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Generalizations of the wave equation. (English) Zbl 0804.26013

The authors introduce the generalized unsymmetric mixed partial derivative \[ \lim_{h,k\to 0} (f(x+ h,y+ k)- f(x+ h,y)- f(x,y+ k)+ f(x,y))/hk \] and the generalized symmetric mixed partial derivative \[ \lim_{h,k\to 0} (f(x+ h,y+ k)- f(x+ h,y- k)- f(x- h,y+ k)+ f(x- h,y- k))/4hk \] for real valued functions on \(\mathbb{R}^ 2\).
From authors’ abstract: “The main result of this paper is a generalization of the property that, for smooth \(u\), \(u_{xy}= 0\) implies \((*)\) \(u(x,y)= a(x)+ b(y)\). Any function having generalized unsymmetric mixed partial derivative identically zero is of the form \((*)\). There is a function with generalized symmetric mixed partial derivative identically zero not of the form \((*)\), but \((*)\) does follow here with the additional assumption of continuity. These results connect to the theory of uniqueness for multiple trigonometric series”.

MSC:

26B05 Continuity and differentiation questions
42B05 Fourier series and coefficients in several variables
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