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Spaces of type \(W\), growth of Hermite coefficients, Wigner distribution, and Bargmann transform. (English) Zbl 0749.46027

Summary: In their famous monograph on generalized functions Gelfand and Shilov introduce the function spaces \(W_ M\), \(W^ \Omega\), and \(W^ \Omega_ M\). These spaces consist of \(C^ \infty\)-functions and holomorphic functions, respectively, with growth behaviour specified by suitable convex functions \(M\) and \(\Omega\). In this paper we study the spaces \(W_ M^{M^ \times}\), where \(M^ \times\) denotes Young’s dual function corresponding to \(M\). We characterize the Hermite expansion coefficients of the functions in \(W_ M^{M^ \times}\), their Fourier transforms, Wigner distributions, and Bargmann transforms. In particular, we prove that \(W_ M^{M^ \times}=W_ M\cap W^{M^ \times}\).

MSC:

46F12 Integral transforms in distribution spaces
46F05 Topological linear spaces of test functions, distributions and ultradistributions
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