# Saturability by smoothness

1. Nov 11, 2007

### Tolya

Sorry for my English. :)
Let function $$f(x)$$ defined on $$[a,b]$$ and its table $$f(x_k)$$ determined in equidistant interpolation nodes $$x_k$$ $$k=0,1,..,n$$ with step $$h=\frac{b-a}{n}$$.
Inaccuracy of piecewise-polynomial interpolation of power $$s$$ (with the help of interpolation polynoms $$P_s(x,f_{kj})$$ on the $$x_k \leq x \leq x_{k+1}$$) when $$f^{(s+1)}(x)$$ exist and limited on $$[a,b]$$ have a $$O(h^{s+1})$$ order.
If all we know about function $$f(x)$$ is that it has limited derivative to some order $$q$$ $$q<s$$, then unavoidable error when we reconstructed the function with the help of its table is $$O(h^{q+1})$$. If we interpolate with $$P_s(x,f_{kj})$$ the order $$O(h^{q+1})$$ reached.
When $$f(x)$$ have limited derivative of the order $$q+1$$, $$q>s$$, then inaccuracy of interpolation with the help of $$P_s(x,f_{kj})$$ remains $$O(h^{q+1})$$, i.e. the order of inaccuracy doesn't react on the supplemented, beyond the $$s+1$$ derivative, smoothness of the function $$f(x)$$.
How can I prove this property called saturability by smoothness.
Thanks for any ideas!

2. Nov 12, 2007

### Tolya

Any references coresponding to this theme?
Any books, links and so on?