On simultaneous Chebyshev approximation in the ``sum'' norm

Author:
William H. Ling

Journal:
Proc. Amer. Math. Soc. **48** (1975), 185-188

MSC:
Primary 41A30; Secondary 41A50

DOI:
https://doi.org/10.1090/S0002-9939-1975-0361555-4

MathSciNet review:
0361555

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Abstract: Let be real valued functions on and let be a nonempty family of real valued functions on . It is shown that the simultaneous approximation of and in the ``sum'' norm by elements of is, with one restriction, equivalent to the approximation of the arithmetic mean, . A complete characterization of best approximations in the ``sum'' norm is given including results for varisolvent families.

**[1]**J. B. Diaz and H. W. McLaughlin,*Simultaneous approximation of a set of bounded real functions*, Math. Comp.**23**(1969), 583-593. MR**40**#1733. MR**0248481 (40:1733)****[2]**-,*Simultaneous Chebyshev approximation of a set of bounded complex-valued functions*, J. Approximation Theory**2**(1969), 419-432. MR**41**#2026. MR**0257375 (41:2026)****[3]**-,*On simultaneous Chebyshev approximation and Chebyshev approximation with an additive weight*, J. Approximation Theory**6**(1972), 68-71. MR**0338631 (49:3395)****[4]**C. B. Dunham,*Simultaneous Chebyshev approximation of functions on an interval*, Proc. Amer. Math. Soc.**18**(1967), 472-477. MR**35**#3334. MR**0212463 (35:3334)****[5]**D. S. Goel, A. S. B. Holland, C. Nasim and B. N. Sahney,*On best simultaneous approximation in normed linear spaces*, University of Calgary, Research Paper No. 186, March 1973. MR**0377384 (51:13556)****[6]**D. S. Goel, A. S. B. Holland, C. Nasim and B. N. Sahney,*Characterization of an element of best -simultaneous approximation*, University of Calgary, Research Paper No. 188, May 1973.**[7]**J. R. Rice,*The approximation of functions*. Vol. II:*Nonlinear and multivariate theory*, Addison-Wesley, Reading, Mass., 1969. MR**39**#5989. MR**0244675 (39:5989)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1975-0361555-4

Keywords:
Simultaneous approximation,
"sum'' norm,
best approximation,
arithmetic mean,
varisolvent family

Article copyright:
© Copyright 1975
American Mathematical Society