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