Asymptotic Interpolating Sequences in Uniform Algebras

T. Hosokawa, K. Izuchi and D. Zheng recently introduced theconcept of asymptotic interpolating sequences (of type 1) inthe unit disk for H(D). It is shown that these sequences coincidewith sequences that are interpolating for the algebra QA. Alsoa characterization is given of the interpolating sequences oftype 1 for H(D), and asymptotic interpolating sequences in thespectrum of H(D) are studied. The existence of asymptotic interpolatingsequences of type 1 for H() on arbitrary domains is verified.It is shown that any asymptotic interpolating sequence in auniform algebra eventually is interpolating.     

Frames and the Feichtinger conjecture

We show that the conjectured generalization of the Bourgain-Tzafriri restricted-invertibility theorem is equivalent to the conjecture of Feichtinger, stating that every bounded frame can be written as a finite union of Riesz basic sequences. We prove that any bounded frame can at least be written as a finite union of linearly independent sequences. We further show that the two conjectures are implied by the paving conjecture. Finally, we show that Weyl-Heisenberg frames over rational lattices are finite unions of Riesz basic sequences.

     

A Method for Obtaining Sequence Transformations

A formulation of a method for obtaining a sequence transformationfrom a certain set of sequences is given. This method, whichis called TSE, was first introduced by Kowalewski. The domainof acceleration and the kernel of a sequence transformationobtained by TSE of a certain type are given. Well-known sequencetransformations such as the Aitken ² process and Lubkin's Wtransformation are obtained by TSE. Applying TSE to a certainset of sequences, the kernel and the domain of accelerationof the W transform are given.     

A p Interpolation in the Unit Ball

We study interpolating sequences in the unit ball for A^–pwith p > 0, the Banach space of holomorphic functions f with(1 – |z|²)^p |f(z)| bounded. The finite unions of A^–p-interpolatingsequences are characterized by a Carleson type condition.     

The Markov Oscillation Problem in Discrete Time

Consider a discrete-time regenerative phenomenon with associatedrenewal sequence u_n. General results for the supremum of u_m+1,..., u_m+n are developed for those renewal sequences {u_n} forwhich the first m + 1 elements match those of a fixed renewalsequence {v_n}, that is, u₀ = v₀, ..., u_m = v_m. A series of associatedlemmas are developed in the process.     

Equivariant Universal Coefficient and Kunneth Spectral Sequences

This paper constructs hyper-homology spectral sequences of Z-gradedand RO_G-graded Mackey functors that compute Ext and Tor overG-equivariant S-algebras A ring spectra) for finite groups G.These specialize to universal coefficient and Künneth spectralsequences. 2000 Mathematics Subject Classification 55N91 (primary),55P43, 55U20, 55U25 (secondary).     

Derived Functors of Inverse Limits Revisited

We prove, correct and extend several results of an earlier paperof ours (using and recalling several of our later papers) aboutthe derived functors of projective limit in abelian categories.In particular we prove that if C is an abelian category satisfyingthe Grothendieck axioms AB3 and AB4* and having a set of generatorsthen the first derived functor of projective limit vanisheson so-called Mittag-Leffler sequences in C. The recent examplesgiven by Deligne and Neeman show that the condition that thecategory has a set of generators is necessary. The conditionAB4* is also necessary, and indeed we give for each integerm 1 an example of a Grothendieck category C_m and a Mittag-Lefflersequence in C_m for which the derived functors of its projectivelimit vanish in all positive degrees except m. This leads toa systematic study of derived functors of infinite productsin Grothendieck categories. Several explicit examples of theapplications of these functors are also studied.     

On A Property Of Minimal Zero-Sum Sequences And Restricted Sumsets

Let G be an additively written abelian group, and let S be asequence in G \ {0} with length |S| 4. Suppose that S is aproduct of two subsequences, say S = BC, such that the elementg + h occurs in the sequence S whenever g.h is a subsequenceof B or of C. Then S contains a proper zero-sum subsequence,apart from some well-characterized exceptional cases. This resultis closely connected with restricted set addition in abeliangroups. Moreover, it solves a problem on the structure of minimalzero-sum sequences, which recently occurred in the theory ofnon-unique factorizations. 2000 Mathematics Subject Classification11B50, 11B75, 11P99.     

Animals, Trees and Renewal Sequences: Corrigendum

IN SECTION 3 of the above we omitted to mention aperiodicity.The period p of the pseudo renewal sequence {a_n: n > 0} isgiven by p = g.c.d. {n > 1: a_n > 0}. We are only concernedwith aperiodic renewal sequences (i.e. where p = 1). As it standsTheorem 3.1 is incorrect and should be restated as: THEOREM 3.1 If a = (a_n: n = 0,1,...) is an aperiodic pseudo-renewalsequence its limit a satisfies g_na^–n > 1 where a^–1 is to be interpreted as; if a = 0.     

On the Distribution of Power Residues and Primitive Elements in Some Nonlinear Recurring Sequences

It is shown that the method of estimation of exponential sumswith nonlinear recurring sequences, invented by the authorsin a recent series of works, can be applied to estimating sumsof multiplicative characters as well. As a consequence, resultsare obtained about the distribution of power residues and primitiveelements in such sequences. 2000 Mathematics Subject Classification11B37, 11L40 (primary), 11A07, 11A15, 11T24 (secondary).     

Dimension functions of Cantor sets

We estimate the packing measure of Cantor sets associated to non-increasing sequences through their decay. This result, dual to one obtained by Besicovitch and Taylor, allows us to characterize the dimension functions recently found by Cabrelli et al for these sets.

     

Randomized Smolyak algorithms based on digital sequences for multivariate integration

Gunther Leobacher ^{In this paper, we consider Smolyak algorithms based on quasi-MonteCarlo rules for high-dimensional numerical integration. Thequasi-Monte Carlo rules employed here use digital (t, , ß,, d)-sequences as quadrature points. We consider the worst-caseerror for multivariate integration in certain Sobolev spacesand show that our quadrature rules achieve the optimal rateof convergence. By randomizing the underlying digital sequences,we can also obtain a randomized Smolyak algorithm. The boundon the worst-case error holds also for the randomized algorithmin a statistical sense. Further, we also show that the randomizedalgorithm is unbiased and that the integration error can beapproximated as well.}     

On the Local and Superlinear Convergence of Quasi-Newton Methods

This paper presents a local convergence analysis for severalwell-known quasi-Newton methods when used, without line searches,in an iteration of the form to solve for x^* such that Fx^* = 0. The basic idea behind theproofs is that under certain reasonable conditions on x_o, Fand x_o, the errors in the sequence of approximations {H_k} toF'(x^*)^–1 can be shown to be of bounded deterioration inthat these errors, while not ensured to decrease, can increaseonly in a controlled way. Despite the fact that H_k is not shownto approach F'(x^*)^–1, the methods considered, includingthose based on the single-rank Broyden and double-rank Davidon-Fletcher-Powellformulae, generate locally Q-superlinearly convergent sequences{x_k}.     

Measures of pseudorandomness for finite sequences: typical values

Mauduit and Sárközy introduced and studied certainnumerical parameters associated to finite binary sequences E_N {–1, 1}^N in order to measure their ‘level of randomness’.Those parameters, the normality measure (E_N), the well-distributionmeasure W(E_N), and the correlation measure C_k(E_N) of order k,focus on different combinatorial aspects of E_N. In their work,amongst others, Mauduit and Sárközy (i) investigatedthe relationship among those parameters and their minimal possiblevalue, (ii) estimated (E_N), W(E_N) and C_k(E_N) for certain explicitlyconstructed sequences E_N suggested to have a ‘pseudorandomnature’, and (iii) investigated the value of those parametersfor genuinely random sequences E_N. In this paper, we continue the work in the direction of (iii)above and determine the order of magnitude of (E_N), W(E_N) andC_k(E_N) for typical E_N. We prove that, for most E_N {–1,1}^N, both W(E_N) and (E_N) are of order N, while C_k(E_N) is oforder for any given 2 k N/4.     

Local Rigidity of Infinite-Dimensional Teichmuller Spaces

This paper presents a rigidity theorem for infinite-dimensionalBergman spaces of hyperbolic Riemann surfaces, which statesthat the Bergman space A¹(M), for such a Riemann surface M,is isomorphic to the Banach space of summable sequence, l¹.This implies that whenever M and N are Riemann surfaces thatare not analytically finite, and in particular are not necessarilyhomeomorphic, then A¹(M) is isomorphic to A¹(N). It is knownfrom V. Markovic that if there is a linear isometry betweenA¹(M) and A¹(N), for two Riemann surfaces M and N of non-exceptionaltype, then this isometry is induced by a conformal mapping betweenM and N. As a corollary to this rigidity theorem presented here,taking the Banach duals of A¹(M) and l¹ shows that the spaceof holomorphic quadratic differentials on M, Q(M), is isomorphicto the Banach space of bounded sequences, l. As a consequenceof this theorem and the Bers embedding, the Teichmüllerspaces of such Riemann surfaces are locally bi-Lipschitz equivalent.     

Random Sequences Interpolating with Probability One

Let {_n} be a sequence of independent random variables uniformlydistributed on [0, 2], and let {r_n} be a sequence of (deterministic)radii in [0, 1). Form points of the unit disc putting z_n = r_ne_n.We characterize those sequences {r_n} for which {z_n} is an interpolatingsequence with probability one.     

Finite Subsequences of Shift Register Sequences

In a stream cipher a cryptogram is produced from a binary datastream by modulo-2-adding it to a keystream sequence. The securityof the system relies on the inability of an interceptor to determinethis keystream sequence. One obvious requirement for such asystem is that there should be sufficiently many possibilitiesfor the keystream sequence that the interceptor cannot possiblytry them all. In this paper we consider the likelihood of an interceptor beingable to decipher the cryptogram correctly even though he maybe trying the wrong keystream sequence. This possibility arisesbecause the length of any particular message is likely to beconsiderably shorter than the period of the keystream sequence,and thus only a comparatively small section of the keystreamsequence is used. Hence, if the interceptor tries a sequencewhich intersects (i.e. agrees) with the keystream sequence inthe appropriate positions, he will deduce the message correctly. A number of the standard methods for generating keystream sequencesuse shift registers as ‘building blocks’. So welook in considerable detail at the number of intersections (ofvarious lengths) for sequences generated by two different shiftregisters. We also show that if a keystream sequence has linearequivalence n, then the local linear equivalence of any subsequenceof length at least 2n is n. This means that if the message haslength at least 2n and the keystream sequence has linear equivalencen, then there is no other sequence of linear equivalence lessthan n+1 which can be used to decipher correctly.     

A continuous path of singular masas in the hyperfinite II1 factor

Using methods of Tauer, we exhibit an uncountable family ofsingular masas in the hyperfinite II₁ factor R all with Pukánszkyinvariant {1}, no pair of which is conjugate by an automorphismof R. This is done by introducing an invariant (A) for a masaA in a II₁ factor N as the maximal size of a projection eA forwhich A e contains non-trivial centralizing sequences for eNe. The masas produced give rise to a continuous map from theinterval [0, 1] into the singular masas in R equipped with thed_{, 2}-metric. A result is also given showing that the Pukánszky invariantis d_{, 2}-upper semi-continuous. As a consequence, the sets ofmasas with Pukánszky invariant {n} are all closed.     

Strong tractability of integration using scrambled Niederreiter points

We study the randomized worst-case error and the randomized error of scrambled quasi-Monte Carlo (QMC) quadrature as proposed by Owen. The function spaces considered in this article are the weighted Hilbert spaces generated by Haar-like wavelets and the weighted Sobolev-Hilbert spaces. Conditions are found under which multivariate integration is strongly tractable in the randomized worst-case setting and the randomized setting, respectively. The -exponents of strong tractability are found for the scrambled Niederreiter nets and sequences. The sufficient conditions for strong tractability for Sobolev spaces are more lenient for scrambled QMC quadratures than those for deterministic QMC net quadratures.

     

Metrical Theorems on Prime Values of the Integer Parts of Real Sequences

Let a_n be an increasing sequence of positive reals with a_n as n . Necessary and sufficient conditions are obtained foreach of the sequences to take on infinitely many prime values for almost all > rß.For example, the sequence a_n is infinitely often prime for almostall > 0 if and only if there is a subsequence of the a_n,say b_n, with b_{n + 1} > b_n + 1 and with the series divergent. Asymptotic formulae areobtained when the sequences considered are lacunary. An earlierresult of the author reduces the problem to estimating the measureof overlaps of certain sets, and sieve methods are used to obtainthe correct order upper bounds. 1991 Mathematics Subject Classification:primary 11N05; secondary 11K99, 11N36.