Universal Padé approximants and their behaviour on the boundary

Zadik, Ilias

doi:10.1007/s00605-016-0935-8

Cited by 6 publications

(6 citation statements)

References 9 publications

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“…In this section we combine and strengthen the results of [23] and [10]. We consider an open set Ω ⊆ C and {L n } n≥1 a sequence of compact subsets of Ω satisfying the following properties:…”

Section: Splitting the Boundarymentioning

confidence: 82%

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Simultaneous Universal Pade-Taylor Approximation

Makridis

2015

Preprint

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Section: Splitting the Boundarymentioning

confidence: 82%

“…Recently the partial sums S N (f, ζ)(•) have been replaced by some rational functions, namely the Padé approximants of f ( [18], [6], [5], [20], [23]). There are two types of universal Padé approximants.…”

Section: Introductionmentioning

confidence: 99%

Simultaneous Universal Pade-Taylor Approximation

Makridis

2015

Preprint

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“…In particular, if K = results are improved by proving the existence of holomorphic functions on (Ω i ∪ S i ). Partially smooth universal Taylor series are not so developed even in one complex variable ( [8], [19]). Our method cannot give the result for all compact sets…”

Section: Introductionmentioning

confidence: 99%

Partially Smooth Universal Taylor Series on products of simply connected domains

Kotsovolis¹

2019

Preprint

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Using a recent Mergelyan type theorem, we show the existence of universal Taylor series on products of planar simply connected domains Ω i that extend continuously onopen in the relative topology. The universal approximation occurs on every product of compact sets K i such that C − K i are connected and for some i 0 it holds K i0 ∩ (Ω i0 ∪ S i0 ) = ∅. Furthermore, we introduce some topological properties of universal Taylor series that lead to the voidance of some families of functions.

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“…Based on the mathematical properties of rational functions (e.g., Bamberger et al, 1988;Trefethen & Halpern, 1986), the Padé rational function exhibits a well-posed approximation behavior for the functions that reduce from higher-order to lower-order terms. Thus, it could be applicable to approach such functions, which tend to be divergent when expanded in a power series (e.g., Feldmann & Freund, 1995;Zadik, 2017). Typical applications of the Padé approximation in geophysics can be found in Cheng (1993) for crack-induced anisotropy, Fu (2006) for wave propagation, and Tang et al (2020) for seismic imaging.…”

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confidence: 99%

Third‐Order Padé Thermoelastic Constants of Solid Rocks

Yang

et al. 2022

JGR Solid Earth

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Classical third‐order thermoelastic constants are generally derived from the theory of small‐amplitude acoustic waves in isotropic materials during heat treatments. Investigating higher‐order thermoelastic constants for higher temperatures is challenging owing to the involvement of the number of unknown parameters. These Taylor‐type thermoelastic constants from the classical thermoelasticity theory are formulated based on the Taylor series of the Helmholtz free energy density for preheated crystals. However, these Taylor‐type thermoelastic models are limited even at low temperatures in characterizing the temperature‐dependent velocities of elastic waves in solid rocks as a polycrystal compound of different mineral lithologies. Thus, we propose using the Padé rational function to the total thermal strain energy function. The resulting Padé thermoelastic model gives a reasonable theoretical prediction for acoustic velocities of solid rocks at a higher temperature. We formulate the relationship between the third‐order Padé thermoelastic constants and the corresponding higher‐order Taylor thermoelastic constants with the same accuracy. Two additional Padé coefficients α1 ${\mathit{\alpha }}_{1}$ and α2 ${\mathit{\alpha }}_{2}$ can be calculated using the second‐, third‐, and fourth‐order Taylor thermoelastic constants associated with the Brugger's constants, which are consistent with those obtained by fitting the experimental data of polycrystalline material. The third‐order Padé thermoelastic model (with four constants) is validated by the fourth‐order Taylor thermoelastic prediction (with six constants) with ultrasonic measurements for polycrystals (olivine samples) and solid rocks (sandstone, granite, and shale). The results demonstrate that the third‐order Padé thermoelastic model can characterize thermally induced velocity changes more accurately than the conventional third‐order Taylor thermoelastic prediction (with four constants), especially for solid rocks at high temperatures. The Padé approximation could be considered a more accurate and universal model in describing thermally induced velocity changes for polycrystals and solid rocks.

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Universal Padé approximants and their behaviour on the boundary

Cited by 6 publications

References 9 publications

Simultaneous Universal Pade-Taylor Approximation

Simultaneous Universal Pade-Taylor Approximation

Partially Smooth Universal Taylor Series on products of simply connected domains

Third‐Order Padé Thermoelastic Constants of Solid Rocks

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