In this paper, we build a multidimensional wavelet decomposition based on polyharmonic B-splines. The prewavelets are polyharmonic splines and so not tensor products of univariate wavelets. Explicit construction of the filters specified by the classical dyadic scaling relations is given and the decay of the functions and the filters is shown. We then design the decomposition/recomposition algorithm by means of downsampling/upsampling and convolution products.

In this paper, we deal with two different problems. First, we provide the convergence rates of multiresolution approximations, with respect to the supremum norm, for the class of elliptic splines defined in Ref. 10, and in particular for polyharmonic splines. Secondly, we consider the problem of recovering a function from a sample of noisy data. To this end, we define a linear and smooth estimator obtained from a multiresolution process based on polyharmonic splines. We discuss its asymptotic properties and we prove that it converges to the unknown function almost surely.

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A new method of inverting the noisy Fourier transform of a periodic function is considered. Results concerning norm approximation in L P (T N ), 1 < p < -foo, or in C(T N ) where T^ is an N-dimensional torus are derived in a form suitable for applications. Then the method is tested by a numerical simulation procedure in the case of the space /^(T).