The commutant of rationally cyclic subnormal operators and rational approximation

Let denote the set of analytic bounded point evaluations forR ^q (K, ). Assume that . In this paper, we first show that if is a finitely connected domain and if the evaluation map fromR ^q (K, )L () toH () is surjective, then | is absolutely continuous with respect to harmonic measure for . This generalizes Olin and Yang's corresponding result for polynomials and the proof we present here is simpler. We also provide an example that shows this absolute continuity property fails in general when is an infinitely connected domain. In the second part, we then offer a solution to a problem of Conway and Elias.     

Convergence proofs and error estimates for an iterative method for conformal mapping

Summary The iterative method as introduced in [8] and [9] for the determination of the conformal mapping of the unit disc onto a domainG is here described explicitly in terms of the operatorK, which assigns to a periodic functionu its periodic conjugate functionK u. It is shown that whenever the boundary curve ofG is parametrized by a function with Lipschitz continuous derivative then the method converges locally in the Sobolev spaceW of 2-periodic absolutely continuous functions with square integrable derivative. If is in a Hölder classC ²⁺, the order of convergence is at least 1+. If is inC ^l+1+ withl1, 0<<1, then the iteration converges inC ^l+. For analytic boundary curves the convergence takes place in a space of analytic functions.For the numerical implementation of the method the operatorK can be approximated by Wittich's method, which can be applied very effectively using fast Fourier transform. The Sobolev norm of the numerical error can be estimated in terms of the numberN of grid points. It isO(N ^1–l–) if is inC ^l+1+, andO (exp (–N/2)) if is an analytic curve. The number in the latter formula is bounded by logR, whereR is the radius of the largest circle into which can be extended analytically such that'(z)0 for |z|<R. The results of some test calculations are reported.     

A q-Analogue of Weber-Schafheitlin Integral of Bessel Functions

In an attempt to find a q-analogue of Weber and Schafheitlin's integral ₀ x ^– J (ax) J (bx) dx which is discontinuous on the diagonal a = b the integral ₀ x ^– J ⁽²⁾ (a(1 – q)x; q)J ⁽¹⁾ (b(1 – q)x; q) dx is evaluated where J ⁽¹⁾ (x; q) and J ⁽²⁾ (x; q) are two of Jackson's three q-Bessel functions. It is found that the question of discontinuity becomes irrelevant in this case. Evaluations of this integral are also made in some interesting special cases. A biorthogonality formula is found as well as a Neumann series expansion for x in terms of J ⁽²⁾ _+1+2n ((1 – q)x; q). Finally, a q-Lommel function is introduced.     

Maximum Monoreflections and Essential Extensions

An extension operator c in a category is an assignment, to each object A a monomorphism c _A : AcA. Seeking to approximate such a c by a functor, in our earlier paper Maximum monoreflections, we showed that with some hypotheses on the category, and on c, there is a monoreflection (c) maximum beneath c. Thus, in a suitable category of rings, using the complete ring of quotients operator Q, each object A has a maximum functorial ring of quotients (Q)A. But the proof gave no hint of how to calculate the general (c)A's, nor the particular (Q)A's. In the present paper, we give an explicit formula (and separate proof of existence) for the (c)A's, under more complicated hypotheses on the category and assuming the c _A 's are essential monomorphisms. We discuss briefly how the formula proves adequate to calculate the (Q)A's in Archimedean f-rings, and some related and necessary constructs in Archimedean l-groups.     

Variational problems in SBV and image segmentation

We show how it is possible to prove the existence of solutions of the Mumford-Shah image segmentation functional F(u,K) = _\K [u² + (u – g)²]dx + ^{n – 1}(K), u W ^1,2(\K), K closed in .We use a weak formulation of the minimum problem in a special class SBV() of functions of bounded variation. Moreover, we also deal with the regularity of minimizers and the approximation of F by elliptic functionals defined on Sobolev spaces. In this paper, we have collected the main results of Ambrosio and others.     

Eisenstein Cohomology and the Construction of p-Adic Analytic L-Functions

Let be a cuspidal automorphic representation of GL₃( ), unramified at pand of cohomological type at infinity. We construct p-adic L-functions, which interpolate the critical values of L(,s) and which satisfy a logarithmic growth condition. We obtain these functions as p-adic Mellin transforms of certain distributions on _p ^* having values in some fixed number field and which are of moderate growth. In the p-ordinary case we obtain the bound |(U)| _p |_Haar(U)| _p for open subsets U _p ^*, where _Haardenotes the invariant distribution on _p ^*.     

The Parameters of Bipartite Q-polynomial Distance-Regular Graphs

Let denote a bipartite distance-regular graph with diameter D 3 and valency k 3. Suppose ₀, ₁, ..., _D is a Q-polynomial ordering of the eigenvalues of . This sequence is known to satisfy the recurrence _{i – 1} – _i + _{i + 1} = 0 (0 > i > D), for some real scalar . Let q denote a complex scalar such that q + q ^–1 = . Bannai and Ito have conjectured that q is real if the diameter D is sufficiently large.We settle this conjecture in the bipartite case by showing that q is real if the diameter D 4. Moreover, if D = 3, then q is not real if and only if ₁ is the second largest eigenvalue and the pair (, k) is one of the following: (1, 3), (1, 4), (1, 5), (1, 6), (2, 4), or (2, 5). We observe that each of these pairs has a unique realization by a known bipartite distance-regular graph of diameter 3.     

Approximation in the mean by bounded analytic functions

Let 1<p< and . LetC _q denote the Bessel capacity in the plane. Let be the set of homomorphisms ofH (G) such that (z)= and letNP denote the set of points in G for which is not a peak set forH (G). In this note, we show that ifC _q (NP)=0, thenH (G) is dense inL _a ^p (G), the Bergman space overG.Partially supported by NSF DMS-9401234     

Convolution Operators with Fractional Measures Associated to Holomorphic Functions

Let be an open set in the complex plane and let be a holomorphic function on . Let K be a compact subset of with nonempty interior such that 0 K. Let be the Borel measure of R ⁴ C ² given by(E = _K E(z, (z))|z|^–2 d(z)where 0 < 2 and d(x ₁ + ix ₂) = dx ₁ dx ₂ denotes the Lebesgue measure on C. Let T be the convolution operator T f = * f. In this paper we characterize the type set E associated to T .     

Bestimmung der Eigenwerte und Eigenvektoren einer Matrix mit Hilfe des Quotienten-Differenzen-Algorithmus

Summary Two previous papers (in Vol. V) describe theory and some applications of the quotient-difference (=QD-) algorithm. Here we give an extension which allows the determination of the eigenvectors of a matrix. Letx ₍₀₎ ¹ , ...,x ₍₀₎ ⁿ be a coordinate system in whichA has Jacobi form (such a system may be constructed with methods ofC. Lanczos orW. Givens). Then the QD-algorithm allows the construction of a sequence of coordinate systemsx ₍₂₎ ¹ , ...,x ₍₂₎ ⁿ , (=0, 1, 2, ...) which converge for to the system of the eigenvectors ofA.     

Semicontinuità inferiore di un funzionale integrale dipendente da funzioni di classeL p a valori in uno spazio di Banach

Summary We study the sequential lower semicontinuity of an integral functional I: L ^s (, X) x L ^q (, Y) [– , ] where X and Y are separable B-spaces and is a finite positive regular complète measure. A necessary and sufficient condition is given on the integrand for the sequential lower semicontinuity of the integral with respect to the strong by weak topology on L ^s (, X)×L ^q (, Y).

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Lavoro svolto nell'ambito del «Laboratorio per la matematica applicata» del C.N.R. presso l'Università di Genova.     

Hermitian and Positive Integrated C-cosine Functions on Banach Spaces

Peculiar properties of hermitian and positive n-times integrated C-cosine functions on Banach spaces are investigated. Among them are: (1) Any nondegenerate positiven -times integrated C-cosine function is infinitely differentiable in operator norm; (2) An exponentially bounded, nondegenerateC -cosine function on L ^p () (1L ¹(), C₀ , in case C has dense range) is positive if and only if its generator is bounded, positive, and commutes with C.     

Asymptotic Expansions for the Congestion Period for the M/M/∞ Queue

We consider the M/M/ queue with arrival rate , service rate and traffic intensity =/. We analyze the first passage distribution of the time the number of customers N(t) reaches the level c, starting from N(0)=m>c. If m=c+1 we refer to this time period as the congestion period above the level c. We give detailed asymptotic expansions for the distribution of this first passage time for , various ranges of m and c, and several different time scales. Numerical studies back up the asymptotic results.     

Reiter’s Condition <Emphasis Type="Italic">P</Emphasis><Subscript>1</Subscript> and Approximate Identities for Polynomial Hypergroups

Let K be a commutative hypergroup with the Haar measure . In the present paper we investigate whether the maximal ideals in L¹(K,) have bounded approximate identities. We will show that the existence of a bounded approximate identity is equivalent to the existence of certain functionals on the space L(K,). Finally we apply the results to polynomial hypergroups and obtain a rather complete solution for this class.The third author was partially supported by KBN (Poland) under grant 5 P03A 034 20 and by Research Training Network Harmonic Analysis and Related Problems Contract HPRN-CT-2001-00273.     

Fractal Characteristics of Measures; An Approach Via Function Spaces

Let be a positive Radon measure in R ⁿ with compact support . Let Q _jm be cubes with side-length 2^-j+1 originating from the canonical tiling of R ⁿ where j\in N ₀ and m\in Z ⁿ. If \in R, 0 < p \le , 0 < q \le , then _pq is the mixed _q – _p -quasi-norm of the sequence 2 ^j (Q _jm ). Quantities of this type are considered in fractal geometry (multifractal formalism) and in the theory of the function spaces B ^s _pq (R ⁿ) and F ^s _pq (R ⁿ). In Theorem 1 we deal with the question when _pq is an equivalent quasi-norm in some of these spaces (-property). If || = 0, then _S consists of those points (t,s) in the t–s-diagram in Figure 1 for which belongs to B ^s _p (R ⁿ) with pt = 1. Theorem 2 deals with the interrelation of S and _pq . Some applications to truncated Riesz potentials, Bessel potentials and Fourier transforms of are given.     

A sharp partitioning-inequality for non-atomic probability measures based on the mass of the infimum of the measures

Summary If ₁, ... , are non-atomic probability measures on the same measurable space (S, ), then there is an -measurable partition {A _i } _{i = 1} ⁿ of S so that _i (A _i )(n – 1 + m)^–1 for all i=1, ..., n, where is the total mass of the largest measure dominated by each of the _i S; moreover, this bound is attained for all n1 and all m in [0, 1]. This result is an analog of the bound (n+1-M) ^-1of Elton et al. [5] based on the mass M of the supremum of the measures; each gives a quantative generalization of a well-known cake-cutting inequality of Urbanik [10] and of Dubins and Spanier [2].Research partly supported by NSF Grants DMS-84-01604 and DMS-86-01608     

Ultrafilter translations

We develop a method for extending results about ultrafilters into a more general setting. In this paper we shall be mainly concerned with applications to cardinality logics. For example, assumingV=L, Gödel's Axiom of Constructibility, we prove that if > then the logic with the quantifier there exist many is (,)-compact if and only if either is weakly compact or is singular of cofinality<. As a corollary, for every infinite cardinals and , there exists a (,)-compact non-(,)-compact logic if and only if either < orcf<cf or < is weakly compact.Counterexamples are given showing that the above statements may fail, ifV=L is not assumed.However, without special assumptions, analogous results are obtained for the stronger notion of [,]-compactness.     

A graded scale of parametric families of distributions,and parameter estimates based on the sample mean

Summary Denote by _k a class of familiesP={P} of distributions on the line R¹ depending on a general scalar parameter , being an interval of R¹, and such that the moments µ₁()=xdP ,...,µ_2k ()=x ^2k dP are finite, ₁ (), ..., _k (), _k+1 () ..., _k () exist and are continuous, with ₁ () 0, and _j +1 ()= ₁ () _j () +[₂() -₁()²] _j ()/ ₁ (), J=2, ..., k. Let ₁=¯x=x ₁ + ... +x _n/n, ₂=x ₁ ² + ... +x _n ²/n, ..., _k =(x ₁ ^k + ... +x _n ^k/n denote the sample moments constructed for a sample x₁, ..., x_n from a population with distribution Pg. We prove that the estimator of the parameter by the method of moments determined from the equation ₁= ₁() and depending on the observations x₁, ..., x_n only via the sample mean ¯x is asymptotically admissible (and optimal) in the class _k of the estimators determined by the estimator equations of the form ₀ () + ₁ () ₁ + ... + _k () _k =0 if and only ifP _k .The asymptotic admissibility (respectively, optimality) means that the variance of the limit, as n (normal) distribution of an estimator normalized in a standard way is less than the same characteristic for any estimator in the class under consideration for at least one 9 (respectively, for every ).The scales arise of classes _{1 2}... of parametric families and of classes _{1 2} ... of estimators related so that the asymptotic admissibility of an estimator by the method of moments in the class _k is equivalent to the membership of the familyP in the class _k .The intersection consists only of the families of distributions with densities of the form h(x) exp {C₀() + C₁() x } when for the latter the problem of moments is definite, that is, there is no other family with the same moments ₁ (), ₂ (), ...Such scales in the problem of estimating the location parameter were predicted by Linnik about 20 years ago and were constructed by the author in [1] (see also [2, 3]) in exact, not asymptotic, formulation.Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei, pp. 41–47, 1981.     

Bounded charges and their ranges

Sunto Si caratterizzano le masse non continue e non atomiche, definite su un campo di insiemi e le masse definite su un -campoA che verificano la proprietá: (A T(A)) è chiuso, per ogni A A. Si dà inoltre, per tali masse, la struttura di (A T(A)), provando che tale insieme numerico è simile al codominio di una misura limitata.

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Research supported by M.P.I. and C.N.R.     

Conditioned super-Brownian motion

Summary We investigate classes of conditioned super-Brownian motions, namely H-transformsP ^H with non-negative finitely-based space-time harmonic functionsH(t, ). We prove thatH ^H is the unique solution of a martingale problem with interaction and is a weak limit of a sequence of rescaled interacting branching Brownian motions. We identify the limit behaviour of H-transforms with functionsH(t, )=h(t, (1)) depending only on the total mass (1). Using the Palm measures of the super-Brownian motion we describe for an additive spacetime harmonic functionH(t, )=h(t, x) (dx) theH-transformP ^H as a conditioned super-Brownian motion in which an immortal particle moves like an h-transform of Brownian motion.