mathematics of computation volume 46, number 174 april 1986. pages 551-565 Least Squares Approximation With Constraints By Gradimir V. Milovanovic and Staffan Wrigge Abstract. In this paper we study two families of functions Fe and F0, and show how to approximate the functions in the interval [-1,1]. The functions are assumed to be real when the argument is real. We define Fe = {f: f(-x) = f(x), f(l) = 0, f z L2[-l,l]} and F0-{/:/(-*)= -/(*),/(l) = 0,/eL2[-l,l]}. Let further £Pm be the set of all real polynomials of degree not higher than m such that the polynomials belong to the set Fe if m is even and to the set F0 if m is odd. We determine the least squares approximation for the function / e Fe (or F„) in the class 9,ln (or á^n + i)' with respect to the norm ||/|| = ((/,/))1/2, where the inner product is defined by (/,g) = j1_lf(x)g(x)w(x)dx, with f,g<= L2[-l,l]- /¿I-1,1] and w(x) = (1_JC2)A-l/2 We also consider the general case when / is neither an even nor an odd function but /sL2[-l,l]and/(-l)=/(l) = 0. Using the theory of Gegenbauer polynomials we obtain the approximating polynomials in the form </>2„(*) = ¿¿„.¿O-*2)* when/sF, and <t>2„ + i(x) = *Y, en,k(l - x2)k when/ef;. *-i We apply the general theory to the functions f(x) = cos(7rx/2) and f(x) = J0(aox), where a0 = {minx > 0: J0(x) = 0}. 0. Introduction. In [12] Wrigge and Fransen considered two families of functions, viz., F and H, and showed how these functions can be approximated on [0, 1] by polynomials of the form k k L cn>,(*(l - *))" and (l-2x)XX,,(*(l-*))". «=1 n=l They used the L2-norm with respect to the weight function w(x) = (x(l - x))q, where q e {0,1,2,...}. This method of approximation can be further generalized, as was shown in Wrigge [11], by using Bernstein polynomials. Received December 13, 1983; revised November 13, 1984 and June 3, 1985. 1980 Mathematics SubjectClassification. Primary 41A10; Secondary 33A45, 65D15, 65D20. Key words and phrases. Approximation theory, Gegenbauer polynomials. ©1986 American Mathematical Society 0025-5718/86 $1.00 + $.25 per page 551 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use