Uniform and Pointwise Shape Preserving Approximation by Algebraic Polynomials

We survey developments, over the last thirty years, in the theory of Shape Preserving Approximation (SPA) by algebraic polynomials on a

[1]  R. Lorentz.  Uniqueness of best approximation by monotone polynomials , 1971 .

[2]  Jackson's theorem in LP, 0, 1979 .

[3]  Igor A. Shevchuk.  One Example in Monotone Approximation , 1996 .

[4]  INEQUALITIES FOR POLYNOMIALS AND THEIR DERIVATIVES II , 1968 .

[5]  Negative Results in Form-Preserving Approximation of Higher Order , 2004 .

[6]  Pointwise estimates for higher order convexity preserving polynomial approximation , 1994, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[7]  Dany Leviatan,et al.  Convex Polynomial Approximation in the Uniform Norm: Conclusion , 2005, Canadian Journal of Mathematics.

[8]  Pointwise estimates for monotone polynomial approximation , 1985 .

[9]  L. Raymon,et al.  Comonotone polynomial approximation , 1974 .

[10]  DEDICATED TO THE MEMORY OF THEODORE S. MOTZKIN , 1972 .

[11]  Kirill A. Kopotun.  Uniform Estimates of Monotone and Convex Approximation of Smooth Functions , 1995 .

[12]  G. Lorentz,et al.  Monotone approximation by algebraic polynomials , 1970 .

[13]  Dany Leviatan,et al.  Shape preserving widths of Sobolev-type classes ofs-monotone functions on a finite interval , 2003 .

[14]  Dany Leviatan,et al.  Some Positive Results and Counterexamples in Comonotone Approximation , 1997 .

[15]  Negative theorems on monotone approximation , 1976 .

[16]  Pointwise Estimates for the Coconvex Approximation of Differentiable Functions , 2005 .

[17]  L. Raymon,et al.  Monotone and comonotone approximation , 1974 .

[18]  ON COMONOTONE APPROXIMATION BY POLYNOMIALS IN Lp SPACE , 1993 .

[19]  Uniform estimates of the coconvex approximation of functions by polynomials , 1992 .

[20]  Nikolskii-type estimates for coconvex approximation of functions with one inflection poin , 2010 .

[21]  D. Leviatan,et al.  The degree of coconvex polynomial approximation , 1999 .

[22]  On Restricted Best Approximation to Functions with Restricted Derivatives , 1976 .

[23]  R. D. Vore.  Monotone Approximation by Polynomials , 1977 .

[24]  G. L. Iliev.  Exact estimates for partially monotone approximation , 1978 .

[25]  Donald J. Newman,et al.  PIECEWISE MONOTONE POLYNOMIAL APPROXIMATION , 1972 .

[26]  L. Raymon,et al.  Exact comonotone approximation , 1978 .

[27]  A. S. Shvedov.  Orders of coapproximation of functions by algebraic polynomials , 1981 .

[28]  Andriy Bondarenko.  Jackson type inequality in 3-convex approximation , 2002 .

[29]  J. Roulier.  Monotone approximation of certain classes of functions , 1968 .

[30]  M. G. Pleshakov,et al.  Piecewise Coapproximation and the Whitney Inequality , 2000 .

[31]  K. R. Gehner.  Characterization theorems for constrained approximation problems via optimization theory , 1975 .

[32]  J. G. Thompson,et al.  A Generalization of a Theorem of , 1973 .

[33]  Uniform estimates for monotonic polynomial approximation , 1993, Ukrainian Mathematical Journal.

[34]  Donald J. Newman,et al.  Efficient co-monotone approximation , 1979 .

[35]  On Comonotone Approximation , 1983, Canadian Mathematical Bulletin.

[36]  G. Lorentz,et al.  Degree of approximation by monotone polynomials I , 1968 .

[37]  On a counterexample in monotone approximation , 1992 .

[38]  Negative theorems on generalized convex approximation , 1976 .

[39]  Igor A. Shevchuk,et al.  Theory of Uniform Approximation of Functions by Polynomials , 2008 .

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

[41]  N. Meyers,et al.  H = W. , 1964, Proceedings of the National Academy of Sciences of the United States of America.

[42]  Ding-Xuan Zhou,et al.  Global smoothness preservation and the variation-diminishing property , 1999 .

[43]  T. Erdélyi.  Extremal Properties of Polynomials , 2006 .

[44]  Coconvex approximation in the uniform norm: the final frontier , 2006 .

[45]  Igor A. Shevchuk,et al.  Piecewise Monotone Pointwise Approximation , 1998 .

[46]  APPROXIMATION OF MONOTONE FUNCTIONS BY MONOTONE POLYNOMIALS , 1993 .

[47]  R. Beatson.  The degree of monotone approximation. , 1978 .

[48]  Pointwise estimates for convex polynomial approximation , 1986 .

[49]  Convex polynomial and spline approximation in C[−1, 1] , 1994 .

[50]  K. Kopotun.  Simultaneous approximation by algebraic polynomials , 1996 .

[51]  D. Leviatan,et al.  Are the degrees of best (co)convex and unconstrained polynomial approximation the same? , 2009 .