Saburou Saitoh

The weierstrass transform and an isometry in the heat equation

The Weiersirass transform comes from a greens Function of the heat equations. In This paper, we obtain a new natural inverse for this transform and establish an interesting isometry in the heat equation.

Approximate real inversion formulas of the gaussian convolution

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.

Analyticity and global existence of small solutions to some nonlinear Schrödinger equations

AbstractIn this paper we will study the nonlinear Schrödinger equations:

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

\begin{gathered} i\partial _t u + \tfrac{1}{2}\Delta u = \left| u \right|^2 u, (t,x) \in \mathbb{R} \times \mathbb{R}_x^n , \hfill \\ u(0,x) = \phi (x), x \in \mathbb{R}_x^n \hfill \\ \end{gathered}

One approach to some general integral transforms and its applications

. It is shown that the solutions of (*) exist and are analytic in space variables fort∈ℝ∖{0} if φ(x) (∈H2n+1,2(ℝxn)) decay exponentially as |x|→∞ andn≧2.

Representations of inverse functions

In this paper we shall give practical real inversion formulas of heat conduction on multidimensional spaces and show their numerical experiments by using computers.

Aveiro Discretization Method in Mathematics: A New Discretization Principle

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.

Natural norm inequalities in nonlinear transforms

Various weighted L p (p >; 1) norm inequalities in convolutions are derived by a simple and general principle whose L 2 version was given by the idea of products in Hilbert spaces introduced through their transforms and obtained by using the theory of reproducing kernels.