Kergin interpolation in Banach spaces

Petersson, Håkan

doi:10.4064/sm153-2-1

Cited by 4 publications

(8 citation statements)

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“…Thanks to Theorem 1, it is possible under certain assumptions, to derive immediately some useful properties of divided differences, in particular: the independence from the ordering of the knots in Θ; its continuity with respect to Θ; its meaning for collapsing points. For further details, one may see [34,17].…”

Section: Proposition 4 Assumementioning

confidence: 98%

“…Some of the results achieved in [36,14,34,17,39] for polynomial interpolation in Banach spaces can be applied, since − → D n is itself a Banach space. These pioneering works were aimed at more theoretical results, whereas here we focus on the numerical analysis; in fact, error estimates are not provided in [36] or demand in [34,17,39] too much regularity. We shall point out that, although the regularity assumptions in [14] are rather weak, the conditions (i), (ii), H1 in [14] for deriving error estimates (in − → D n ) still demand research.…”

Section: Introductionmentioning

confidence: 97%

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Set-valued Hermite interpolation

Baier

Perria

2011

Journal of Approximation Theory

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The problem of interpolating a set-valued function with convex images is addressed by means of directed sets. A directed set will be visualised as a usually non-convex set in R n consisting of three parts together with its normal directions: the convex, the concave and the mixed-type part. In the Banach space of the directed sets, a mapping resembling the Kergin map is established. The interpolating property and error estimates similar to the point-wise case are then shown; the representation of the interpolant through means of divided differences is given. A comparison to other set-valued approaches is presented. The method developed within the article is extended to the scope of the Hermite interpolation by using the derivative notion in the Banach space of directed sets. Finally, a numerical analysis of the explained technique corroborates the theoretical results.

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Section: Proposition 4 Assumementioning

confidence: 98%

Section: Introductionmentioning

confidence: 97%

Set-valued Hermite interpolation

Baier

Perria

2011

Journal of Approximation Theory

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“…In [8] (see also [9]) Petersson has settled a convenient formalism (using the concept of pairing of Banach spaces) and extended results of [3] to Banach spaces. Our Theorem 1 is likely to have a similar infinite dimensional counterpart.…”

Section: Introductionmentioning

confidence: 99%

Polynomial projectors preserving homogeneous partial differential equations

Dũng

Calvi

Trung

2005

Journal of Approximation Theory

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A polynomial projector of degree d on H (C n ) is said to preserve homogeneous partial differential equations (HPDE) of degree k if for every f ∈ H (C n ) and every homogeneous polynomial of degree k, q(z) = | |=k a z , there holds the implication: q(D)f = 0 ⇒ q(D) (f ) = 0. We prove that a polynomial projector preserves HPDE of degree k, 1 k d, if and only if there are analytic functionals k , k+1 , . . . , d ∈ H (C n ) with i (1) = 0, i = k, . . . , d, such that is represented in the following formwith some a 's ∈ H (C n ), | | < k, where u (z) := z / !. Moreover, we give an example of polynomial projectors preserving HPDE of degree k (k 1) without preserving HPDE of smaller degree. We also give a characterization of Abel-Gontcharoff projectors as the only Birkhoff polynomial projectors that preserve all HPDE.

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“…Petersson [12], Filipsson [6] and Simon [14] have extended Kergin interpolation and approximation to the Banach space setting. Hence, the results obtained in this paper are not entirely new.…”

Section: Introductionmentioning

confidence: 99%