The Sylvester-Ramanujan System of Equations and The Complex Power Moment Problem

Lyubich, Yuri I.

doi:10.1023/b:rama.0000027196.19661.b7

Cited by 11 publications

(25 citation statements)

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“…The system (1.6) with unknown µ k , λ k and given s m is well known as the discrete moment problem. Classical works of Prony, Sylvester, Ramanujan and papers of many contemporary researchers are devoted to the problem of its solvability (see [24,[29][30][31]33,38]). Note that the system (1.6) is bound up with Hankel forms, orthogonal polynomials, continued fractions, Gaussian quadratures and Padé approximants (a detailed review of these connections is given in [29,30] and also in Section 2).…”

Section: Introduction and Statement Of The Problemmentioning

confidence: 99%

Approximation by amplitude and frequency operators

Chunaev¹,

Danchenko²

2016

Journal of Approximation Theory

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We study Pad\'{e} interpolation at the node $z=0$ of functions $f(z)=\sum_{m=0}^{\infty} f_m z^m$, analytic in a neighbourhood of this node, by amplitude and frequency operators (sums) of the form $$ \sum_{k=1}^n \mu_k h(\lambda_k z), \qquad \mu_k,\lambda_k\in \mathbb{C}. $$ Here $h(z)=\sum_{m=0}^{\infty} h_m z^m$, $h_m\ne 0$, is a fixed (basis) function, analytic at the origin, and the interpolation is carried out by an appropriate choice of amplitudes $\mu_k $ and frequencies $\lambda_k$. The solvability of the $2n$-multiple interpolation problem is determined by the solvability of the associated moment problem $$ \sum_{k=1}^n\mu_k \lambda_k^m={f_m}/{h_m}, \qquad m=\overline{0,2n-1}. $$ In a number of cases, when the moment problem is consistent, it can be solved by the classical method due to Prony and Sylvester, moreover, one can easily construct the corresponding interpolating sum too. In the case of inconsistent moment problems, we propose a regularization method, which consists in adding a special binomial $c_1z^{n-1}+c_2 z^{2n-1}$ to an amplitude and frequency sum so that the moment problem, associated with the sum obtained, can be already solved by the method of Prony and Sylvester. This approach enables us to obtain interpolation formulas with $n$ nodes $\lambda_k z$, being exact for the polynomials of degree $\le 2n-1$, whilst traditional formulas with the same number of nodes are usually exact only for the polynomials of degree $\le n-1$. The regularization method is applied to numerical differentiation and extrapolation.Comment: We added several examples and remarks recommended by the referees and also corrected minor misprints found in the previous version

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Section: Introduction and Statement Of The Problemmentioning

confidence: 99%

Approximation by amplitude and frequency operators

Chunaev¹,

Danchenko²

2016

Journal of Approximation Theory

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“…Furthermore, in spite of the huge bibliography related to the Prony method and general amplitude and frequency sums (see [2,3,5,12,20,23,24] and references therein), we could not find any more or less general estimates for amplitudes and frequencies similar to those in Theorem 5. Probably, they just do not exist because of the above-mentioned divergence examples from [12, Section 7] and the results from [2].…”

Section: Comparison With the Original Prony Exponential Interpolationmentioning

confidence: 84%

Interpolation by generalized exponential sums with equal weights

Chunaev

2020

Journal of Approximation Theory

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“…Отсюда сразу следует первое неравенство в (31). Далее, выбрав ϕ так, чтобы в одной из точек t k (ϕ) функция |ρ n (x)| принимала наибольшее значение, получим и второе неравенство в (31). Аналогичные неравенства справедливы и для ρ − n .…”

Section: 5unclassified

“…Для решения регулярных систем существует классический метод Прони (см. [30,31,39]). Ключевую роль в этом методе играет производящий многочлен…”

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