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Kergin interpolation in Banach spaces

100%

Petersson H.

Studia Mathematica

2002

tom 153

nr 2

101-114

We study the Kergin operator on the space $H_{Nb}(E)$ of nuclearly entire functions of bounded type on a Banach space E. We show that the Kergin operator is a projector with interpolating properties and that it preserves homogeneous solutions to homogeneous differential operators. Further, we show that the Kergin operator is uniquely determined by these properties. We give error estimates for approximating a function by its Kergin polynomial and show in this way that for any given bounded sequence of interpolation points and any nuclearly entire function, the corresponding sequence of Kergin polynomials converges.

Rings of PDE-preserving operators on nuclearly entire functions

100%

Petersson H.

Studia Mathematica

2004

tom 163

nr 3

217-229

Let E,F be Banach spaces where F = E' or vice versa. If F has the approximation property, then the space of nuclearly entire functions of bounded type, $ℋ_{Nb}(E)$, and the space of exponential type functions, Exp(F), form a dual pair. The set of convolution operators on $ℋ_{Nb}(E)$ (i.e. the continuous operators that commute with all translations) is formed by the transposes $φ(D) ≡ ^{t}φ$, φ ∈ Exp(F), of the multiplication operators φ :ψ ↦ φ ψ on Exp(F). A continuous operator T on $ℋ_{Nb}(E)$ is PDE-preserving for a set ℙ ⊆ Exp(F) if it has the invariance property: T ker φ(D) ⊆ ker φ(D), φ ∈ ℙ. The set of PDE-preserving operators 𝒪(ℙ) for ℙ forms a ring and, as a starting point, we characterize 𝒪(ℍ) in different ways, where ℍ = ℍ(F) is the set of continuous homogeneous polynomials on F. The elements of 𝒪(ℍ) can, in a one-to-one way, be identified with sequences of certain growth in Exp(F). Further, we establish a kernel theorem: For every continuous linear operator on $ℋ_{Nb}(E)$ there is a unique kernel, or symbol, and we characterize 𝒪(ℍ) by describing the corresponding symbol set. We obtain a sufficient condition for an operator to be PDE-preserving for a set ℙ ⊇ ℍ. Finally, by duality we obtain results on operators that preserve ideals in Exp(F).

Ograniczanie wyników

2 Studia Mathematica

2 Petersson H.

1 2004

1 2002

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Kergin interpolation in Banach spaces

Rings of PDE-preserving operators on nuclearly entire functions