Recovery of bivariate band-limited functions using scattered translates of the Poisson kernel

Ledford, Jeff

doi:10.1016/j.jat.2014.10.013

Cited by 4 publications

(8 citation statements)

References 8 publications

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“…The sum over j on the right hand side is majorized by a constant (depending on ε and k) times h −k+1 e − 2π 3h , which in turn is bounded by a constant depending only on k. Consequently, (29) Σ 3 ≤ C f Lp .…”

Section: Now the Discrete Version Of Young's Convolution Inequality I...mentioning

confidence: 99%

“…If p = 1 and k > 1 or p = ∞ and k ∈ N, there is a constant C, independent of h, so that for every g ∈ W k p (R), I h α g − g Lp ≤ C(1 + | ln h|)h k g W k p . For general results on RBF interpolation of bandlimited functions at nonuniform sequences of points, refer to [19,28,29,40], while [17] contains a nonuniform analogue to Theorem 3.5 for Gaussian interpolation in one dimension. Also, a recent work of Buhmann and Dai [9] examined pointwise error estimates for quasi-interpolation schemes involving RBFs.…”

Section: Remarkmentioning

confidence: 99%

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Cardinal interpolation with general multiquadrics: convergence rates

Hamm

Ledford

2017

Adv Comput Math

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This article pertains to interpolation of Sobolev functions at shrinking lattices hZ d from Lp shift-invariant spaces associated with cardinal functions related to general multiquadrics, φα,c(x) := (|x| 2 + c 2 ) α . The relation between the shift-invariant spaces generated by the cardinal functions and those generated by the multiquadrics themselves is considered. Additionally, Lp error estimates in terms of the dilation h are considered for the associated cardinal interpolation scheme. This analysis expands the range of α values which were previously known to give such convergence rates (i.e. O(h k ) for functions with derivatives of order up to k in Lp, 1 < p < ∞). Additionally, the analysis here demonstrates that some known best approximation rates for multiquadric approximation are obtained by their cardinal interpolants.

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Section: Now the Discrete Version Of Young's Convolution Inequality I...mentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

Cardinal interpolation with general multiquadrics: convergence rates

Hamm

Ledford

2017

Adv Comput Math

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“…For additional convergence phenomena similar to the ones listed in this section, the interested reader is referred to [28,29,32,45,47,48,51,61].…”

Section: Proof Plancherel's Identity Allows Us To Check This Results mentioning

confidence: 99%

On the structure and interpolation properties of quasi shift-invariant spaces

Hamm

Ledford

2018

Journal of Functional Analysis

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The structure of certain types of quasi shift-invariant spaces, which take the form V (ψ, X ) := span L 2 {ψ(· − x j ) : j ∈ Z} for an infinite discrete set X = (x j ) ⊂ R is investigated. Additionally, the relation is explored between pairs (ψ, X ) and (φ, Y) such that interpolation of functions in V (ψ, X ) via interpolants in V (φ, Y) solely from the samples of the original function is possible and stable. Some conditions are given for which the sampling problem is stable, and for which recovery of functions from their interpolants from a family of spaces V (φα, Y) is possible.

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“…Other functions e may be considered as well, both for the present net moment estimation problem or for higher order or local moments recovery. One can also take the components e i to be appropriate band-limited basis functions, of Slepian type [22]. Particularly interesting magnetizations are the unidirectional ones, of which the moment recovery using the above linear estimator process still requires a specific study.…”

Section: Perspectives Conclusionmentioning

confidence: 99%

Solutions to inverse moment estimation problems in dimension 2, using best constrained approximation

Leblond

Pozzi

2021

Journal of Approximation Theory

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We study an inverse problem that consists in estimating the first (zero-order) moment of some R 2 -valued distribution m supported within a closed intervalS ⊂ R, from partial knowledge of the solution to the Poisson-Laplace partial differential equation with source term equal to the divergence of m on another interval parallel to and located at some distance from S. Such a question coincides with a 2D version of an inverse magnetic "net" moment recovery question that arises in paleomagnetism, for thin rock samples. We formulate and constructively solve a best approximation problem under constraint in L 2 and in Sobolev spaces involving the restriction of the Poisson extension of the divergence of m. Numerical results obtained from the described algorithms for the net moment approximation are also furnished.

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Recovery of bivariate band-limited functions using scattered translates of the Poisson kernel

Cited by 4 publications

References 8 publications

Cardinal interpolation with general multiquadrics: convergence rates

Cardinal interpolation with general multiquadrics: convergence rates

On the structure and interpolation properties of quasi shift-invariant spaces

Solutions to inverse moment estimation problems in dimension 2, using best constrained approximation

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