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
Let E ⊂ C be a Borel set such that 0 < H 1 (E) < ∞. David and Léger proved that the Cauchy kernel 1/z (and even its coordinate parts Re z/|z| 2 and Im z/|z| 2 , z ∈ C \ {0}) has the following property: the L 2 (H 1 ⌊E)-boundedness of the corresponding singular integral operator implies that E is rectifiable. Recently Chousionis, Mateu, Prat and Tolsa extended this result to any kernel of the form (Re z) 2n−1 /|z| 2n , n ∈ N. In this paper, we prove that the above-mentioned property holds for operators associated with the much wider class of the kernels (Re z) 2N −1 /|z| 2N +t·(Re z) 2n−1 /|z| 2n , where n and N are positive integer numbers such that N n, and t ∈ R \ (t 1 , t 2 ) with t 1 , t 2 depending only on n and N .
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