Advanced Problems in Constructive Approximation pp 129-141 | Cite as

# On the Degree of Approximation in Multivariate Weighted Approximation

## Abstract

Let *s* ≥ 1 be an integer, *f* ∈ L^{ P }(**R** ^{ s }) for some *p*, 1 ≤ *p* ∞ or be a continuous function on **R** ^{ S } vanishing at infinity. We consider the degree of approxima-tion of *f* by expressions of the form exp \( ( - {\text{ }}\sum\limits_{k = 1}^s {{Q_k}\left( {{x_k}} \right)} )P\left( {{x_1},...,{x_s}} \right) \) where each exp(—*Q* _{ k }(·)) is a Freud type weight function, and *P* is a polynomial of specific degrees in each coordinate. Direct and converse theorems are stated. In particular, it is shown that if each *Q* _{ k } = l*x*l^{ α } for some even, positive integer a, and the degree of approximation has a power decay, then the same property holds when the weight function is replaced by exp \( ( - \sum\limits_{k = 1}^s {{a_k}{Q_k}\left( {{x_k}} \right)} ) \) for any positive constants *a* _{ k }.

## Keywords

Weight Function Multivariate Setting Bold Face Letter Converse Theorem Weighted Polynomial## Preview

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## References

- [1]S.B. DAMELIN,
*Converse and smoothness theorems for Erdós weights in LP*(0 <*p ≤∞)*,*J*. Approx. Theory, 93 (1998), 349–398.MathSciNetzbMATHCrossRefGoogle Scholar - [2]Z. DITZIAN AND D.S. LUBINSKY,
*Jackson and smoothness theorems for Freud weights in LP*(0 <*p ≤ ∞)*, Constr. Approx., 11 (1997), 99–152.MathSciNetGoogle Scholar - [3]M.M. DZRBASYAN AND A.B. TAVADYAN,
*On weighted uniform approximation by polynomials of functions of several variables*, Mat. Sb., N.S., 43 (85) (1957), 227–256.MathSciNetGoogle Scholar - [4]G. FREUD,
*Direct and converse theorems in the theory of weighted polynomial approximation*, Math. Zeitschrift, 126 (1972), 123–134.MathSciNetzbMATHCrossRefGoogle Scholar - [5]A. KROO AND J. SZABADOS,
*Weighted polynomial approximation on the real line*, J. Approx. Theory, 83 (1995), 41–64.MathSciNetzbMATHCrossRefGoogle Scholar - [6]D.S. LUBINSKY,
*Jackson and Bernstein theorems for exponential weights*, BSMS: Approximation theory and function series, 5 (1996), 85–115.MathSciNetGoogle Scholar - [7]H.N. MHASKAR, “Introduction to the theory of weighted polynomial approximation”, World Scientific, Singapore, 1996.Google Scholar
- [8]E.B. SAFF AND V. TOTIK, “Logarithmic potentials with external fields”, Springer Verlag, New York, 1997.Google Scholar