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.