Canonical sets of best L1-approximation

In mathematics, the term approximation usually means either interpolation on a point set or approximation with respect to a given distance. There is a concept, which joins the two approaches together, and this is the concept of characterization of the best approximants via interpolation. It turns out that for some large classes of functions the best approximants with respect to a certain distance can be constructed by interpolation on a point set that does not depend on the choice of the function to be approximated. Such point sets are called canonical sets of best approximation. The present paper summarizes results on canonical sets of best L1-approximation with emphasis on multivariate interpolation and best L1-approximation by blending functions. The best L1-approximants are characterized as transfinite interpolants on canonical sets. The notion of a Haar-Chebyshev system in the multivariate case is discussed also. In this context, it is shown that some multivariate interpolation spaces share properties of univariate Haar-Chebyshev systems. We study also the problem of best one-sided multivariate L 1-approximation by sums of univariate functions. Explicit constructions of best one-sided L1-approximants give rise to well-known and new inequalities

On a polynomial inequality of P. Erdős and T. Grünwald

<p/> <p>Let <inline-formula><graphic file="1029-242X-1999-379264-i1.gif"/></inline-formula> be a polynomial with only real zeros having <inline-formula><graphic file="1029-242X-1999-379264-i2.gif"/></inline-formula>, <inline-formula><graphic file="1029-242X-1999-379264-i3.gif"/></inline-formula> as consecutive zeros. It was proved by P. Erd&#337;s and T. Gr&#252;nwald that if <inline-formula><graphic file="1029-242X-1999-379264-i4.gif"/></inline-formula> on <inline-formula><graphic file="1029-242X-1999-379264-i5.gif"/></inline-formula>, then the ratio of the area under the curve to the area of the tangential rectangle does not exceed <inline-formula><graphic file="1029-242X-1999-379264-i6.gif"/></inline-formula>. The main result of our paper is a multidimensional version of this result. First, we replace the class of polynomials considered by Erd&#337;s and Gr&#252;nwald by the <it>wider</it> class <inline-formula><graphic file="1029-242X-1999-379264-i7.gif"/></inline-formula> consisting of functions of the form <inline-formula><graphic file="1029-242X-1999-379264-i8.gif"/></inline-formula>, where <inline-formula><graphic file="1029-242X-1999-379264-i9.gif"/></inline-formula> is logarithmically concave on <inline-formula><graphic file="1029-242X-1999-379264-i10.gif"/></inline-formula>, and show that their result holds for all functions in <inline-formula><graphic file="1029-242X-1999-379264-i11.gif"/></inline-formula>. More generally, we show that if <inline-formula><graphic file="1029-242X-1999-379264-i12.gif"/></inline-formula> and <inline-formula><graphic file="1029-242X-1999-379264-i13.gif"/></inline-formula>, then for all <inline-formula><graphic file="1029-242X-1999-379264-i14.gif"/></inline-formula>, the integral <inline-formula><graphic file="1029-242X-1999-379264-i15.gif"/></inline-formula> does not exceed <inline-formula><graphic file="1029-242X-1999-379264-i16.gif"/></inline-formula>. It is this result that is extended to higher dimensions. Our consideration of the class <inline-formula><graphic file="1029-242X-1999-379264-i17.gif"/></inline-formula> is crucial, since, unlike the narrower one of Erd&#337;s and Gr&#252;nwald, its definition does not involve the distribution of zeros of its elements; besides, the notion of logarithmic concavity makes perfect sense for functions of several variables.</p