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 interpolation of functions of very many variables

The difficulty is first shown in the nonlinear interpolation of functions defined in a space of very many dimensions. There is a method using a sampling technique [J. ACM, Vol. 17, pp. 420–425, July 1970] that works fairly well in the regime ofk~10?20 (k=number of dimensions). The sampling errors, however, increase exponentially with increasingk, so that fork greater than the above values the computation is no more feasible. This is due to the subtraction between two large sums of about the same magnitude, each of which suffers stochastic fluctuations accompanying the samplings. To avoid this difficulty, a “pairwise” sampling method is devised where one draws two samples at a time when required, one from each of these two sums of terms. With the use of this technique, standard errors are reduced by orders of magnitude. Some details of algorithm are given together with typical computed examples.     

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.     

Numerical integration of functions with a very small significant support

In some applications, one has to deal with the problem of integrating, over a bounded interval, a smooth function taking significant values, with respect to the machine precision or to the accuracy one wants to achieve, only in a very small part of the domain of integration. In this paper, we propose a simple and efficient numerical approach to compute or discretize integrals of this type. We also consider a class of second kind integral equations whose integral operator has the above behavior. Some numerical testing is presented.     

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.     

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.     

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.