Numerical integration of functions of very many variables

Haselgrove's method is currently considered to be the best for the numerical integration of smooth functions in very many (say, 6 or more) dimensions. This method, however, does not warrant the practically required accuracy of 3 significant decimals for integrands of remarkable variability. The method proposed in this paper introduces a skillful use of samplings to a multidimensional interpolatory quadrature scheme, and is shown to guarantee the above practical accuracy even where Haselgrove's method fails. This method has been devised especially for multivariable composite functions made up of complicated element functions of fewer variables.     

Numerical differentiation of functions of very many variables

It is highly desirable that the numerical differentiation at arbitrary points of a multidimensional space can be done with a reasonable amount of computational labor. A variant of the recently developed method of nonlinear interpolation is shown adequate for numerical differentiation. Computed results indicate that the proposed scheme is indeed feasible. This paper provides a strong case that, to overcome the vastness of hyperspace, random samplings are the only and mandatory choice in the numerical analysis of functions of very many variables.     

Nonlinear approximations of classes of periodic functions of many variables

Order-sharp estimates are established for the best N-term approximations of functions in the classes $B_{pq}^{sm} (\mathbb{T}^k )$ and $L_{pq}^{sm} (\mathbb{T}^k )$ of Nikol’skii-Besov and Lizorkin-Triebel types with respect to the multiple system of Meyer wavelets in the metric of $L_r (\mathbb{T}^k )$ for various relations between the parameters s, p, q, r, and m (s = (s ₁, ..., s _n ) ∈ ? ₊ ⁿ , 1 ≤ p, q, r ≤ ∞, m = (m ₁, ..., m _n ) ∈ ? ⁿ , and k = m ₁ + ... + m _n ). The proof of upper estimates is based on variants of the so-called greedy algorithms.     

Interpolation by L-spline functions of many variables

The choice of function space allows us to make conclusions in the multidimensional case that are analogous to results in the theory of spline functions of one variable. We establish the minimum norm property, the existence and uniqueness of a solution of the interpolation problem, the property of best approximation, and the convergence of interpolation processes.Translated from Matematicheskie Zametki, Vol. 14, No. 1, pp. 11–20, July, 1973.     

Smoothing by L-spline functions of many variables

We investigate the problem of the smoothing of experimental data by cell-like L-spline functions of many variables from the point of view of the theory of such functions proposed by the author. Given values of a function and its derivatives up to some order are smoothed on a rectangular network of nodes. Existence and uniqueness of the solution are proved and equations are derived.     

Asymptotics of approximation of Ψ-differentiable functions of many variables

We investigate the approximation characteristics of classes of Ψ-differentiable functions of many variables introduced by Stepanets. We present the asymptotics of approximation of functions of these classes. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 8, pp. 1051–1057, August, 2008.     

Approximation of some classes of periodic functions of many variables

We obtain the exact order of deviations of Fejér sums on the class of continuous functions. This order is determined by a given majorant of the best approximations.     

Representation of holomorphic functions of many variables by Cauchy-Stieltjes-type integrals

We consider functions of many complex variables that are holomorphic in a polydisk or in the upper half-plane. We give necessary and sufficient conditions under which a holomorphic function is a Cauchy-Stieltjes-type integral of a complex charge. We present several applications of this criterion to integral representations of certain classes of holomorphic functions. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 4, pp. 522–542, April, 2006.     

Kolmogorov-type inequalities for mixed derivatives of functions of many variables

Let = (₁,...,_d) be a vector with positive components and let D be the corresponding mixed derivative (of order _j with respect to the jth variable). In the case where d > 1 and 0 < k < r are arbitrary, we prove that

Correction and representation theorems for functions of countably many variables

Some similarities of the well-known theorems on correction and representation of functions of one and several variables are proved for functions of countably many variables.     

Best cubature formulas for some classes of functions of many variables

The problem of minimizing the error in the cubature formula for a given class of functions is considered. For cubature formulas with a lattice arrangement of points this problems is solved exactly for a wide class of functions of m variables.Basic contents of this paper presented with proofs at the Seminar on Theory of Functions at Dnepropetrovsk State University, December, 1965.Translated from Matematicheskie Zametki, Vol. 3, No. 5, pp. 565–576, May, 1968.     

Approximative characteristics of the isotropic classes of periodic functions of many variables

Exact-order estimates are obtained for the best orthogonal trigonometric approximations of the Besov (B _p,θ ^r ) and Nukol’skii (H _p ^r ) classes of periodic functions of many variables in the metric of L _q , 1 ≤ p, q ≤ ∞. We also establish the orders of the best approximations of functions from the same classes in the spaces L ₁ and L _∞ by trigonometric polynomials with the corresponding spectrum.     

On weighted Hilbert spaces and integration of functions of infinitely many variables

We study aspects of the analytic foundations of integration and closely related problems for functions of infinitely many variables _x1,_x2,…∈D$x_1,x_2,\dots \in D$. The setting is based on a reproducing kernel k$k$ for functions on D$D$, a family of non-negative weights _γu${\gamma }_u$, where u$u$ varies over all finite subsets of N$\mathbb{N}$, and a probability measure ρ$\rho$ on D$D$. We consider the weighted superposition K=_∑uγuku$K={\sum }_u{\gamma }_u{k}_u$ of finite tensor products _ku$k_u$ of k$k$. Under mild assumptions we show that K$K$ is a reproducing kernel on a properly chosen domain in the sequence space D^N$D^{\mathbb{N}}$, and that the reproducing kernel Hilbert space H(K)$H\left(K\right)$ is the orthogonal sum of the spaces H(_γuku)$H\left({\gamma }_u{k}_u\right)$. Integration on H(K)$H\left(K\right)$ can be defined in two ways, via a canonical representer or with respect to the product measure ρ^N${\rho }^{\mathbb{N}}$ on D^N$D^{\mathbb{N}}$. We relate both approaches and provide sufficient conditions for the two approaches to coincide.