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Characterization of Non-Linear Transformations Possessing Kernels

Published online by Cambridge University Press:  20 November 2018

Victor J. Mizel
Victor J. Mizel*
Affiliation:
Carnegie-Mellon University, Pittsburgh, Pennsylvania
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Recently, in collaboration with Martin [10] and Sundaresan [11], I obtained a characterization of certain classes of non-linear functionals defined on spaces of measurable functions (see also [12]). The functionals in question had the form

(1.1)

with a continuous “kernel” φ: R → R,or

(1.2)

with a separately continuous kernel φ: R2 → R. There are direct applications of this work to the theory of generalized random processes in probability (see [8]) and to the theory of fading memory in continuum mechanics [3]. However, the main motivation for these studies was an interest in possible application to the functional analytic study of non-linear differential equations. From the standpoint of this latter application it would also be desirable to characterize the broader class of functionals having the form

(1.3)

where the kernel φ: R × T → R satisfies “Carathéodory conditions”.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 3 , 01 June 1970 , pp. 449 - 471
Copyright
Copyright © Canadian Mathematical Society 1970

References

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