m at h . FA ] 2 6 Fe b 20 07 Convoluted C - cosine functions and semigroups . Relations with ultradistribution and hyperfunction sines
نویسندگان
چکیده
Convoluted C-cosine functions and semigroups in a Banach space setting extending the classes of fractionally integrated C-cosine functions and semigroups are systematically analyzed. Structural properties of such operator families are obtained. Relations between convoluted C-cosine functions and analytic convoluted C-semigroups, introduced and investigated in this paper are given through the convoluted version of the abstract Weierstrass formula which is also proved in the paper. Ultradistri-bution and hyperfunction sines are connected with analytic convoluted semigroups and ultradistribution semigroups. Several examples of operators generating convo-luted cosine functions, (analytic) convoluted semigroups as well as hyperfunction and ultradistribution sines illustrate the abstract approach of the authors. As an application, it is proved that the polyharmonic operator (−∆) 2 n , n ∈ N, acting on L 2 [0, π] with appropriate boundary conditions, generates an exponentially bounded K n-convoluted cosine function, and consequently, an exponentially bounded analytic K n+1-convoluted semigroup of angle π 2 , for suitable exponentially bounded kernels K n and K n+1 .
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