Least squares approximations for dual trigonometric series

Abstract

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.

The convergence of least squares approximations for dual orthogonal series

“…that have appeared in the literature. Our results establish for such dual series the existence of a sequence of functions satisfying in the L 2 norm the dual series relation, with an error tending to zero and, in particular, rigorously justify the calculations in [2] which showed least squares to be a practical approximation procedure for dual trigonometric equations. In fact, the desire to provide a rigorous convergence theorem for [2] motivated this study.…”

“…In a manuscript in preparation entitled " Dual orthogonal series ", we show that, if ajb n tends to a positive limit as n -» oo, the sequence of least squares approximations converges to a vector {j\Ji> • • •) with square summable components. The proof is more elaborate than the straightforward analysis given here and does not apply to the examples here, in [2], and, in general, to dual orthogonal series associated with mixed boundary value problems in which one of the mixed conditions is a Dirichlet condition.…”

“…Let {b n } be a sequence of positive numbers. Let {a n } be such that either (i) a n > 0 (n = l,2,...), (2) or (ii) there is a positive integer K such that a n = 0 (n = l , 2 , . .…”

A Dirichlet-Jordan theorem for dual trigonometric series

“…These dual series have primarily been examined in connection with applications, especially in mechanical engineering as explained in [19] (see [1; 2; 9; 12; 20; 16] and references [6; 8] in [10] for more recent applications). Understandably this has led to the development of formal answers and special methods with little information on the limitations needed to insure their validity, cf., [19; 4; 7; 10; 24] and references [1; 2; 9-17] in [10]. This paper was motivated by the desire to present a unified approach to these equations and to answer basic mathematical questions of existence, uniqueness, and behavior raised earlier.…”

“…The need for this was made more urgent by Srivastav's interesting formal construction [21] showing eq. (4A-B) does not have a unique solution and the discussion that has occurred for some solutions e.g., reference [1] in [10] and Math. Rev.…”