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Suppose I know the values of a continuous function on $[a,b]$ in some finite number of points $x_0,x_1 \ldots x_n$. I can form the Lagrange interpolating polynomial, $p$. I am curious if there is any interesting estimate of the expression $|f(x)-p(x)|$ for an arbitrary point $x$ that does not assume $f$ is smooth? (Perhaps it would involve moduli of continuity of higher order.) The textbooks on approximation theory I have seen only treat differentiable case.

To clarify what I mean by interesting: I can rather trivially bound $|f(x)-p(x)|$ from above by $\sum |\omega(x-x_i)||L_i(x)|$, where $\omega$ is the modulus of continuity and $L_i$ are the normal Lagrange basis polynomials. I do not view this as interesting, among other things because it does not reduce to the normal Lagrange error term when $f$ is smooth.

Suppose now further that I know that the function is in fact nowhere differentiable and I want to interpolate between the given values of the function. Could one give lower bounds for the error now when using Lagrange interpolation of various orders?

The background to the question is that I been thinking about if one could interpolate in a table of the Weierstrass function, such as the one given here: http://www.ams.org/journals/mcom/1961-15-074/S0025-5718-1961-0122011-X/home.html. (Do not ask me why I have been thinking about that.)

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One can get error estimates for taylor series in banach spaces, so it should be possible to get analogous formula for the Sobolev space $H^k([a,b])$ where you only have weak derivatives. The Weierstrauss function is pretty crazy though and I'm dubious if you could put it in a space where "derivatives" make sense. The following site has the remainder formula: myyn.org/m/article/taylor-polynomials-in-banach-spaces –  Nick Alger Dec 11 '10 at 15:06
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