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Representing and interpolating sequences on parabolic Bloch type spaces. (English) Zbl 1252.35011
Summary: Let \(H\) be the upper half-space of the Euclidean space. The \(\alpha \)-parabolic Bloch type space \(\mathcal B_{\alpha }(\sigma )\) on \(H\) is the set of all solutions \(u\) of the parabolic equation \((\partial /\partial t + ( - \Delta _{x})^{\alpha })u = 0\) with \(0 < \alpha \leq 1\) which belong to \(C^{1}(H)\) and have finite Bloch norm with weight \(t^{\sigma }\). In this paper, we study representing and interpolating sequences on parabolic Bloch type spaces. In our paper [Hiroshima Math. J. 41, No. 1, 55–87 (2011; Zbl 1233.35199)], the reproducing formula on \(\mathcal B_{\alpha }(\sigma )\) was given. A representing sequence gives a discrete version of the reproducing formula on \(\mathcal B_{\alpha }(\sigma )\). Interpolating sequences are closely related to representing sequences, and such sequences are very interesting in their own right.
MSC:
35A08 Fundamental solutions to PDEs
35K05 Heat equation
31B10 Integral representations, integral operators, integral equations methods in higher dimensions
32A18 Bloch functions, normal functions of several complex variables
35R11 Fractional partial differential equations
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