Universal approximation by translates of fundamental solutions of elliptic equations

Nestoridis, Vassili; Smyrlis, Yiorgos–Sokratis

doi:10.1524/anly.2011.1061

Cited by 2 publications

(2 citation statements)

References 19 publications

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“…Special cases of Theorem 2.7 regarding universal series whose terms are translates of fundamental solutions of elliptic operators, translates of approximate identities or translates of fundamental solutions of the ∂-operator can be found in [11,12,14,16].…”

Section: Examplesmentioning

confidence: 99%

Universal series in ∩_p
>1 ℓ^p

Koumandos¹,

Nestoridis²,

Smyrlis³

et al. 2009

Bulletin of the London Mathematical Society

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In this paper an abstract condition is given yielding universal series defined by sequences a = {a j } ∞ j=1 in ∩p>1 p but not in 1 . We obtain a unification of some known results related to approximation by translates of specific functions including the Riemann ζ-function, or a fundamental solution of a given elliptic operator in R ν with constant coefficients or an approximate identity as, for example, the normal distribution. Another application gives universal trigonometric series in R ν simultaneously with respect to all σ-finite Borel measures in R ν . Stronger results are obtained by using universal Dirichlet series.

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Section: Examplesmentioning

confidence: 99%

Universal series in ∩_p
>1 ℓ^p

Koumandos¹,

Nestoridis²,

Smyrlis³

et al. 2009

Bulletin of the London Mathematical Society

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“…The existence of universal series of translates of fundamental solutions of Laplace's equation with singularities on a prescribed pseudo-boundary was established in[21].…”

mentioning

confidence: 99%

Density results with linear combinations of translates of fundamental solutions

Smyrlis

2009

Journal of Approximation Theory

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In the present work, we investigate the approximability of solutions of elliptic partial differential equations in a bounded domain Ω by linear combinations of translates of fundamental solutions of the underlying partial differential operator. The singularities of the fundamental solutions lie outside of Ω . The domains under consideration may possess holes and they are required to satisfy a rather mild boundary regularity requirement, namely the segment condition. We study approximations with respect to the norms of the spaces C k (Ω ) and the spaces of uniformly Hölder continuous functions lip k,σ (Ω ), and we establish density and non-density results for elliptic operators with constant coefficients. We also provide applications of our density results related to the method of fundamental solutions and to the theory of universal series.

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Universal approximation by translates of fundamental solutions of elliptic equations

Cited by 2 publications

References 19 publications

Universal series in ∩_p
>1 ℓ^p

Universal series in ∩_p
>1 ℓ^p

Density results with linear combinations of translates of fundamental solutions

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Universal approximation by translates of fundamental solutions of elliptic equations

Cited by 2 publications

References 19 publications

Universal series in ∩ p >1 ℓ p

Universal series in ∩ p >1 ℓ p

Density results with linear combinations of translates of fundamental solutions

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Universal series in ∩_p
>1 ℓ^p

Universal series in ∩_p
>1 ℓ^p