Application of the AAK theory and Prony-like Methods for sparse approximation of exponential sums

We derive a new method for optimal ℓ²-approximation of discrete signals on ℓ²(ℕ₀) whose entries can be represented as an exponential sum of finite length. Our approach employs Prony's method in a first step to recover the exponential sum that is determined by the signal. In the second step we use the theory of Adamjan, Arov and Krein (AAK) to derive an algorithm for computing a shorter exponential sum that approximates the original signal in the ℓ²-norm well. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mean approximation of functions by Fourier-Gegenbauer sums

Necessary and sufficient conditions for best approximations of functions in the (–1,1) metric, –1/2<1/2 to zero at a certain rate are established (for =–1/2 known results are obtained). Inequalities for algebraic polynomials are used in the reasoning.Translated from Matematicheskie Zametki, Vol. 3, No. 5, pp. 587–596, May, 1968.I wish to thank my research director G. I. Natanson for his valuable advices.     

On approximation of periodic functions by Fourier sums

Let L _p , 1 ≤ p< ∞, be the space of 2π-periodic functions f with the norm || f ||_p = ( ò _{- p}^p | f |^p )^{1 \mathord}

A limit theorem for sums of exponentials

Translated from Matematicheskie Zametki, Vol. 46, No. 3, pp. 50–57, September, 1989.     

On the approximation of discrete functions by fourier-hahn sums

— f: {0,1, ..., N–1}R. .     

Limit theorems for sums of random exponentials

We study limiting distributions of exponential sums as t→∞, N→∞, where (X_i) are i.i.d. random variables. Two cases are considered: (A) ess sup X_i = 0 and (B) ess sup X_i = ∞. We assume that the function h(x)= -log P{X_i>x} (case B) or h(x) = -log P {X_i>-1/x} (case A) is regularly varying at ∞ with index 1 < ϱ <∞ (case B) or 0 < ϱ < ∞ (case A). The appropriate growth scale of N relative to t is of the form , where the rate function H₀(t) is a certain asymptotic version of the function (case B) or (case A). We have found two critical points, λ₁<λ₂, below which the Law of Large Numbers and the Central Limit Theorem, respectively, break down. For 0 < λ < λ₂, under the slightly stronger condition of normalized regular variation of h we prove that the limit laws are stable, with characteristic exponent α = α (ϱ, λ) ∈ (0,2) and skewness parameter β ≡ 1.Research supported in part by the DFG grants 436 RUS 113/534 and 436 RUS 113/722.Mathematics Subject Classification (2000): 60G50, 60F05, 60E07     

Asymptotic estimates of approximation of continuous periodic functions by Fourier sums

Asymptotic estimates, expressed in terms of the value of the modulus of continuity of r-th order (r2) at the point t=/n of a functionf C ₂ or of the (, )-derivative of a functionf C _B C, are established for the deviations of continuous periodic functions from their Fourier sums.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 6, pp. 747–755, June, 1990.     

Some inverse theorems for approximation of functions by the Fourier-Laguerre sums

In this paper we prove two inverse theorems for approximation of functions of two variables by Fourier-Laguerre sums in the space L ₂(ℝ₊²;x ^α y ^β e ^−x−y ).     

A uniform asymptotic expansion for weighted sums of exponentials

We consider the random variable Z_n,α=Y₁+2^αY₂+?+n^αY_n, with α∈R and Y₁,Y₂,… independent and exponentially distributed random variables with mean one. The distribution function of Z_n,α is in terms of a series with alternating signs, causing great numerical difficulties. Using an extended version of the saddle point method, we derive a uniform asymptotic expansion for P(Z_n,α<x) that remains valid inside (α≥−1/2) and outside (α<−1/2) the domain of attraction of the central limit theorem. We discuss several special cases, including α=1, for which we sharpen some of the results in Kingman and Volkov (2003).     

The approximation of continuous periodic functions of two variables by Faward sums

An estimate is given of the magnitude of the exact upper bound of the errors of double Faward sums on classes of continuous periodic functions, and asymptotic equations are found in the case of the classes H _A,B ^, for these quantities, expressed in terms of the exact upper bounds of the errors of Faward sums on the classes H _A and H _B of functions of one variable.Translated from Matematicheskie Zametki, Vol. 13, No. 5, pp. 655–666, May, 1973.     

Linear extrapolation by rational functions,exponentials and logarithmic functions

In this paper linear extrapolation by rational functions with given poles is considered from an arithmetical point of view. It is shown that the classical interpolation algorithms of Lagrange, Neville-Aitken and Newton which are well known for polynomial interpolation can be extended in a natural way to this problem yielding recursive methods of nearly the same complexity. The proofs are based upon explicit representations of generalized Vandermonde-determinants which are calculated by the elimination method combined with analytical considerations. As an application a regularity criterion for certain linear sequence-transformations is given. Also, by the same method simplified recurrence relations for linear extrapolation by exponentials and logarithmic functions at special knots are derived.