nowhere greater than one in absolute value. This happens to be true for all the cardinal functions, ensuring a good condition number for the interpolation. But the story for global polynomials is very different.

clf, plot(t,y,'.') % ignore this line

c = polyfit(t,y,n);

hold on, plot(x,polyval(c,x))

title('Polynomial cardinal function') % ignore this line

xlabel('x'), ylabel('p(x)') % ignore this line

From the figure we can see that the condition number for polynomial interpolation on these nodes is at least 500.