1. Introduction. A systematic and easily automated least squares procedure, not using integral equations or special functions, is presented for approximating the solutions of general dual trigonometric equations. This is desirable, since current analytic methods apply only to special equations, require the use of integral equation and special function theory, and do not lend themselves easily to numerical work; see, e.g. [1,2,6,8,9,10,11,12,13,14,15,16,17].The series are described in § 2. The equation for least squares approximation is derived in § 3 and used to develop the computer program DUTSA (DUal Trigonometric Series Analyser). A few examples are presented in § 4 from amongst the several dozen dual trigonometric series to which DUTSA has been applied. These include examples from classes of dual equations for which solutions are not now available (save possibly for very special cases), e.g. arbitrary series not connected with applications, series associated with harmonic mixed boundary value problems in (bounded) rectangles and series with one of the mixed boundary conditions corresponding to a (linear) radiation condition.! The evidence from these computations indicates that most dual trigonometric series from applications can be solved with a relative least squares error (defined in § 5) smaller than 4% in 10 seconds or fewer on a computer with a 6 microsecond multiplication time.Our analysis, in common with earlier studies, is heuristic: the numerical evidence is suggestive, but it does not rigorously describe the limitations of the method. The least squares approach is so simple that we searched the literature carefully, but found no evidence of its prior use. We expect that it will be helpful in studying other dual and similar series.
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