On the Structure of Hyperfunctions with Compact Supports

Akira KANEKO
1971 Proceedings of the Japan Academy  
We discuss an analogue of the classical structure theorem of distributions on a compact set. We mainly treat the case of one variable (n=1). The case of several variables with some applications will be discussed by a somewhat different method in a paper under preparation (see [3]). Theorem 1. Let u be a hyperf unction o f one variable with support in the interval K= [a, b]. Then u can be expressed as follows: 2, 3 are measures with supports in [a, b], and J(D), i=1, 2, 3 are local operators
more » ... constant coefficients. (Local operators with constant coefficients are differential operators of infinite order in the theory of hyperf unctions ; see, e. g., [1], § 2. On the operation of J(D), the measures are considered as hyperfunctions.) We prepare two lemmas. Let 9[K] denote the space of hyperfunctions with support in K. Let HK(~) denote the supporting function sup Re of K (i=/1). xSK Lemma 2. The Fourier transform u(~) o f u e B[K] is an entire function which satisfies the following growth condition: From the above estimates it is easily seen that iJr1(r)-oo when r--oo. Thus the function cp(r) = max {inf fr1(r),1} serves our purpose. q.e. d. s>r Lemma 3. Assume that the function cp(r) has the properties mentioned in Lemma 2. Then for any prescribed constants A, C, c1, c2 there exists a local operator J(D) whose Fourier transform J(~) satisfies the following estimate from below: ' Partially supported by Fujukai .
doi:10.2183/pjab1945.47.supplementii_956 fatcat:hjg4ipeqxrba3evrccurqa3x5u