Sharp estimate of the Laplacian of a polyharmonic function and applications

Author:
Ognyan Iv. Kounchev

Journal:
Trans. Amer. Math. Soc. **332** (1992), 121-133

MSC:
Primary 35J30; Secondary 31B05, 35B45, 41A27

DOI:
https://doi.org/10.1090/S0002-9947-1992-1068930-8

MathSciNet review:
1068930

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Abstract: The classical sharp inequality of Markov estimates the values of the derivative of the polynomial of degree in the interval through the uniform norm of the polynomial in the same interval multiplied by . In the present paper we provide an exact estimate for the values of the Laplacian of a polyharmonic function of degree by the uniform norm of the polyharmonic function multiplied by where is the distance from the point to the boundary of the domain. The inequality of Markov (and the similar inequality of Bernstein about trigonometric polynomials) finds many applications in approximation theory for functions of one variable. We prove analogues to some of these results in the multivariate case.

**[1]**O. Kounchev,*Exact estimation of the Laplacian of a polyharmonic function and approximation through polyharmonic functions*, C. R. Acad. Bulgare Sci.**43**(1990), 27-30. MR**1067372 (92j:35042a)****[2]**N. Aronszajn, T. M. Creese, and L. J. Lipkin,*Polyharmonic functions*, Clarendon Press, Oxford, 1983. MR**745128 (86g:31001)****[3]**S. N. Bernstein,*On the best approximation of continuous functions through polynomials of fixed degree*, Collected Works, vol. 1, Publ. Acad. Sci. USSR, 1952, pp. 11-104.**[4]**S. L. Sobolev,*Introduction to the theory of cubature formulas*, "Nauka," Moscow, 1974. MR**0478560 (57:18037)****[5]**L. L. Helms,*Introduction to potential theory*, Wiley, 1963.**[6]**V. K. Dzyadyk,*Introduction to the theory of uniform approximation of functions through polynomials*, "Nauka", Moscow, 1977. (Russian)**[7]**M. Riesz,*Formule d'interpolation pour la derivée d'un polynome trigonométrique*, Comptes Rendus**158**(1914), 1154.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1992-1068930-8

Keywords:
Polyharmonic function,
Markov inequality,
Bernstein inequality,
inverse theorems,
Kolmogorov type inequality

Article copyright:
© Copyright 1992
American Mathematical Society