✦ LIBER ✦

On Boundedness of Lagrange Interpolation inLp,p<1

✍ Scribed by D.S. Lubinsky

Publisher: Elsevier Science
Year: 1999
Tongue: English
Weight: 96 KB
Volume: 96
Category: Article
ISSN: 0021-9045
DOI: 10.1006/jath.1998.3249

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✦ Synopsis

We estimate the distribution function of a Lagrange interpolation polynomial and deduce mean boundedness in L p , p<1.

1999 Academic Press

1. THE RESULT

There is a vast literature on mean convergence of Lagrange interpolation, see [4 8] for recent references. In this note, we use distribution functions to investigate mean convergence. We believe the simplicity of the approach merits attention.

Recall that if g: R Ä R, and m denotes Lebesgue measure, then the distribution function m g of g is

One of the uses of m g is in the identity [1, p. 43]

( 2 )

Moreover, the weak L 1 norm of g may be defined by

) If & g& L p (R) < , then for p< , it is easily seen that m g (*) * & p & g& p L p (R) , *>0, (4)

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