an:01188852
Zbl 0919.42023
Ehrich, Sven
On localised error bounds for orthogonal approximation from shift invariant spaces
EN
Proceedings of the 3rd international conference on functional analysis and approximation theory, Acquafredda di Maratea (Potenza), Italy, September 23--28, 1996. Vols. I and II. Palermo: Circolo Matem??tico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 52, 393-407 (1998).
1998
a
42C15 41A30 41A50
shift invariant spaces; reproducing kernel; wavelets; Riesz basis; error bounds; Peano kernel
A shift invariant space \(S(N)\) of functions on \(\mathbb{R}\) generated by a compactly supported function \(N\) whose integer translates are a Riesz basis of \(S(N)\) has a unique orthonormal basis generated by a fundamental function of exponential decay. This result is extended by showing how the localization carries over to the error of the orthogonal (i.e., best \(L_2\)) approximations \(P[f]\) from \(S(N)\) to a class of functions \(f\): the paper obtains localized error bounds of the type
\[
| f(x)- P_h[f](x)|\leq C_{s,p, \rho,\omega} h^{s-{1\over p}}\| \omega^{-1}\rho(h^{-1} x- h^{-1}\cdot)f^{(s)}\|_p,
\]
where \(\omega\) is any function in \(L_q(\mathbb{R})\) \((p^{-1}+ q^{-1}= 1)\) such that the norm on the right is finite, and \(\rho\) is a determined radial function of exponential decay. Optimal choices of \(\rho\) are characterized using the zeros of the generalized Euler-Frobenius polynomial. The main tool is provided by the associated Peano kernel whose decay properties are studied.
For the entire collection see [Zbl 0892.00039].
A.L.Brown (Chandigarh)