Convolution equations for vector-valued entire functions of nuclear bounded type
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- by Thomas A. W. Dwyer PDF
- Trans. Amer. Math. Soc. 217 (1976), 105-119 Request permission
Abstract:
Given two complex Banach spaces E and F, convolution operators “with scalar coefficients” are characterized among all convolution operators on the space ${H_{Nb}}(E’;F)$ of entire mappings of bounded nuclear type of E’ into F. The transposes of such operators are characterized as multiplication operators in the space $Exp(E;F’)$ of entire mappings of exponential type of E into F’. The division theorem for entire functions of exponential type of Malgrange and Gupta is then extended to the case when one factor is vector-valued. With this tool the following “vector-valued” existence and approximation theorems for convolution equations are proved: THEOREM 1. Nonzero convolution operators “of scalar type” are surjective on ${H_{Nb}}(E’;F)$. THEOREM 2. Solutions of homogeneous convolution equations of scalar type can be approximated in ${H_{Nb}}(E’;F)$ by exponential-polynomial solutions.References
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Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 217 (1976), 105-119
- MSC: Primary 46G99; Secondary 32H15
- DOI: https://doi.org/10.1090/S0002-9947-1976-0487450-3
- MathSciNet review: 0487450