Natural outputs and global inputs of linear systems with a finite number of input data

Abstract

We shall introduce a new concept for the treatment of certain general linear systems: for a finite number of input data, we would like to consider some natural global output and the corresponding global natural input satisfying the input and output relation. We will be able to formulate such a problem by considering the generalized inverse by means of the Tikhonov regularization in the framework of reproducing kernel Hilbert spaces. We shall give concrete examples for the heat conduction problem and the wave equation case.

Keywords:

• forward problem
• sensibility of system
• discrete data
• reproducing kernel
• sampling theory
• Tikhonov regularization
• Gaussian convolution
• Weierstrass transform
• heat conduction
• wave equation
• Paley–Wiener space
• Sobolev space
• approximation of function
• generalized inverse
• approximate inverse

AMS Subject Classifications::

• 44A15
• 46E22
• 35K05
• 30C40
• 35L05

Acknowledgements

This work was supported in part by Center for Research and Development in Mathematics and Applications of Universidade de Aveiro through the Portuguese Science Foundation (FCT–Fundação para a Ciência e a Tecnologia). Saitoh was supported in part by the Grant-in-Aid for the Scientific Research (C)(2) (No. 21540111) from the Japan Society for the Promotion of Science.