Restriction operators, balayage and doubly orthogonal systems of analytic functions

Systems of analytic functions which are simultaneously orthogonal over each of two domains were apparently first studied in particular cases by Walsh and Szegö, and in full generality by Bergman. In principle, these are very interesting objects, allowing application to analytic continuation that is not restricted (as Weierstrassian continuation via power series) either by circular geometry or considerations of locality. However, few explicit examples are known, and in general one does not know even gross qualitative features of such systems. The main contribution of the present paper is to prove qualitative results in a quite general situation.It is by now very well known that the phenomenon of “double orthogonality” is not restricted to Bergman spaces of analytic functions, nor even indeed has it any intrinsic relation to analyticity; its essence is an eigenvalue problem arising whenever one considers the operator of restriction on a Hilbert space of functions on some set, to a subset thereof, provided this restriction is injective and compact. However, in this paper only Hilbert spaces of analytic functions are considered, especially Bergman spaces. In the case of the Hardy spaces Fisher and Micchelli discovered remarkable qualitative features of doubly orthogonal systems, and we have shown how, based on the classical potential-theoretic notion of balayage, and its modern generalizations, one can deduce analogous results in the Bergman space set-up, but with restrictions imposed on the geometry of the considered domains and measures; these were not needed in the Fisher-Micchelli analysis, but are necessary here as shown by examples.From a more constructive point of view we study the Bergman restriction operator between the unit disk and a compactly contained quadrature domain and show that the representing kernel of this operator is rational and it is expressible (as an inversion followed by a logarithmic derivative) in terms of the polynomial equation of the boundary of the inner domain.     

Lattice measures and associated outer measures

In this paper, we consider lattice measures and introduce certain associated outer measures (not the usual induced outer measures), study their properties, and investigate the associated classes of measureable sets. We utilize some of these outer measures to characterize normality and investigate lattice separation properties; also, to extend the notion of regularity of measures to weak regularity of measures. We give applications of our results to specific topological lattices.     

A combinatorial problem on translation-invariant extensions of the Lebesgue measure

It is proved that, for every natural number k≥2, there exist k subsets of the real line such that any k−1 of them can be made measurable with respect to a translation-invariant extension of the Lebesgue measure, but there is no nonzero σ-finite translation-quasi-invariant measure for which all of these k subsets become measurable. In connection with this result, a related open problem is posed.     

On sums of real-valued functions with extremely thick graphs

We consider some properties of those functions acting from the real line R$\mathbf{R}$ into itself, whose graphs are extremely thick subsets of the Euclidean plane R²${\mathbf{R}}^2$. The structure of sums of such functions is studied and the obtained results are applied to certain measure extension problems.     

Properties of a special class of doubly stochastic measures

Summary A measure on the unit squareI } I is doubly stochastic if(A } I) = (I } A) = the Lebesgue measure ofA for every Lebesgue measurable subsetA ofI = [0, 1]. By the hairpinL L ^–1, we mean the union of the graphs of an increasing homeomorphismL onI and its inverseL ^–1. By the latticework hairpin generated by a sequence {x _n :n Z} such thatx _n-1 < x_n (n Z), x _n = 0 and x _n = 1, we mean the hairpinL L ^–1 , whereL is linear on [x _n-1 ,x _n ] andL(n) =x _n-1 forn Z. In this note, a characterization of latticework hairpins which support doubly stochastic measures is given. This allows one to construct a variety of concrete examples of such measures. In particular, examples are given, disproving J. H. B. Kemperman's conjecture concerning a certain condition for the existence of doubly stochastic measures supported in hairpins.     

Some remarks on the existence of doubly stochastic measures with latticework hairpin support

Summary In their paper Properties of a special class of doubly stochastic measures, which appeared in volume 36 of this journal (pp. 212–229), A. Kaminski et al. introduced latticework hairpins and used them to provide a counterexample to a conjecture of J. H. B. Kemperman. However, their paper contains an error and, as a consequence, a umber of incorrect statements (which fortunately do not invalidate its main results). We point out the error and the incorrect statements. More importantly, we modify the argument, present correct versions wherever possible, and provide a valid characterization of those latticework hairpins that support doubly stochastic measures. We also construct a number of new examples, including another counterexample to Kemperman's conjecture.     

Lévy group and density measures

We will deal with finitely additive measures on integers extending the asymptotic density. We will study their relation to the Lévy group G of permutations of N. Using a new characterization of the Lévy group G we will prove that a finitely additive measure extends density if and only if it is G-invariant.     

Convexity properties of harmonic measures

It is shown that any convex combination of harmonic measures , where U₁,…,Uk are relatively compact open neighborhoods of a given point x∈Rd, d?2, can be approximated by a sequence of harmonic measures such that each Wn is an open neighborhood of x in U₁∪?∪Uk.This answers a question raised in connection with Jensen measures. Moreover, it implies that, for every Green domain X containing x, the extremal representing measures for x with respect to the convex cone of potentials on X (these measures are obtained by balayage of the Dirac measure at x on Borel subsets of X) are dense in the compact convex set of all representing measures.This is achieved approximating balayage on open sets by balayage on unions of balls which are pairwise disjoint and very small with respect to their mutual distances and then reducing the size of these balls in a suitable manner.These results, which are presented simultaneously for the classical potential theory and for the theory of Riesz potentials, can be sharpened if the complements or the boundaries of the open sets have a capacity doubling property. The methods developed for this purpose (continuous balayage on increasing families of compact sets, approximation using scattered sets with small capacity) finally lead to answers even in a very general potential-theoretic setting covering a wide class of second order partial differential operators (uniformly elliptic or in divergence form, or sums of squares of vector fields satisfying Hörmander's condition, for example, sub-Laplacians on stratified Lie algebras).     

Martin compactification for discrete potential theory and the mean value property

A converse of the well-known theorem on themean value property of harmonic functions is given. It is shown that a positive measurable function is harmonic if it possesses arestricted mean value property. Earlier proofs obtained using the probabilistic techniques were given by Veech, Heath and Baxter. Our approach is based on a Martin type compactification built up with the help of some quite elementarya priori inequalities foraveraging kernels.     

A fixed point theorem for moment matrices of self-similar measures

We consider the self-similar measure on the complex plane C$\mathbb{C}$ associated to an iterated function system (IFS) with probabilities. From this IFS we define an operator in a complete metric space of infinite matrices. Using the expression obtained in a previous work of the authors, we prove that this operator has as fixed point the moment matrix of the self-similar measure. As a consequence, we obtain a very efficient algorithm to compute the moment matrix of the self-similar measure. Finally, in order to estimate the rate of convergence of the algorithm, we find an upper bound of the norm of this contractive operator.     

On thick subgroups of uncountable σ-compact locally compact commutative groups

We show that, for any uncountable commutative group (G,+), there exists a countable covering where each Gj is a subgroup of G satisfying the equality card(G/Gj)=card(G). This purely algebraic fact is used in certain constructions of thick and nonmeasurable subgroups of an uncountable σ-compact locally compact commutative group equipped with the completion of its Haar measure.     

Quasi-boundedness and subtractivity; Applications to excessive measures

We study the quasi-boundedness and subtractivity in a general frame of cones of potentials (more precisely in H-cones). Particularly we show that the subtractive elements are strongly related to the existence of recurrent balayages. In the special case of excessive measures we improve results of P. J. Fitzsimmons and R. K. Getoor from [13], obtained with probabilistic methods.     

Pseudo q-Engel expansions and Rogers-Ramanujan type identities

Abstract

Andrews, Knopfmacher and Knopfmacher have used the Schur polynomials to consider the celebrated Rogers-Ramanujan identities in the context of q-Engel expansions. We extend this view using similar polynomials, provided by Sills, in the context of Slater's list of 130 Rogers-Ramanujan type identities.     

Box dimensions of Riemann-Liouville fractional integrals of continuous functions of bounded variation

If a continuous function f(x) has bounded variation on the unit interval [0,1], the box dimension of f(x) is 1. Furthermore, the box dimension of a Riemann-Liouville fractional integral of f(x) is still 1.