Abstract:
The goal of the paper is to establish cubature formulas on finite combinatorial graphs. Two types of cubature formulas are developed. Cuba- ture formulas of the first type are exact on spaces of variational splines on graphs. Since badlimited functions can be obtained as limits of variational splines we obtain cubature formulas which are "essentially" exact on spaces of bandlimited functions. Cubature formulas of the second type are exact on spaces of bandlimited functions. Accuracy of a cubature formulas is given in terms of smoothness of functions. The results have potential applications to problems that arise in data mining.

Abstract:
We provide cubature formulas for the calculation of derivatives of expected values in the spririt of Terry Lyons and Nicolas Victoir. In financial mathematics derivatives of option prices with respect to initial values, so called Greeks, are of particular importance as hedging parameters. Cubature formulas allow to calculate these quantities very quickly. Simple examples are added to the theoretical exposition.

Abstract:
We consider a sequence of composite bivariate Bernstein operators and the cubature formula associated with them. The upper bounds for the remainder term of the cubature formula are described in terms of moduli of continuity of order two. Also we include some results showing how non-multiplicative the integration functional is.

Abstract:
A family of minimal cubature rules is established on an unbounded domain, which is the first such family known on unbounded domains. The nodes of such cubature rules are common zeros of certain orthogonal polynomials on the unbounded domain, which are also constructed.

Abstract:
We study the properties of hierarchical bases in the space of continuous functions with bounded domain and construct the hierarchical cubature formulas. Hierarchical systems of functions are similar to the well-known Faber-Schauder basis. It is shown that arbitrary hierarchical basis generates a scale of Hilbert subspaces in the space of continuous functions. The scale in many respects is similar to the usual classification of functional spaces with respect to moothness. By integration over initial domain the standard interpolation formula for the given continuous integrand, we construct the hierarchical cubature formulas and prove that each of these formulas is optimal simultaneously in all Hilbert subspaces associated with the initial hierarchical basis. Hence, we have constructed the universally optimal cubature formulas.

Abstract:
Based on a novel point of view on 1-dimensional Gaussian quadrature, we present a new approach to the computation of d-dimensional cubature formulae. It is well known that the nodes of 1-dimensional Gaussian quadrature can be computed as eigenvalues of the so-called Jacobi matrix. The d-dimensional analog is that cubature nodes can be obtained from the eigenvalues of certain mutually commuting matrices. These are obtained by extending (adding rows and columns to) certain noncommuting matrices A_1,...,A_d, related to the coordinate operators x_1,...,x_d, in R^d. We prove a correspondence between cubature formulae and "commuting extensions" of A_1,...,A_d, satisfying a compatibility condition which, in appropriate coordinates, constrains certain blocks in the extended matrices to be zero. Thus the problem of finding cubature formulae can be transformed to the problem of computing (and then simultaneously diagonalizing) commuting extensions. We give a general discussion of existence and of the expected size of commuting extensions and describe our attempts at computing them, as well as examples of cubature formulae obtained using the new approach.

Abstract:
We provide a necessary and sufficient condition for existence of Gaussian cubature formulas. It consists of checking whether some overdetermined linear system has a solution and so complements Mysovskikh's theorem which requires computing common zeros of orthonormal polynomials. Moreover, the size of the linear system shows that existence of a cubature formula imposes severe restrictions on the associated linear functional. For fixed precision (or degree), the larger the number of variables the worse it gets. And for fixed number of variables, the larger the precision the worse it gets. Finally, we also provide an interpretation of the necessary and sufficient condition in terms of existence of a polynomial with very specific properties.

Abstract:
Several cubature formulas on the cubic domains are derived using the discrete Fourier analysis associated with lattice tiling, as developed in \cite{LSX}. The main results consist of a new derivation of the Gaussian type cubature for the product Chebyshev weight functions and associated interpolation polynomials on $[-1,1]^2$, as well as new results on $[-1,1]^3$. In particular, compact formulas for the fundamental interpolation polynomials are derived, based on $n^3/4 +\CO(n^2)$ nodes of a cubature formula on $[-1,1]^3$.