Cyclotomic units over finite fields

Let ζ be a primitivesp-th root of unity for a primep>2, and consider the group Ω(ζ) of cyclotomic units in the ringR(ζ)=ℒ[ζ+ζ^-1]. This paper deals with the image of Ω(ζ) in the unit group ofR(ζ)/qR(ζ), whereq is a prime ≠p. In particular, it obtains criteria for this image to be essentially everything, and a lower bound on the density of primesp (withq fixed) for which it cannot be. These results have a direct bearing on previous work about units in integral group rings for cyclic groups of orderpq. Work supported in part by an operating grant from NSERC (Canada).     

Wilson's functional equations on groups

We study properties of solutionsf, g, h C(G) of the functional equation

On translation planes of orderq 3 that admit a collineation group of orderq 3, II

Let II be a translation plane of orderq ³ with kernel GF(q) that admits a collineation groupG of orderq ³ in the linear translation complement such thatG fixes a point at infinity and acts transitively on the remaining points at infinity.In this paper, we show that any such translation plane II is one of the following types of planes:     

The lower estimate for linear positive operators,I

A method to prove lower estimates for linear operators is introduced. As a result the best lower estimate for certain convolution operators, for the multivariate Bernstein-Durrmeyer operators in part I and the Bernstein polynomial operators in part II (see [10]), are obtained.Communicated by Hubert Berens     

Large deviations for martingales with some applications

Let(X _i ) be a martingale difference sequence. LetY be a standard normal random variable. We investigate the rate of uniform convergence

Shape-preserving approximation inL p

We prove a direct theorem for shape preservingL _p -approximation, 0p, in terms of the classical modulus of smoothnessw ₂(f, t _p ¹ ). This theorem may be regarded as an extension toL _p of the well-known pointwise estimates of the Timan type and their shape-preserving variants of R. DeVore, D. Leviatan, and X. M. Yu. It leads to a characterization of monotone and convex functions in Lipschitz classes (and more general Besov spaces) in terms of their approximation by algebraic polynomials.Communicated by Ron DeVore.     

Degree of approximation of analytic functions by “near-best” polynomial approximants

We study the rate of approximation of a functionfA(K) in the interior of the compact setK by polynomials, that are close to best polynomial approximants on the whole setK. Lower and upper estimates of possible improvement of convergence (depending on the geometry ofK) are obtained.Communicated by Vilmos Totik.     

Necessary and sufficient conditions for mean convergence of orthogonal expansions for Freud weights

A classic 1970 paper of B. Muckenhoupt established necessary and sufficient conditions for weightedL _p convergence of Hermite series, that is, orthogonal expansions corresponding to the Hermite weight. We generalize these to orthogonal expansions for a class of Freud weightsW ²:=e ^–2Q , by first proving a bound for the difference of the orthonormal polynomials of degreen+1 andn–1 of the weightW ². Our identical necessary and sufficient conditions close a slight gap in Muckenhoupt's conditions for the Hermite weight at least forp>1. Moreover, our necessary conditions apply whenQ(x)=|x|, >1 while our sufficient conditions apply at least for =2,4,6,....Communicated by Vilmos Totik.     

A counterexample to subexponential growth of orthogonal polynomials

We construct an example of polynomialsp _n orthonormal with respect to a measure such that the sequencep _n (x) has an exponential lower bound for a pointx in the support of .Communicated by Paul Nevai.     

The Dold-Lashof-Fuchs construction revisited

Sia dato uno spazio topologicoE con azione di un monoide topologicoH e siaE→B una funzione continue che, su ogni apertoU di una partizione dell'unità diB, sia, a meno di omotopia, la proiezioneU×H→U (ovvero una fibrazione numerabile). Un classico risultato di A. Dold e R. Lashof (1959) permette di costruire, a partire daE→B, una funzione continuaE _∞→B_∞, conE _∞ debolmente contraibile e munito di azione diH: laH-fibrazione universale associata daH. Tale funzione, in generale, non è purtroppo numerabile e quindi non si presta alla classificazione delleH-fibrazioni numerabili. Successivamente (1971), M. Fuchs ha trovato un modo di modificare la costruzione di Dold-Lashof per recuperare la numerabilità. La costruzione di Dold-Lashof-Fuchs è, da allora, uno dei principali strumenti nella teoria degli spazi classificanti di monoidi topologici, anche se vi è un uso di topologie alquanto complesse e quindi poco maneggevoli. In questo lavoro ci proponiamo di mostrare come, lavorando nella categoria conveniente deik-spazi, sia possibile descrivere la costruzione di Dold-Lashof-Fuchs in modo estremamente semplificato ed adattarla anche alla classificazione delleF-fibrazioni numerabili.

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Conferenza tenuta da R. Piccinini il 15 maggio 1995