Approximation by amplitude and frequency operators

Abstract

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

Interpolation by generalized exponential sums with equal weights

“…For further discussion we recall how to find the set Λ n (see (12)) from the following system for their power sums S m = S m (Λ n ):…”

“…Namely, it is proved in [12] that for the functions f and h satisfying the assumptions of Theorem 3, it holds with uniquely determined {µ k , λ k } n k=1 that…”

“…If we take h(z) = 1/(z − 1) in (4) and (2), then we correspondingly get rational hsums called simple partial fractions whose properties are actively studied [15] and the well known [n − 1, n]-type Padé fractions (see e.g. [12,Subsection 2.3]) as interpolants to f . In comparison with H rat n , the calculation of the h-sums require more arithmetic operations whilst the Padé fractions may not exist for some f and n (see Section 3.2).…”

“…For (33) with ρ(x) ≡ 1, one can find quadratures based on the h-sums in [13] but they still require more arithmetic operations than H n (see Section 3.2). If the integral (33) with ρ(x) ≡ 1 is interpolated by amplitude and frequency sums, then one obtains the well known Gauss quadrature (see [12,Sunsection 2.2]). If ρ(x) = (1 − x 2 ) −1/2 in (33), the corresponding Gauss type quadrature (usually called Gauss-Chebyshev or Hermite quadrature) has equal amplitudes as H n does (see […”

Rate of approximation of zf′(z) by special sums associated with the zeros of the Bessel polynomials

“…here w k , u k satisfy d 0 = d 2 = · · · = d n = 0, d 1 = 1 and |d n+1 | = minimum, where d m := n k=1 w k u m k (see also References in [1,3]).…”