Solving Boundary Value Problems with "Average" Data Sampled on an Arc

Function-theoretic methods are employed with appropriate regularity conditions to solve boundary value problems of the type (V 2 + P(1 x 1 2 ))iz ( x ) = 0 (x 6 D; u(-o) = 1(xo), x0 E aD). The solutions are uniquely determined from the mean values of the boundary data 1(x) sampled at equally spaced points of the circumference; or, along an arc on the circumference of the disk D. Equivalent problems are formulated and solved from the point of view of conformal equivalence of domains and transmutation of differential operators. Solutions to inverse problems are reconstructed from the mean values of data taken along the boundary.