Infinite order sobolev spaces, analytic continuation and polynomial expansions

Abstract

Let

and

. We consider the following questions:

(A) Under what condition on e _j , does every f ∊ W(e _j ;R) have an analytic continuation to a strip S(r)={z: ∣Imz∣<r}?

(B) Under what condition on e _j , does every f ∊ W(e _j ;R⁺) have an analytic continuation to a sector ∊(α)={z∣Imz∣<α}?

Problem (B) is more subtle. In fact, we prove that for any c _j with W(e _j ;R⁺)¦{0}, there is f ∊ W(c _j R⁺) that fails to have an analytic continuation to a “concave” sector ∊(α), α→π/2. Certain polynomial expansions play important roles in connection with (B).

AMS No:

• 46E35
• 30B50

^∗This research was partially supported by Grant-in-Aid for Encouragement of Young Scientist (No. 03740071) of Ministry of Education of Japan.

^∗This research was partially supported by Grant-in-Aid for Encouragement of Young Scientist (No. 03740071) of Ministry of Education of Japan.

Notes

^∗This research was partially supported by Grant-in-Aid for Encouragement of Young Scientist (No. 03740071) of Ministry of Education of Japan.