Smooth polynomial expansions of piecewise-differentiable functions

Approximating Piecewise-Smooth Functions

We consider the possibility of using locally supported quasi-interpolation operators for the approximation of univariate non-smooth functions. In such a case one usually expects the rate of approximation to be lower than that of smooth functions. It is shown in this paper that prior knowledge of the type of ’singularity’ of the function can be used to regain the full approximation power of the ...

Piecewise-smooth Refinable Functions

Univariate piecewise-smooth refinable functions (i.e., compactly supported solutions of the equation φ( 2 ) = ∑N k=0 ckφ(x−k)) are classified completely. Characterization of the structure of refinable splines leads to a simple convergence criterion for the subdivision schemes corresponding to such splines, and to explicit computation of the rate of convergence. This makes it possible to prove a...

Spectral Reconstruction of Piecewise Smooth Functions

Nonparametric estimation of piecewise smooth regression functions

Estimation of univariate regression functions from bounded i.i.d. data is considered. Estimates are deened by minimizing a complexity penalized residual sum of squares over all piecewise polynomials. The integrated squared error of these estimates achieves for piecewise p-smooth regression functions the rate (ln 2 (n)=n) 2p 2p+1 .

Finding Discontinuities of Piecewise-smooth Functions

Formulas for stable differentiation of piecewise-smooth functions are given. The data are noisy values of these functions. The locations of discontinuity points and the sizes of the jumps across these points are not assumed known, but found stably from the noisy data.