Vol. 80, No. 2, 1979

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Examples of solvable and nonsolvable convolution equations in š’¦pā€², p ā‰„ 1

Olaf von Grudzinski

Vol. 80 (1979), No. 2, 561ā€“574

For p [1,+) let 𝒦′p be the space of distributions on Rn not growing faster than some power of exp(|⋅|p), and let 𝒦be the space of distributions on Rn of finite order. For every p (1,+] the existence of convolutors f is proved such that f ∗𝒦p= 𝒦pbut f ∗𝒦s𝒦sfor every s < p. The main step in the proof is a construction of slowly decreasing entire functions which satisfy suitable estimates of Paley-Wiener type and which have countably many zeros of orders as high as possible.

Mathematical Subject Classification 2000
Primary: 45E10
Secondary: 46F05
Received: 20 March 1978
Published: 1 February 1979
Olaf von Grudzinski