Bivariate monotone approximation

Author:
George A. Anastassiou

Journal:
Proc. Amer. Math. Soc. **112** (1991), 959-964

MSC:
Primary 41A29; Secondary 41A25

DOI:
https://doi.org/10.1090/S0002-9939-1991-1069682-2

MathSciNet review:
1069682

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a two variable continuously differentiable real-valued 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 two-dimensional polynomials with , so that is approximated simultaneously and uniformly by . This approximation is accomplished quantitatively by the use of a suitable two-dimensional first modulus of continuity.

**[1]**G. A. Anastassiou and O. Shisha,*Monotone approximation with linear differential operators*, J. Approx. Theory**44**(1985), 391-393. 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), 11-17. MR**938778 (89c:41017)****[3]**R. A. DeVore,*Monotone approximation by polynomials*, SIAM J. Math. Anal.**8**(1977), 906-921. MR**0510582 (58:23252)****[4]**R. A. DeVore and X. M. Yu,*Pointwise estimates for monotone polynomial approximation*, Constr. Approx.**1**(1985), 323-331. MR**891762 (88h:41010)****[5]**D. Leviatan,*Pointwise estimates for convex polynomial approximation*, Proc. Amer. Math. Soc.**98**(1986), 471-474. MR**857944 (88i:41010)****[6]**-,*Monotone and comonotone polynomial approximation revisited*, J. Approx. Theory**53**(1988), 1-16. MR**937138 (89h:41017)****[7]**G. G. Lorentz,*Monotone approximation*, Inequalities III (O. Shisha, ed.), Academic Press, New York, 1972, pp. 201-215. MR**0346375 (49:11100)****[8]**G. G. Lorentz and K. Zeller,*Degree of approximation by monotone polynomials*. I, J. Approx. Theory**1**(1968), 501-504. MR**0239342 (39:699)****[9]**D. J. Newman,*Efficient co-monotone approximation*, J. Approx. Theory**25**(1979), 189-192. MR**531408 (80j:41011)****[10]**E. Passow and L. Raymon,*Monotone and comonotone approximation*, Proc. Amer. Math. Soc.**42**(1974), 340-349. MR**0336176 (49:952)****[11]**E. Passow, L. Raymon, and J. A. Roulier,*Comonotone polynomial approximation*, J. Approx. Theory**11**(1974), 221-224. MR**0352807 (50:5293)****[12]**J. A. Roulier,*Monotone approximation of certain classes of functions*, J. Approx. Theory**1**(1968), 319-324. MR**0236580 (38:4875)****[13]**O. Shisha,*Monotone approximation*, Pacific J. Math.**15**(1965), 667-671. MR**0185334 (32:2802)****[14]**D. D. Stancu,*Studii Si Cercetari Stiintifice*,**XI**(2) (1960), 221-233.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
41A29,
41A25

Retrieve articles in all journals with MSC: 41A29, 41A25

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1991-1069682-2

Article copyright:
© Copyright 1991
American Mathematical Society