Analytic Continuation and Projection of Hyperfunctions

Abstract

Let us begin with the function f(x) defined by $f(x) = \left\{ {\begin{array}{*{20}{c}} {{{(x - a)}^\alpha }{{(x - b)}^\beta },{\text{ a < x < b}},} \\ {0,{\text{ }}x < a,x > b.} \end{array}} \right.$. If Re α > -1, Re β > -1, then f(x) is absolutely integrable, so f(x) can be reinterpreted as a hyperfunction. If Re α ≤ -1 and/or Re β ≤ -1, f(x) cannot be reinterpreted as a hyperfunction as it stands. What can be done in such as case? We might consider the following method. Notice that f(x) has only two singularities x = a and x = b. So, we choose a point c such that a < c < b and consider the functions

$${{\text{f}}_{\text{1}}}{\text{(}}x{\text{) = }}\left\{ {\begin{array}{*{20}{c}} {f(x),{\text{ }}x < c,} \\ {0,{\text{ }}x > c,} \end{array}} \right.{\text{ }}$$

(1.2)

$${{\text{f}}_2}{\text{(}}x{\text{) = }}\left\{ {\begin{array}{*{20}{c}} {f(x),{\text{ }}x < c,} \\ {0,{\text{ }}x > c.} \end{array}} \right.{\text{ }}$$

(1.3)

Then,

$${\text{f}}(x) = {f_1}(x) + {\text{ }}{f_2}(x).$$

(1.4)

. The singular points of f ₁(x) are x = a and x = c, but x = c is a simple discontinuity. The singular point x = a corresponds to the singularity of xα at x = O. Similarly, the singular points of f ₂(x) are the discontinuity at x = c and x = b which corresponds to the singularity of x _β at x = O. Therefore, f ₁(x) and f ₂(x) are simpler than f(x) itself, so that it may be convenient to consider hyperfunctions corresponding to f ₁ (x) and f ₂(x) and to combine them to obtain the hyperfunction corresponding to f(x).