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Convoluted \(C\)-cosine functions and semigroups. Relations with ultradistribution and hyperfunction sines. (English) Zbl 1171.47036
Different extensions of the concepts of \(C_0\)-semigroups and cosine functions have appered in the last forty years. In this nice paper, the authors consider local convoluted \(C\)-cosine functions and semigroups. Local convoluted semigroups were introduced by I. Ciorănescu [Lect. Notes Pure Appl. Math. 168, 107–122 (1994; Zbl 0818.47038)] in order to extend the theory of integrated semigroups introduced by W. Arendt and M. Hieber in 1986.
In the present paper, the notions of subgenerator and integral generator are used for local convoluted \(C\)-cosine functions and semigroups. Interesting examples which fall under this approach are ultradistributions in the Roumieu–Gevrey and Beurling classes and hyperfunctions sines which were first handled by S. Ōuchi [Proc. Japan Acad. 47, 541–544 (1971; Zbl 0234.46045)] and J. Chazarain [J. Funct. Anal. 7, 386–446 (1971; Zbl 0211.12902)].
They also consider the well-known abstract Weierstrass formula to show the connection between convoluted \(C\)-semigroups and analytic convoluted \(C\)-semigroups. In these ideas, they show that generators of hyperfunction and ultradistribution sines are also generators of analytic convoluted semigroups. Nice examples are also given: they consider the polyharmonic operator \(\Delta^{2^n}\) for \(n\in \mathbb N\) on the space \(L^2([0, 1])\) with suitable boundary conditions. Under these conditions, these operators generate a \(K_n\)-convoluted semigroup for a certain kernel \(K_n\).

MSC:
47D09 Operator sine and cosine functions and higher-order Cauchy problems
34G10 Linear differential equations in abstract spaces
46F10 Operations with distributions and generalized functions
47D60 \(C\)-semigroups, regularized semigroups
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