Title of Your Presentation

On a multivariate Multi-Resolution Analysis

using generalized (non homogeneous) polyharmonic splines
Christophe Rabut
University of Toulouse (INSA, IMT, IREM), France

[e-mail]

Keywords. Multi-resolution Analysis, polyharmonic splines, Whittle-Matérn kernel, Fourier Transform.
Polyharmonic splines use the kernel defined as an inverse Fourier transform of the function û()= 1/ ||||²ⁿ . The Matérn kernel is the inverse Fourier transform of û()= 1/ (||||²+²) .We generalize these kernels by regularizing û in the form function û()= 1/ Product_i(||||²+k_i²)^i, (some k_i may be, or not be, zero, the _I may be integer or non integer positive numbers) and we build the corresponding multi-resolution analysis. In particular we build the associated Lagrangean function, “B-spline”, semi-orthogonal wavelets and orthogonal wavelets. Then we determine some of the properties of the so-derived functions and wavelet decomposition (stability, decay at infinity). Throughout this work we make an extensive use of the Fourier Transform of the various introduced functions.

The role of the k_i and of the _i are examined. To be short, the so-obtained spaces can be considered as regularized spaces of fractional polyharmonic splines under tension, the _i being the value of (fractional) orders while the k_i are weights directly connected with the tension associated to the order _i (precise connection will be given during the talk). Examples are given, showing the influence of the _i and f the k_i , which is easier seen in one dimension but is similar in any dimension.

Acknowledgements.

This is a joint work with Milvia Rossini and Mira Bozzini (Milano)

References

[1] Charles Micchelli, Christophe Rabut, Florencio Utreras (1991) Using the refinement equation for the construction of pre-wavelets III: elliptic splines Numerical Algorithms, vol 1, pp331-352.

[2] Barbara Bacchelli, Mira Bozzini, Christophe Rabut, Maria-Leonor Varas (2005) Decomposition and reconstruction of multidimensional signals using polyharmonic pre-wavelets. Appl. Comput. Harmon. Anal., pp. 282-299

[3] Christophe Rabut, Milvia Rossini (2008) Polyharmonic multiresolution analysis: an overview and some new results. Numerical Algorithms 48, pp.135-160.

[4] Mira Bozzini, Milvia Rossini, Robert Schaback (2013) Generalized Whittle-Matérn and polyharmonic kernels. Advances in Computational Mathematics, 39, 129-141