On Variation{diminishing Schoenberg Operators: New Quantitative Statements ⁄

We give quantitative results for variation{diminishing splines, focusing on the case of equidistant knots. New direct inequalities are obtained, both in terms of the classical second modulus of continuity and in terms of the second Ditzian{Totik modulus. These new results are based upon a detailed analysis of the second moments and very recent theorems for positive linear operator approximation. The potential for simultaneous approximation is described by means of an estimate involving both the flrst and the second classical modulus of continuity. The topic of global smoothness preservation is also addressed. Furthermore, we discuss the degree of simultaneous approximation in the multivariate case, namely for Boolean sums and tensor products of Schoenberg splines.

[1]  I. J. Schoenberg,et al.  On Pólya frequency functions IV: The fundamental spline functions and their limits , 1966 .

[2]  Sätze vom Jackson-Bernstein-Typ für die Approximation mit Spline-Funktionen , 1969 .

[3]  M. Marsden.  An identity for spline functions with applications to variation-diminishing spline approximation☆ , 1970 .

[4]  M. Marsden.  On uniform spline approximation , 1972 .

[5]  R. DeVore.  The Approximation of Continuous Functions by Positive Linear Operators , 1972 .

[6]  O. Shisha.  A characterization of functions having Zygmund's property , 1973 .

[7]  On the representation of the remainder in the variation-diminishing spline approximation , 1973 .

[8]  Richard Franklin Riesenfeld,et al.  Applications of b-spline approximation to geometric problems of computer-aided design. , 1973 .

[9]  D. Newman.  The Zygmund condition for polygonal approximation , 1974 .

[10]  G. Coman.  On the Approximation of Multivariate Functions. , 1974 .

[11]  L. Schumaker,et al.  Some multidimensional spline approximation methods , 1974 .

[12]  On pointwise convergence estimates for positive linear operators on C[a, b] , 1976 .

[13]  H. Knoop,et al.  Ein Satz vom Korovkin-Typ fürCk-Räume , 1976 .

[14]  Variations on a theme by Schoenberg , 1976 .

[15]  Manfred W. Müller.  Degree of Lp-approximation by integral Schoenberg splines , 1977 .

[16]  Carl de Boor,et al.  A Practical Guide to Splines , 1978, Applied Mathematical Sciences.

[17]  G. Coman,et al.  On some bivariate spline operators , 1979 .

[18]  S. L. Lee,et al.  Spline approximation operators of Bernstein-Schoenberg type in one and two variables , 1981 .

[19]  L. Schumaker.  Spline Functions: Basic Theory , 1981 .

[20]  R. Beatson.  Monotone and Convex Approximation by Splines: Error Estimates and a Curve Fitting Algorithm , 1982 .

[21]  R. Riesenfeld,et al.  A geometric proof for the variation diminishing property of B-spline approximation , 1983 .

[22]  L. Rubel,et al.  Constructive Function Theory , 1984 .

[23]  Quantitative Korovkin type theorems on simultaneous approximation , 1984 .

[24]  T. Goodman,et al.  A PROPERTY OF BERNSTEIN-SCHOENBERG SPLINE OPERATORS , 1985 .

[25]  R. DeVore,et al.  A geometric proof of total positivity for spline interpolation , 1985 .

[26]  A Voronovskaya Theorem for Variation-Diminishing Spline Approximation , 1986, Canadian Journal of Mathematics.

[27]  V. Totik,et al.  Moduli of smoothness , 1987 .

[28]  P. Sablonnière.  Positive spline operators and orthogonal splines , 1988 .

[29]  I. J. Schoenberg,et al.  On Variation Diminishing Spline Approximation Methods , 1988 .

[30]  T. Goodman.  Shape preserving representations , 1989 .

[31]  M. Tiwari,et al.  On the degree of simultaneous approximation by modified Bernstein polynomials , 1989 .

[32]  Gerald Farin,et al.  Curves and surfaces for computer aided geometric design , 1990 .

[33]  H. Gonska.  Degree of simultaneous approximation of bivariate functions by Gordon operators , 1990 .

[34]  GLOBAL SMOOTHNESS OF APPROXIMATING FUNCTIONS , 1991 .

[35]  Z. Ditzian,et al.  Strong converse inequalities , 1993 .

[36]  B. Sahakian,et al.  Spline Functions and Multivariate Interpolations , 1993 .

[37]  V. Totik.  APPROXIMATION BY BERNSTEIN POLYNOMIALS , 1994 .

[38]  Xinlong Zhou,et al.  The Lower Estimate for Linear Positive Operators (II) , 1994 .

[39]  T. Goodman.  Asymptotic formulas for multivariate Bernstein-Schoenberg operators , 1995 .

[40]  Josef Hoschek,et al.  Fundamentals of computer aided geometric design , 1996 .

[41]  Exact bounds for simultaneous approximation of functions of two variables and their derivatives by bilinear splines , 1996 .

[42]  R. Păltănea.  Optimal Estimates with Moduli of Continuity , 1997 .

[43]  Gerald E. Farin,et al.  Curves and surfaces for computer-aided geometric design - a practical guide, 4th Edition , 1997, Computer science and scientific computing.

[44]  H. Wenz.  On the Limits of (Linear Combinations of) Iterates of Linear Operators , 1997 .

[45]  M. Felten.  Local and Global Approximation Theorems for Positive Linear Operators , 1998 .

[46]  Direct and Inverse Estimates for Bernstein Polynomials , 1998 .

[47]  George A. Anastassiou,et al.  Approximation theory - moduli of continuity and global smoothness preservation , 1999 .

[48]  G. Tachev.  Piecewise Linear Interpolation With Nonequidistant Nodes , 2000 .

[49]  Gershon Elber,et al.  Geometric modeling with splines - an introduction , 2001 .

[50]  H. Gonska,et al.  The Second Ditzian-Totik modulus revisited: refined estimates for positive linear operators , 2003, Journal of Numerical Analysis and Approximation Theory.

[51]  A. Pinkus,et al.  Approximation of Multivariate Functions , 2004 .

[52]  Laura Beutel,et al.  Quantitative Inheritance Properties for Simultaneous Approximation by Tensor Product Operators , 2004, Numerical Algorithms.