Bivariate monotone approximation
Author:
George A. Anastassiou
Journal:
Proc. Amer. Math. Soc. 112 (1991), 959964
MSC:
Primary 41A29; Secondary 41A25
MathSciNet review:
1069682
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Abstract 
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Abstract: Let be a two variable continuously differentiable realvalued function of certain order on and let be a linear differential operator involving mixed partial derivatives and suppose that . Then there exists a sequence of twodimensional polynomials with , so that is approximated simultaneously and uniformly by . This approximation is accomplished quantitatively by the use of a suitable twodimensional first modulus of continuity.
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 [1]
 G. A. Anastassiou and O. Shisha, Monotone approximation with linear differential operators, J. Approx. Theory 44 (1985), 391393. MR 804853 (86m:41026)
 [2]
 I. Badea and C. Badea, On the order of simultaneous approximation of bivariate functions by Bernstein operators, Anal. Numér. Théor. Approx. 16 (1987), 1117. MR 938778 (89c:41017)
 [3]
 R. A. DeVore, Monotone approximation by polynomials, SIAM J. Math. Anal. 8 (1977), 906921. MR 0510582 (58:23252)
 [4]
 R. A. DeVore and X. M. Yu, Pointwise estimates for monotone polynomial approximation, Constr. Approx. 1 (1985), 323331. MR 891762 (88h:41010)
 [5]
 D. Leviatan, Pointwise estimates for convex polynomial approximation, Proc. Amer. Math. Soc. 98 (1986), 471474. MR 857944 (88i:41010)
 [6]
 , Monotone and comonotone polynomial approximation revisited, J. Approx. Theory 53 (1988), 116. MR 937138 (89h:41017)
 [7]
 G. G. Lorentz, Monotone approximation, Inequalities III (O. Shisha, ed.), Academic Press, New York, 1972, pp. 201215. MR 0346375 (49:11100)
 [8]
 G. G. Lorentz and K. Zeller, Degree of approximation by monotone polynomials. I, J. Approx. Theory 1 (1968), 501504. MR 0239342 (39:699)
 [9]
 D. J. Newman, Efficient comonotone approximation, J. Approx. Theory 25 (1979), 189192. MR 531408 (80j:41011)
 [10]
 E. Passow and L. Raymon, Monotone and comonotone approximation, Proc. Amer. Math. Soc. 42 (1974), 340349. MR 0336176 (49:952)
 [11]
 E. Passow, L. Raymon, and J. A. Roulier, Comonotone polynomial approximation, J. Approx. Theory 11 (1974), 221224. MR 0352807 (50:5293)
 [12]
 J. A. Roulier, Monotone approximation of certain classes of functions, J. Approx. Theory 1 (1968), 319324. MR 0236580 (38:4875)
 [13]
 O. Shisha, Monotone approximation, Pacific J. Math. 15 (1965), 667671. MR 0185334 (32:2802)
 [14]
 D. D. Stancu, Studii Si Cercetari Stiintifice, XI (2) (1960), 221233.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199110696822
PII:
S 00029939(1991)10696822
Article copyright:
© Copyright 1991
American Mathematical Society
