Acta Mathematica Academiae Scientiarum Hungaricae 62. (1993)
1993 / 1-2. szám - Prestin, J.: Lagrange interpolation for functions of bounded variation
(1) Acta Math. Hung. 62 (1-2) (1993), 1-13. LAGRANGE INTERPOLATION FOR FUNCTIONS OF BOUNDED VARIATION J. PRESTIN (Rostock) 1. Introduction. We investigate the Lagrange interpolation on Jacobi abscissas for functions of bounded variation on [—1,1]. Error estimates in weighted i^-norms are established, which ensure convergence for a wide class of weights and Jacobi parameters a,ß. The order of convergence is in most cases best possible. For some Jacobi nodes the results can be improved, if the function is of bounded variation and continuous. It turns out that the estimates depend essentially on the behaviour of the error near the endpoints of the interval. 2. Notations and preliminary results. Let be the roots of the Jacobi polynomial P^*'^ (a,ß > —1, n = 1,2,...; see e.g. G. Szegő [5]). In the sequel we will omit the superfluous notations a, ß. Further we write xn+i = — 1, xq = 1 and xk = cost?*, x = cost? with 0 < t?*, 1? < 7Г. For a function / defined in the interval [—1,1] we denote, as usual, L„f(x) = J2f(xkMx) k=l the Lagrange interpolatory polynomials of degree n - 1 based on the nodes (1). Here lk are the A;-th fundamental polynomials of the Lagrange interpolation lk(x) Pn{x) Pn(xk)(x - Xfc)’ к = 1,... , те. In [6] P. Vértesi proved for / 6 С П BV (/ is continuous and of bounded variation in [-1,1]) and for — 1 < a,ß < 1/2 that (2) max |/(x) - L„/(x)| = o(l), n-> oo. *€[-1,1] Furthermore, it is proved in [7] that (2) does not hold in general for max(a,/3) > 1/2. However, for arbitrary a,ß > -1 and / G BV П C, J. L. Geronimus [1] obtained pointwise convergence, too, but only in a compact subinterval of ( — 1,1). Let us mention here that pointwise estimates -1 < x(a>0)< X(a’0) П —1< ... < X(«./3) 2 < X1 < 1