NEW TWO-RADII THEOREMS IN THE THEORY OF HARMONIC FUNCTIONS

Functions satisfying the mean value equation over balls of several fixed radii are investigated. A number of substantial amplifications of known theorems of Delsarte and Flatto are obtained. Considered also is the case when the mean value equation is satisfied only approximately (restriction on the growth of the difference between the value of the function at the center of a ball and the mean value over that ball), but nevertheless allows deduction of harmonicity of the function under certain conditions.