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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Generalizations of the wave equation


Authors: J. Marshall Ash, Jonathan Cohen, C. Freiling and Dan Rinne
Journal: Trans. Amer. Math. Soc. 338 (1993), 57-75
MSC: Primary 35L05; Secondary 26B40, 42B99
MathSciNet review: 1088475
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Abstract: The main result of this paper is a generalization of the property that, for smooth $ u$, $ {u_{xy}} = 0$ implies $ (\ast)$

$\displaystyle u(x,y) = a(x) + b(y).$

Any function having generalized unsymmetric mixed partial derivative identically zero is of the form $ (\ast)$. There is a function with generalized symmetric mixed partial derivative identically zero not of the form $ (\ast)$, but $ (\ast)$ does follow here with the additional assumption of continuity.

These results connect to the theory of uniqueness for multiple trigonometric series. For example, a double trigonometric series is the $ {L^2}$ generalized symmetric mixed partial derivative of its formal $ (x,y)$-integral.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1993-1088475-X
PII: S 0002-9947(1993)1088475-X
Article copyright: © Copyright 1993 American Mathematical Society