### Abstract

The subject of this paper is the investigation of the linear two variable functional equation {Mathematical expression} where g_{0}, ⋯, g_{n}, h_{0}, ⋯, h_{n} and F are given real valued functions on an open set Ω ⊂R^{2}, further f_{0}, ⋯, f_{n} are unknown real functions. Assuming differentiability of sufficiently large order, we construct a linear partial differential operator {Mathematical expression} where α_{ij} is defined on Ω, so that {Mathematical expression} for all sufficiently smooth functions φ{symbol} defined on {Mathematical expression}. Then, applying D to (*), we obtain {Mathematical expression}, which is a k-th order linear differential-functional equation for the unknown function f_{0}. Using Járai's regularity theorems (see [3], [4], [5]) one can see that, if the given functions g_{t}, h_{t}, F, are differentiable up to an order k (1 ≤k ≤ ∞), then the measurability of the unknown functions f_{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.

Original language | English |
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Pages (from-to) | 236-247 |

Number of pages | 12 |

Journal | Aequationes Mathematicae |

Volume | 43 |

Issue number | 2-3 |

DOIs | |

Publication status | Published - Apr 1 1992 |

### Keywords

- AMS (1990) subject classification: 39B22

### ASJC Scopus subject areas

- Mathematics(all)
- Discrete Mathematics and Combinatorics
- Applied Mathematics