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A new representation of the Cauchy kernel for an arbitrary acute convex cone Γ in ℝⁿ is found. The domain of holomorphy of is described. An estimation of the growth of near the singularities is given.
The space of Laplace ultradistributions supported by a convex proper cone is introduced. The Seeley type extension theorem for ultradifferentiable functions is proved. The Paley-Wiener-Schwartz type theorem for Laplace ultradistributions is shown. As an application, the structure theorem and the kernel theorem for this space of ultradistributions are given.
A kernel theorem for spaces of Laplace ultradistributions supported by an n-dimensional cone of product type is stated and proved.
We give a new characterisation of Borel summability of formal power series solutions to the n-dimensional heat equation in terms of holomorphic properties of the integral means of the Cauchy data. We also derive the Borel sum for the summable formal solutions.
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