A Note on Functions of Exponential Type
نویسنده
چکیده
for all sufficiently large \z\. The following three theorems were proved respectively by D. V. Widder [4], I. J. Schoenberg [2], and H. Poritsky [l] and J. M. Whittaker [3]. THEOREM 1. (Widder.) If a real f unction f (z), of class C ° ° w 0 ^ x ^ l , satisfies the condition (2) ( l)*/ (*) è O , 0 ^ x S 1; n = 0, 1, 2, • • • , then f (x) coincides on (0, 1) with an entire function of exponential type at most ir. THEOREM 2. (Schoenberg.) If f{z) is an entire function of exponential type at most T1 and if (3) ƒ<»»>((>) =/<*»>(l) = 0 , # = 0, 1, 2, . . , then f(z) is a sine polynomial of order at most T/TT :
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تاریخ انتشار 2007