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On reduction of linear two variable functional equations to differential equations without substitutions

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Summary

The subject of this paper is the investigation of the linear two variable functional equation

$$h_0 (x,y)f_0 (g_0 (x,y)) + \cdots + h_n (x,y)f_n (g_n (x,y)) = F(x,y),$$
((*))

whereg 0, ⋯,g n,h 0, ⋯,h n andF are given real valued functions on an open setΩR 2, furtherf 0, ⋯,f n are unknown real functions.

Assuming differentiability of sufficiently large order, we construct a linear partial differential operator

$$D = \sum\limits_{i,j \geqslant 0,i + j \leqslant k} {\alpha _{ij} (x,y)} \partial _x^i \partial _y^i ,$$

whereα ij is defined onΩ, so that

$$D[h_i (x,y)\varphi (g_i (x,y))] = 0, (x,y) \in \Omega , i = 1,...,n$$

for all sufficiently smooth functionsϕ defined on\( \cup _{l = 1}^n g_i (\Omega )\). Then, applyingD to (*), we obtain

$$D[h_0 (x,y)f_0 (g_0 (x,y))] = DF(x,y), (x,y) \in \Omega ,$$

, which is ak-th order linear differential-functional equation for the unknown functionf 0.

Using Járai's regularity theorems (see [3], [4], [5]) one can see that, if the given functionsg t,h t,F, are differentiable up to an orderk (1 ⩽k ⩽ ∞), then the measurability of the unknown functionsf t imply their differentiability up to the same order. In this paper we prove an analogous result, namely that the analyticity of the given functions implies that the unknown functions are also analytic provided that they are measurable.

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References

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Páles, Z. On reduction of linear two variable functional equations to differential equations without substitutions. Aeq. Math. 43, 236–247 (1992). https://doi.org/10.1007/BF01835706

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  • DOI: https://doi.org/10.1007/BF01835706

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