Interpolation by generalized exponential sums with equal weights

Abstract

In this paper we solve Padé-(i.e. multiple) and Prony (i.e. simple exponential) interpolation problems for the amplitude and frequency sums with equal amplitudes,

Convergence rate of one class of differentiating sums

“…Theorem 2.1 will be proved in Section 3. We recall that according [2], the parameters in formula (1.3) satisfy the estimate…”

“…A simple analysis of the proof shows that a) this estimate holds true also for 𝑡 −1 if we replace 𝑡 by |𝑡| in it, b) as |𝑡| ≍ 𝑛 we can state more, for instance, as 𝑡 = 𝑛, by Theorem 4(b) in [2] with 𝑟 0 = 𝑛, 𝛾 = 𝑎 = 1 we obtain an estimate for error in formula (1.2):…”

“…In the present work, for each fixed 𝑡 < 0, we construct a new estimate for the error in formula (1.3), which can be applied in a circle of form |𝑧| < 𝑟, where 𝑟 = 𝑟(𝑡, 𝑛) → 1 as 𝑛 → ∞. The way of the proof differs substantially from the approaches in works [1], [2] and it can not be generalized for the case 𝑡 > 0.…”

“…In works [1], [2], the domains of applicability of differentiation formulae (1.2), (1.3) were established and the errors were estimated. For instance, it was shown in [2] that if…”

“…Nevertheless, the domains of applicability of formulae (1.2), (1.3) found in [1], [2] are essentially less than the analyticity domain of the function 𝑓 , which contains the unit circle. In the present work, for each fixed 𝑡 < 0, we construct a new estimate for the error in formula (1.3), which can be applied in a circle of form |𝑧| < 𝑟, where 𝑟 = 𝑟(𝑡, 𝑛) → 1 as 𝑛 → ∞.…”