Acta Mathematica Academiae Scientiarum Hungaricae 62. (1993)

1993 / 1-2. szám - Prestin, J.: Lagrange interpolation for functions of bounded variation

J. PRESTIN for trigonometric interpolation of functions from BVi* П Сг» on equidistant nodes in [0,2л-] are due to W. Quade [4]. An X^-error estimate is given by K. Zacharias in [8]. For the general Lpv-case see [2] and [3]. Now we define for 1 < p < oo, a, b > —1 and 0 < e < 1 the norm p,a,b,e —( f \f(x)\pw -l+£ where w(x) = (1 - x)“(l + x)b. Obviously, the weight w is only interesting for asymptotic results in the case e = 0. 3. Results. The main result here is the following theorem. Theorem 1. Let f € BV. If e > 0, then there exists a constant C, depending only on a,ß,a,b,p and e, such that for arbitrary a,ß > —1 and II/ - £„/lip,„A. S c • v(f) ■ n-4' ■ I f n * I = ^< oo, where V(f) denotes the total variation of f on [—1,1]. If e = 0, then there exists a constant C, depending only on a, ß,a,b and p such that II/ - £n/||p,a,b,o ^ C • V(f) ■ (nd^D(p,a,a,n) + nd^D(p,b,ß, n)), where d(p, a, a For а ф 1/2 this means D(p,a,a,n In n 1 if d(p, a, a) = — 1, if d(p,a,a) > —1. We remark here that one can easily deduce the corresponding estimates for the one-sided Xp-norms on [—1,1 — s] and [—1 + £,1] with £ > 0. To illustrate the case e = 0 we consider possible choices of the parameters. Corollary 1. Let f € BV be given. For every a,ß > —1 one can choose a,b, namely a > —1/2 if a < 1/2, a > —1/2 if a — 1/2, a ^ >(a — l/2)p/2 - 1/2 if a > 1/2 and b respectively, such that II/ - A./IUM S C ■ V(f n-V». I” if p= 1, if 1 < p < OO. In2 n In n D(p,a,a,n) = < 1 if p = 1, a = 1/2, a = —1/2, if (p = 1, a ^ 1/2, a > —1/2, a < 2a + 3/2) or if (p = 1, a = 1/2, a ф —1/2) or if (p> 1, a = 1/2, a g —1/2), otherwise. Acta Mathematica Hungarica 62, 1993

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