- #1
Tolya
- 22
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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!
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!
