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Several approaches to solving elliptic problems numerically are based on hierarchical Riesz bases in Sobolev spaces. We are interested in determining the exact range of Sobolev exponents for which a system of compactly supported functions derived from a multiresolution analysis forms such a Riesz basis. This involves determining the smoothness of the dual system. The elements of the dual system typically consist of noncompactly supported functions, whose smoothness can be treated by extending the results of Cohen and Daubechies (1996), Cohen et al. (1999), and Jia (to appear). We show how to determine the exact range of Sobolev exponents in the multivariate case, both theoretically and numerically, from spectral properties of transfer operators. This technique is applied to several bases deriving from linear finite elements which have been proposed in the literature. For hierarchical basis, we find that it forms a Riesz basis in H s (R d ) for −0.990236 . . . < s < 3/2.

We introduce the sequence of generalized Gončarov polynomials, which is a basis for the solutions to the Gončarov interpolation problem with respect to a delta operator. Explicitly, a generalized Gončarov basis is a sequence (tn(x)) n≥0 of polynomials defined by the biorthogonality relation εz i (d i (tn(x))) = n! δi,n for all i, n ∈ N, where d is a delta operator, Z = (zi) i≥0 a sequence of scalars, and εz i the evaluation at zi. We present algebraic and analytic properties of generalized Gončarov polynomials and show that such polynomial sequences provide a natural algebraic tool for enumerating combinatorial structures with a linear constraint on their order statistics.2010 Mathematics Subject Classification. Primary 05A10, 41A05. Secondary 05A40.

This paper is devoted to bivariate interpolation. The problem is to find a polynomial P(x, y) whose values and the values of whose derivatives at given points match given data. Methods of Birkhoff interpolation are used throughout. We define interpolation matrices E, their regularity, their almost regularity, and finally the regularity of the pair E, Z for a given set of knots Z.Many concrete examples and applications are possible.