Analytical and numerical inversion formulas in the Gaussian convolution by using the Paley–Wiener spaces

Abstract

We shall discuss the relations among sampling theory (Sinc method), reproducing kernels and the Tikhonov regularization. Here, we see the important difference of the Sobolev Hilbert spaces and the Paley–Wiener spaces when we use their reproducing kernel Hibert spaces as approximate spaces in the Tikhonov regularization. Further, by using the Paley–Wiener spaces, we shall illustrate numerical experiments for new inversion formulas for the Gaussian convolution as a much more powerful and improved method by using computers. In this article, we shall be able to give practical numerical and analytical inversion formulas for the Gaussian convolution that is realized by computers.

Keywords:

• Sampling theory
• Tikhonov regularization
• Gaussian convolution
• Weierstrass transform
• Heat conduction
• Paley–Wiener space
• Approximation of functions
• Reproducing kernel
• Sobolev space
• Generalized inverse
• Approximate inverse
• Numerical experiments

Keywords:

• Mathematics Subject Classifications (2000): Primary 44A15
• 35K05
• 30C40

Acknowledgments

T. M. was partially supported by the Gunma University Foundation for Promotion of Science and Engineering. S. S. was partially supported by the Grant-in-Aid for the Scientific Research (C)(2) (No. 16540137) from the Japan Society for the Promotion Science and by the Mitsubishi Foundation, Natural Sciences, Vol. 36, No. 20 (2005–2006).