Certain problems of analytic continuation from interior sets (Q1078703)

Let D be a bounded domain in the (x,y) plane. A set \(M\subset D\) is called a set of uniqueness for D if every function f(x,y) real-analytic in x and in y is uniquely determined by its values on M. For such an M let \((x_ 0,y_ 0)\) be a point such that \(M_ r=M\cap \{(x-x_ 0)^ 2+(y-y_ 0)^ 2<r^ 2\}\) is a set of uniqueness for the domain \(\{(x- x_ 0)^ 2+(y-y_ 0)^ 2\leq r'{}^ 2\}\) whenever \(0<r<r'\). The author gives a geometrical condition for a point \((x_ 0,y_ 0)\in M\) to be a point with the property described above. For this purpose the notion of analytic density of the set M at \((x_ 0,y_ 0)\) is introduced.