Integrals involving products of bessel functions

Summary The integrals and , where n is any positive integer, are evaluated in terms ofMacRobert E-functions and generalized hypergeometric functions.     

Entropies with the branching property

Summary It is proved that every entropy with the property of branching has the form , where J is a compositive information measure under a regular binary operation. In particular all the classical entropies with the branching property such as that of Shannon have the form .

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Sunto Si dimostra che ogni entropia che goda della proprietà di diramazione ha necessariamente la forma dove J è una misura di informazione compositiva con una operazione binaria regolare. In particolare tutte le entropie classiche che godono della proprietà di diramazione quale quella di Shannon hanno la forma .

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Entrata in Redazione il 9 aprile 1973.     

Differential equations of infinite order for orthogonal polynomials

Summary Differential equations of infinite order of the form , where M_k is a polynomial of degree ≦k, have polynomial solutions for suitable values of λ. Necessary and sufficient conditions are given in order that these solutions form a set of orthogonal polynomials. The cases where { degree M_k }=1 and 2 are studied in detail. Supported by N.S.F. Grant GP-5311.     

Inequalities of Paley type for noncommutative martingales

Summary The purpose of this paper is to study the validity of the Paley inequality on square function, for noncommutative martingales. Let be a regular gage space, and a sequence of von-Neumann algebras such that we prove that for every , where ɛ_n(F) is the conditional expectation of F with respect to the subalgebra : We also consider the case of a martingale arising in the context of harmonic analysis on noncommutative discrete groups, in analogy to the theorem of R.E.A.C. Paley on Fourier-Walsh series. Entrata in Redazione il 26 gennaio 1977. Partially sponsored by C.N.R.     

An asymptotic method for calculating limits of maximal means

For a continuous almost periodic function , we show that the function

On the space l P(β)

Let be a sequence of positive numbers and 1 ≤p < ∞. We consider the spacel ^P(β) of all power series such that . We give a necessary and sufficient condition for a polynomial to be cyclic inl ^P(β) and a point to be bounded point evaluation onl ^P(β).     

On a certain basis inc 0

A basis is constructed inc ₀ such that there exists no bounded linear projection ofc ₀ onto the subspace spanned by a certain subsequence of . This is part of the author’s Ph.D. thesis prepared at the Hebrew University of Jerusalem under the suppervision of Professor A. Dvoretzky and Dr. J. Lindenstrauss. The author wishes to thank Dr. Lindenstrauss for his helpful advice.     

Tensor sequences and inductive limits with local partition of unity

In this note we characterize those locally convex hulls for which the -tensor product is topologically isomorphic to the locally convex hull for all normed spaces E. By using -sequences in the sense of W.Kaballo and D.Vogt [18] it is shown that is commutable with -tensor products if and only if F admits a local partition of unity. In this paper we give examples of locally convex hulls with local partition of unity. From this one obtains conditions under which the -tensor product of two bornological (DF)-spaces is again a bornological (DF)-space. Applications to sequence spaces are given.     

A criterion for the existence of bounded nonoscillatory solutions for nonlinear retarded differential equatory

Summary For the equation (⋆) below we give a condition which guarantees the existence of a bounded nonoscillatory solution x with Entrata in Redazione il 19 luglio 1977.     

ON A CLASS OF BESICOVITCHFUNCTIONS TO HAVE EXACT BOX DIMENSION： A NECESSARY AND SUFFICIENT CONDITION

This paper summarized recent achievements obtained by the authors about the box dimensions of the Besicovitch functions given byB(t) := ∞∑k=1 λs-2k sin(λkt),where 1 ＜ s ＜ 2, λk ＞ 0 tends to infinity as k →∞ and λk satisfies λk 1/λk ≥λ＞ 1. The results show thatlimk→∞ log λk 1/log λk = 1is a necessary and sufficient condition for Graph(B(t)) to have same upper and lower box dimensions.For the fractional Riemann-Liouville differential operator Du and the fractional integral operator D-v,the results show that if λ is sufficiently large, then a necessary and sufficient condition for box dimension of Graph(D-v(B)),0 ＜ v ＜ s - 1, to be s - v and box dimension of Graph(Du(B)),0 ＜ u ＜ 2 - s, to be s uis also lim k→∞logλk 1/log λk = 1.     

Characterization of certain classes of orthogonal polynomials related to elliptic functions

Summary In this paper we determine all orthogonal polynomials U_n(x) such that U_n(x)=x^−1/2 F _2n+1 (x ^1/2 ) and where f(t), u(t) have Taylor series expansions. Supported in part by N. S. F. grant GP-1593.     

Asymptotic normality of a quadratic measure of orthogonal series type density estimate

LetX ₁,...,X _n be i.i.d. random variable with a common densityf. Let be an estimate off(x) based on a complete orthonormal basis {φ _k :k≧0} ofL ₂[a, b]. A Martingale central limit theorem is used to show that , where and .     

Strong consistency of nearest neighbor kernel regression estimation for stationary dependent samples

Under quite mild conditions onk _n . the strong consistency is proved for the nearest neighbor density, the nearest neighbor kernel regression and the modified nearest neighbor kernel regression of an a-mixing stationary sequence in time series context. The condition imposed on the mixing coefficients is , . which is simple and weak.     

Asymptotic properties of the norm of the extremum of a sequence of normal random functions

Under additional conditions on a bounded normally distributed random function X = X( t), t ∈ T, we establish a relation of the form

Structure of bounded solutions for a class of abstract differential equations

Summary The paper is concerned with bounded solutions of an equationu′(t)=Bu(t) in Hilbert spaces, . A representation formula is obtained depending on the zeros of Rez(θ).

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Riassunto è studiata la struttura delle soluzioni limitate di una equazione differenzialeu′(t)=Bu(t) in uno spazio di Hilbert, ove , in funzione dei zeri di Rez(θ).

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This research is supported through a grant of the National Research Council Canada.     

Bounded solutions for 119-01119-01119-01-problem in pseudo-Siegel domains

Summary We study the problem of the existence of bounded solutions for the equation on pseudo-Siegel domains when the data satisfies the condition .

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Sunto Si studia il problema dell'esistenza di soluzioni limitate per l'equazione sui domini pseudo-Siegel quando il dato soddisfa alla condizione .

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Work done within the Project 40% M.U.R.S.T. «Geometria Reale e Complessa».     

The true order of magnitude of Lamé rounding error sums

Let (z ∈ ℝ). Further let λ denote a large real parameter. We show that for arbitrary real numbersk and α withk>=2.7013 and 0<α≦1,

Trigonometric series of classesL^p (\mathbb{T}), p \in ]1;\infty [ and their conservative meansand their conservative means

Suppose that a lower triangular matrix μ:[μ _m ⁽ⁿ⁾ ] defines a conservative summation method for series, i.e.,

Regolarità hölderiana delle derivate dei minimi di funzionali con parte principale quadratica

Riassunto In questo lavoro si prova la regolarità h?lderiana delle derivate, fino all'ordinek, dei minimi locali dei funzionali sotto opportune ipotesi suA _ij ^αβ e sug.

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Summary In this paper we prove h?lder-continuity of the derivates, up to orderk, of local minima of functionals under suitable hypotheses forA _ij ^αβ andg.