Approximate real inversion formulas of the gaussian convolution

Abstract

We shall give very and surprisingly simple approximate real inversion formulas of the Gaussian convolution (the Weierstrass transform) for the first-order Sobolev Hilbert space on the whole real line by using best approximations and the theory of reproducing kernels and by using a good connection with the Tikhonov regularization.

Keywords:

• Gaussian convolution
• Weierstrass transform
• Approximation of functions
• Reproducing kernel
• Tikhonov regularization
• Sobolev space
• Generalized inverse
• Approximate inverse
• Mathematics Subject Classifications (2000): Primary 44A15
• 35K05
• 30C40

Acknowledgement

The author wishes to express his deep thanks to Professor Masaru Ikehata for valuable discussions on the Tikhonov regularization.