APM  Vol.4 No.11 , November 2014
Approximation Theorems for Exponentially Bounded a-Times Integrated Cosine Function
Abstract: In this paper, based on the theories of α-times Integrated Cosine Function, we discuss the approximation theorem for α-times Integrated Cosine Function and conclude the approximation theorem of exponentially bounded α-times Integrated Cosine Function by the approximation theorem of n-times integrated semigroups. If the semigroups are equicontinuous at each point   , we give different methods to prove the theorem.
Cite this paper: Ling, L. (2014) Approximation Theorems for Exponentially Bounded a-Times Integrated Cosine Function. Advances in Pure Mathematics, 4, 580-584. doi: 10.4236/apm.2014.411066.

[1]   Arendt, W. (1987) Vector-Valued Laplace Transforms and Cauchy Problems. Israel Journal of Mathematics, 59, 327-352.

[2]   Arent, W. and Kellermaan, H. (1989) Integrated Solutions of Volterra Integro-Differential Equations and Applications. Pitman Research Notes in Mathematics, 190, 21-51.

[3]   Davies, E.B. and Pang, M.M.H. (1987) The Cauchy Problem and a Generalization of the Hill-Yosida Theorem. Proceedings of the London Mathematical Society, 55, 181-208.

[4]   Zheng, Q. and Lei, Y.S. (1993) Exponentially Bounded C-Semigroup and Integrated Semigroup with Nondensely Defined Generators I: Approximation. Acta Mathematica Scientia, 13, 251-260.

[5]   Lizama, C. (1994) On the Convergence and Approximation of Integrated Semigroups. Journal of Mathematical Analysis and Applications, 181, 89-103.

[6]   Shaw, S.-Y. and Liu, H. (2002) Convergence Rates of Regularized Approximation Processes. Journal of Approximation Theory, 115, 21-43.

[7]   Campiti, M. and Tacelli, C. (2008) Approximation Processes for Resolvent Operators. Calcolo, 45, 235-245.

[8]   Cao, D.-X., Song, X.-Q. and Zhang, X.-Z. (2007) The Approximations of m-Times Integrated Cosine Functions. Mathematics in Practice and Theory, 37, 164-167.

[9]   Xiao, T.-J. and Liang, J. (2000) Approximation of Laplace Transforms and Integrated Semigroups. Journal of Functional Analysis, 172, 202-220.

[10]   Zhang, J.Z. (1997) α-Times Integrated Cosine Function. Acta Mathematica Scientia, 17, 33-38.