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The aim of the paper is to construct a multiresolution analysis of L2(IRd) based on generalized kernels which are fundamental solutions of differential operators of the form $\boldsymbol {\prod }_{\ell = 1}^{m}(-{\Delta }+\kappa _{\ell }^{2}\,I)$ . We study its properties and provide a set of pre-wavelets associated with it, as well as the filters which are indispensable to perform decomposition and reconstruction of a given signal, being very useful in applied problems thanks to the presence of the tension parameters κl.

We have presented in Bozzini et al. (2011) a procedure in spaces of m-harmonic splines in Rd that starts from a simple generator φ0 and recursively defines generators φ1, φ2,.,φm-1 with corresponding quasi-interpolation operators reproducing polynomials of degrees 3, 5,.,2m-1 respectively. In this paper we study the properties of generators φj, and we prove that these new generators are positive definite functions, and are scaling functions whenever φ0 has those properties. Moreover φ0 and φj generate the same multiresolution analysis. We show that it is possible to define a convergent subdivision scheme, and to provide in this way a fast computation of the quasi-interpolant.

We generate various new radial kernels by taking derivatives of known kernels with respect to scale. This is different from the well-known scale mixtures used before. In addition, we provide a simple recipe that explicitly constructs new kernels from the negative Laplacian of known kernels. Surprisingly, these two methods for generating new kernels can be proven to coincide for certain standard classes of radial kernels. The resulting radial kernels are positive definite, and a few illustrations are provided.

We present a numerical method for the detection of faults and gradient faults of two dimensional functions when uniformly scattered and noisy data are given. The method recognizes the presence of a discontinuity curve by a suitable statistic. Moreover, we discuss the recovery of such curves and provide a computational bound for the pointwise error and convergence properties.

Within kernel-based interpolation and its many applications, the handling of the scaling or the shape parameter is a well-documented but unsolved problem. We consider native spaces whose kernels allow us to change the kernel scale of a d-variate interpolation problem locally, depending on the requirements of the application. The trick is to define a scale function c on the domain ? ? Rd to transform an interpolation problem from data locations xj in Rd to data locations (xj, c(xj)) and to use a fixed-scale kernel on Rd+1 for interpolation there. The (d+1)-variate solution is then evaluated at (x, c(x)) for x ? Rd to give a d-variate interpolant with a varying scale. A large number of examples show how this can be done in practice to get results that are better than the fixed-scale technique, with respect to both condition number and error. The background theory coincides with fixed-scale interpolation on the submanifold of Rd+1 given by the points (x, c(x)) of the graph of the scale function c.

This paper simultaneously generalizes two standard classes of radial kernels, the polyharmonic kernels related to the differential operator (???Δ) m and the Whittle---Matern kernels related to the differential operator (???Δ?+?I) m . This is done by allowing general differential operators of the form $\prod_{j=1}^m(-\Delta+\kappa_j^2I)$ with nonzero ? j and calculating their associated kernels. It turns out that they can be explicity given by starting from scaled Whittle---Matern kernels and taking divided differences with respect to their scale. They are positive definite radial kernels which are reproducing kernels in Hilbert spaces norm-equivalent to $W_2^m(\ensuremath{\mathbb{R}}^d)$ . On the side, we prove that generalized inverse multiquadric kernels of the form $\prod_{j=1}^m(r^2+\kappa_j^2)^{-1}$ are positive definite, and we provide their Fourier transforms. Surprisingly, these Fourier transforms lead to kernels of Whittle---Matern form with a variable scale ?(r) between ? 1,...,? m . We also consider the case where some of the ? j vanish. This leads to conditionally positive definite kernels that are linear combinations of the above variable-scale Whittle---Matern kernels and polyharmonic kernels. Some numerical examples are added for illustration.

The aim of the paper is to provide a computationally effective way to construct stable bases on general non-degenerate lattices. In particular, we define new stable bases on hexagonal lattices and we give some numerical examples which show their usefulness in applications.

Under very mild additional assumptions, translates of conditionally positive definite radial basis functions allow unique interpolation to scattered multivariate data, because the interpolation matrices have a symmetric and positive definite dominant part. In many applications, the data density varies locally according to the signal behaviour, and then the translates should get different scalings that match the local data density. Furthermore, if there is a local anisotropy in the data, the radial basis functions should possibly be distorted into functions with ellipsoids as level sets. In such cases, the symmetry and the definiteness of the matrices are no longer guaranteed. However, this brief note is the first paper to provide sufficient conditions for the unique solvability of such interpolation processes. The basic technique is a simple matrix perturbation argument combined with the Ball-Narcowich-Ward stability results. © Springer-Verlag 2004.

In this paper the authors develope a new Montecarlo method for many variables interpolation when the values of the function are known on a lattice. This technique uses cardinal splines functions. Numerical results are also reported for some problems. © 1975 Instituto di Elaborazione della Informazione del CNR

The paper presents a simple procedure for the construction of quasi-interpolation operators in spaces of m-harmonic splines in R(d), which reproduce polynomials of high degree. The procedure starts from a generator phi(0), which is easy to derive but with corresponding quasi-interpolation operator reproducing only linear polynomials, and recursively defines generators phi(1), phi(2), ... , phi(m-1) with corresponding quasi-interpolation operators reproducing polynomials of degree up to 3. 5 2m - 1 respectively. The construction of phi(j); from phi(j-1) is explicit, simple and independent of m. The special case d = 1 and the special cases d = 2. m = 2, 3,4 are discussed in details.

For clinical application of heart potential mapping, a reduced set of leads is determined, yet suitable to represent all possible phenomenologies. The analysis of the reliability of the model is done and some numerical results are presented.

In this paper a local type method is proposed to smooth noisy data. Criteria of convergence and error bounds are given. An applciation is also presented for a biomedical problem in ℝ3, which is usually solved only in ℝ2. © 1984 Springer-Verlag.