On the completeness of algebraic polynomials in the spaces L p (ℝ, dμ) (original) (raw)

Almost-everywhere convergence and polynomials

Journal of Modern Dynamics, 2008

Denote by Γ the set of pointwise good sequences. Those are sequences of real numbers (a k ) such that for any measure preserving flow (Ut) t∈R on a probability space and for any f ∈ L ∞ , the averages 1 n P n k=1 f (Ua k x) converge almost everywhere. We prove the following two results.

The space of polynomials with roots of bounded multiplicity

2012

We describe an alternative approach to some results of Vassiliev ([Va1]) on spaces of polynomials, by using the "scanning method" which was used by Segal ([Se2]) in his investigation of spaces of rational functions. We explain how these two approaches are related by the Smale-Hirsch Principle or the h-Principle of Gromov. We obtain several generalizations, which may be of interest in their own right.

On finiteness theorems of polynomial functions

European Journal of Mathematics

Let d be a positive integer. We show a finiteness theorem for semialgebraic \mathscr {RL}RLtrivialityofaNashfamilyofNashfunctionsdefinedonaNashmanifold,generalisingBenedetti–Shiota’sfinitenesstheoremforsemialgebraicRL triviality of a Nash family of Nash functions defined on a Nash manifold, generalising Benedetti–Shiota’s finiteness theorem for semialgebraicRLtrivialityofaNashfamilyofNashfunctionsdefinedonaNashmanifold,generalisingBenedetti–Shiota’sfinitenesstheoremforsemialgebraic\mathscr {RL}RLequivalenceclassesappearinginthespaceofrealpolynomialfunctionsofdegreenotexceedingd.WealsoproveFukuda’sclaim,Theorem1.3,anditssemialgebraicversionTheorem1.4,onthefinitenessofthelocalRL equivalence classes appearing in the space of real polynomial functions of degree not exceeding d. We also prove Fukuda’s claim, Theorem 1.3, and its semialgebraic version Theorem 1.4, on the finiteness of the localRLequivalenceclassesappearinginthespaceofrealpolynomialfunctionsofdegreenotexceedingd.WealsoproveFukuda’sclaim,Theorem1.3,anditssemialgebraicversionTheorem1.4,onthefinitenessofthelocal{\mathscr {R}}$$ R types appearing in the space of real polynomial functions of degree not exceeding d.