BEST ONE-SIDED L-1-APPROXIMATION BY HARMONIC-FUNCTIONS

Let f is an element of C((B) over bar), where B is the open unit ball in R-n (n greater than or equal to 2), and let U-B(f) denote the colle ction of functions h in C((B) over bar) which are harmonic on B and sa tisfy h less than or equal to f on B. A function h in U-B(f) is calle d a best harmonic one-sided L-1-approximant to f if integral((B) over bar)\f - h\ less than or equal to integral\ f - h\ for all h in U-B(f ). This paper characterizes such approximants and discusses questions of existence and uniqueness. Corresponding results for approximation o n the cylinder B x R are also established, but the proofs in this case are more difficult and rely on recent work concerning tangential harm onic approximation. The characterizations are quite different in natur e from those recently obtained for harmonic L-1-approximation without a one-sidedness condition.