Approximation Theorems for Exponentially Bounded *a*-Times Integrated Cosine Function

Affiliation(s)

School of Mathematics and Information Science, Shangqiu Teachers College, Shangqiu, Henan, China.

School of Mathematics and Information Science, Shangqiu Teachers College, Shangqiu, Henan, China.

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.

In this paper, based on the theories of α-times Integrated Cosine Function, we discuss the approximation theorem for

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.

Ling, L. (2014) Approximation Theorems for Exponentially Bounded

References

[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.

http://dx.doi.org/10.1112/plms/s3-55.1.181

[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. http://dx.doi.org/10.1006/jmaa.1994.1007

[6] Shaw, S.-Y. and Liu, H. (2002) Convergence Rates of Regularized Approximation Processes. Journal of Approximation Theory, 115, 21-43. http://dx.doi.org/10.1006/jath.2001.3650

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

http://dx.doi.org/10.1007/s10092-008-0152-5

[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.

[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.

http://dx.doi.org/10.1112/plms/s3-55.1.181

[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. http://dx.doi.org/10.1006/jmaa.1994.1007

[6] Shaw, S.-Y. and Liu, H. (2002) Convergence Rates of Regularized Approximation Processes. Journal of Approximation Theory, 115, 21-43. http://dx.doi.org/10.1006/jath.2001.3650

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

http://dx.doi.org/10.1007/s10092-008-0152-5

[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.