A unified theory for classical triple trigonometric series

Abstract

An existence and uniqueness theory is established for the classical triple trigonometric series having the kernels {cos(n-i)x}, {sinnx}, or {cosnx} when the right hand sides of the equations are given functions of bounded variation. It is shown that these series do not, in general, converge in the ordinary sense on any set of positive measure but do converge at all points in the sense of Abel-Poisson The key to the proof is use of a generalized reflection principle to establish uniqueness and thus prove the equivalence of different representations of the solution. Best possible asymptotic estimates for growth and uniqueness of the coefficients are derived

^†The author acknowledges with thanks that this work was supported by the U.S. Army Research Office under Grant No.DAH CO474 G0140

^†The author acknowledges with thanks that this work was supported by the U.S. Army Research Office under Grant No.DAH CO474 G0140

Notes

^†The author acknowledges with thanks that this work was supported by the U.S. Army Research Office under Grant No.DAH CO474 G0140