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Interpolation of vector-valued real analytic functions. (English) Zbl 1027.46048
The vector-valued interpolation problem considered in this deep and important paper reads as follows: Given a domain \(\omega \subset \mathbb{R}^d\); a discrete sequence \((z_n)_n\) in \(\omega\); a sequentially complete locally convex space \(E\); a sequence \((k_n)_n\) of nonnegative integers; and a family \(\{x_{n,\alpha}: \alpha\in \mathbb{N}^d\), \(|\alpha|\leq k_n\), \(n \in \mathbb{N}\}\subset E\), find a real analytic function \(f :\omega \to E\) such that \[ \frac{\partial^{|\alpha|}}{\partial x^\alpha}f(z_n)=x_{n,\alpha} \quad\text{for all }\alpha\in \mathbb{N}^d, \text{ with }|\alpha|\leq k_n, \text{ and }n \in \mathbb{N}. \] In the especially interesting case when \(E\) is a function space, this can be interpreted as a system of equations where the \(x_{n,\alpha}\) depend nicely on a parameter and we want the solution to depend also nicely on the parameter.
Two notions of real analyticity for an \(E\)-valued function can be considered, namely, \(f :\omega\to E\) is said to be real analytic, and we put \(f\in A(\omega,E)\), if for all \(x'\in E'\) the scalar function \(x'\circ f\) is real analytic, whereas \(f\) as above is said to be topologically real analytic, and we put \(f\in A_t(\omega,E)\), if for all \(x\in\omega\) the function \(f\) has a Taylor series convergent to \(f\) around \(x\) in the topology of \(E\). The sets \(A(\omega,E)\) and \(A_t(\omega,E)\) are different for some Fréchet spaces \(E\), but they coincide for sequentially complete DF-spaces [see J. Bonet and P. Domański, Monatsh. Math. 126, 13-36 (1998, Zbl 0918.46034)].
The main result of this paper states that for a sequentially complete DF-space \(E\) each interpolation problem as above (with arbitrary data) has a solution \(f\in A(\omega,E)\) if and only if there exists a fundamental sequence \(B_0\subset B_1\subset\cdots\) of bounded Banach discs in \(E\) and positive numbers \(\delta_k\) such that \(B_k\subset rB_0+\frac 1{r^{\delta_k}}B_{k+1}\) for all \(k \in \mathbb{N}\) and \(r>0.\) This necessary and sufficient condition is a topological invariant, known as condition \((\underline{\text{A}})\), which is closely related to the well-known \((\underline{\text{DN}})\) condition for Fréchet spaces. The proofs consist in careful knittings of techniques involving short exact sequences, topological invariants, tensor products and complex analysis. The main examples of application are products of duals of power series spaces of finite and infinite type and, in particular, the space of distributions \(E={\mathcal D}'\).

MSC:
46G20 Infinite-dimensional holomorphy
26E05 Real-analytic functions
46E40 Spaces of vector- and operator-valued functions
46E10 Topological linear spaces of continuous, differentiable or analytic functions
46A13 Spaces defined by inductive or projective limits (LB, LF, etc.)
32E30 Holomorphic, polynomial and rational approximation, and interpolation in several complex variables; Runge pairs
46A63 Topological invariants ((DN), (\(\Omega\)), etc.) for locally convex spaces
46A04 Locally convex Fréchet spaces and (DF)-spaces
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