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Integral representation of holomorphic mappings on fully nuclear spaces. (English) Zbl 1134.46019

Let \(E\) be a fully nuclear space with a basis \((e_n)\). Recall that a holomorphic function \(f:E \to \mathbb C\) is said to be of \(\eta\)-exponential type for some sequence \(\eta=(\eta_n)\) of non-negative numbers \(\eta_n\) if it satisfies an estimate \(| f(z)| \leq A\cdot e^{B\cdot \| z\| _\eta}\) for some constants \(A,B>0\) and all \(z \in E\), where \(\| z\| _\eta=\sum_n \eta_n\cdot | z_n| \). For those functions, the authors prove the following integral representation
\[ f(z)=\int_{E_\beta'} e^{\langle z,w \rangle} \cdot (f \circ D)(w)\,d\mu_\gamma(w), \] where \(D\) is a densely defined diagonal operator from \(E'_\beta\) into \(E\) and \(\mu_\gamma\) is a Gaussian measure on \(E_\beta'\) with variance \(\gamma\), \(\eta/\gamma \in \ell_1\). This formula can be considered as an infinite version of the Cauchy integral formula.
For Banach spaces, a similar result was obtained before by D. Pinasco and I. Zalduendo [J. Math. Anal. Appl. 308, No. 1, 159–174 (2005; Zbl 1086.46033)].

MSC:

46G20 Infinite-dimensional holomorphy
32A26 Integral representations, constructed kernels (e.g., Cauchy, Fantappiè-type kernels)

Citations:

Zbl 1086.46033
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References:

[1] Bourbaki, N., Éléments de Mathématique, Intégration IX, Act. Sc. et Ind., vol. 1343 (1969), Hermann: Hermann Paris · Zbl 0189.14201
[2] Dineen, S., Complex Analysis on Infinite Dimensional Spaces, Monogr. Math. (1999), Springer-Verlag · Zbl 1034.46504
[3] D. Pinasco, Integral representations of holomorphic functions on Banach spaces II, preprint; D. Pinasco, Integral representations of holomorphic functions on Banach spaces II, preprint
[4] Pinasco, D.; Zalduendo, I., Integral representations of holomorphic functions on Banach spaces, J. Math. Anal. Appl., 308, 159-174 (2005) · Zbl 1086.46033
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