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The method of self-similar factor approximants is completed by defining the approximants of odd orders, constructed from the power series with the largest term of an odd power. It is shown that the method provides good approximations for…
The problem of reconstructing functions from their asymptotic expansions in powers of a small variable is addressed by deriving a novel type of approximants. The derivation is based on the self-similar approximation theory, which presents…
An approach is suggested defining effective sums of divergent series in the form of self-similar exponential approximants. The procedure of constructing these approximants from divergent series with arbitrary noninteger powers is developed.…
The problem is analyzed of extrapolating power series, derived for an asymptotically small variable, to the region of finite values of this variable. The consideration is based on the self-similar approximation theory. A new method is…
We propose a general method for optimization with semi-infinite constraints that involve a linear combination of functions, focusing on the case of the exponential function. Each function is lower and upper bounded on sub-intervals by…
The use of approximants of Pad\`e type are employed to develop a method aimed at opening new perspectives in the theory of Appell polynomials $a_n(x)$, specified by the generating function \sum_{n=0}^{\infty} \frac{t^n}{n!} a_n(x) = A(t)…
We present a new method for approximating real-valued functions on ${\mathbb R}^+$ by linear combinations of exponential functions with complex coefficients. The approach is based on a multi-point Pad\'e approximation of the Laplace…
In this paper we are interested in the approximation of fractional powers of self-adjoint positive operators. Starting from the integral representation of the operators, we apply the trapezoidal rule combined with a single-exponential and a…
In this survey we describe some modifications of Prony's method. In particular, we consider the recovery of general expansions into eigenfunctions of linear differential operators of first order and show, how these expansions can be…
The method of self-similar factor approximants is shown to be very convenient for solving different evolution equations and boundary-value problems typical of physical applications. The method is general and simple, being a straightforward…
Power series representations for special functions are computationally satisfactory only in the vicinity of the expansion point. Thus, it is an obvious idea to use instead Pad\'{e} approximants or other rational functions constructed from…
A novel method of summation for power series is developed. The method is based on the self-similar approximation theory. The trick employed is in transforming, first, a series expansion into a product expansion and in applying the…
Pad\'e approximants are rational functions whose series expansion match a given series as far as possible. These approximants are usually written under a rational form. In this paper, we will show how to write them also under two different…
A novel type of approximants is introduced, being based on the ideas of self-similar approximation theory. The method is illustrated by the examples possessing the structure typical of many problems in applied mathematics. Good numerical…
Virial expansions are the series in powers of density assumed to be small. However, the equations of state require to consider finite densities for which virial expansions, as a rule, diverge. In order to extrapolate a virial expansion to…
We propose an explicit recursive method to approximate a power-law with a finite sum of weighted exponentials. Applications to moving averages with long memory are discussed in relationship with stochastic volatility models.
We investigate the rational approximation of fractional powers of unbounded positive operators attainable with a specific integral representation of the operator function. We provide accurate error bounds by exploiting classical results in…
A novel method of summing asymptotic series is advanced. Such series repeatedly arise when employing perturbation theory in powers of a small parameter for complicated problems of condensed matter physics, statistical physics, and various…
Complicated physical problems usually are solved by resorting to perturbation theory leading to solutions in the form of asymptotic series in powers of small parameters. However, finite, and even large values of the parameters often are of…
In this paper we propose a method for proving some exponential inequalities based on power series expansion and analysis of derivations of the corresponding functions. Our approach provides a simple proof and generates a new class of…