An uncertainty principle for convolution operators on discrete groups

Consider a discrete group and a bounded self-adjoint convolution operator on ; let be the spectrum of . The spectral theorem gives a unitary isomorphism between and a direct sum , where , and is a regular Borel measure supported on . Through this isomorphism corresponds to multiplication by the identity function on each summand. We prove that a nonzero function and its transform cannot be simultaneously concentrated on sets , such that and the cardinality of are both small. This can be regarded as an extension to this context of Heisenberg's classical uncertainty principle.

     

The Furuta inequality in Banach -algebras

Let be real numbers with and Furuta (1987) proved that if bounded linear operators on a Hilbert space satisfy , then . This inequality is called the Furuta inequality and has many applications. In this paper, we prove that the Furuta inequality holds in a unital hermitian Banach -algebra with continuous involution.

     

Kuttner's problem and a Pólya type criterion for characteristic functions

Let be a continuous function with and . If is convex, then , , is the characteristic function of an absolutely continuous probability distribution. The criterion complements Pólya's theorem and applies to characteristic functions with various types of behavior at the origin. In particular, it provides upper bounds on Kuttner's function , , which gives the minimal value of such that is a characteristic function. Specifically, . Furthermore, improved lower bounds on Kuttner's function are obtained from an inequality due to Boas and Kac.

     

A characterization of total reflection orders

Let be a Coxeter system with set of reflections . It is known that if is a total reflection order for , then, for each , and its complement are stable under conjugation by . Moreover the upper and lower -conjugates of are still total reflection orders. For any total order on , say that is stable if is stable under conjugation by for each . We prove that if and all orders obtained from by successive lower or upper -conjugations are stable, then is a total reflection order.

     

Yang index of the deleted product

For any a -dimensional polyhedron is constructed such that the Yang index of its deleted product equals . This answers a question of Izydorek and Jaworowski (1995). For any a -dimensional closed manifold with involution is constructed such that , but can be mapped into a -dimensional polyhedron without antipodal coincidence.

     

A generalization of a theorem of Heins

Let be the family of holomorphic selfmaps of the unit disk in the complex plane . Heins established the continuity of the functional which assigns to ( denotes the identity map) either (i) the fixed point of or (ii) the limit of its iterations or (iii) if ( represents the boundary of ). Using an Abate extension of the Denjoy-Wolff lemma to strongly convex domains, we extend this result of Heins to selfmaps of strongly convex domains in with boundary.

     

-sequences

A sequence of positive integers is called a -sequence if every integer has at most representations with all in and . A -sequence is also called a -sequence or Sidon sequence. The main result is the following

Theorem. Let be a -sequence and for an integer . Then there is a -sequence of size , where .

Corollary. Let . The interval then contains a -sequence of size , when .

     

Integer-valued polynomials over Krull-type domains and Prüfer -multiplication domains

Let be a domain with quotient field . The ring of integer-valued polynomials over is . We characterize the Krull-type domains such that is a Prüfer -multiplication domain.

     

On the excess of sets of complex exponentials

For let be complex numbers such that is bounded. For define , where . Then the excesses in the sense of Paley and Wiener satisfy .

     

Uniqueness of the least-energy solution for a semilinear Neumann problem

We prove that the least-energy solution of the problem

where is a ball, and if , if , is unique (up to rotation) if is small enough.

     

On a theorem of Barbara Schmid

Let be a finite group and a complex representation. Barbara Schmid has shown that the algebra of invariant polynomial functions on the vector space is generated by homogeneous polynomials of degree at most , where is the largest degree of a generator in a minimal generating set for , and is the complex regular representation of . In this note we give a new proof of this result, and at the same time extend it to fields whose characteristic is larger than , the order of the group .

     

On the divergence of the means of double Walsh-Fourier series

In 1992, Móricz, Schipp and Wade proved the a.e. convergence of the double means of the Walsh-Fourier series () for functions in ( is the unit square). This paper aims to demonstrate the sharpness of this result. Namely, we prove that for all measurable function we have a function such as and does not converge to a.e. (in the Pringsheim sense).

     

Bounded composition operators with closed range on the Dirichlet space

For composition operators on spaces of analytic functions it is well known that norm estimates can be converted to Carleson measure estimates. The boundedness of the composition operator becomes equivalent to a Carleson measure inequality. The measure corresponding to a composition operator on the Dirichet space is , where is the cardinality of the preimage . The composition operator will have closed range if and only if the corresponding measure satisfies a ``reverse Carleson measure' theorem: for all . Assuming is bounded, a necessary condition for this inequality is a reverse of the Carleson condition: (C) for all Carleson squares . It has long been known that this is not sufficient for a completely general measure. Here we show that it is also not sufficient for the special measures . That is, we construct a function such that is bounded and satisfies (C) but the composition operator does not have closed range.

     

Inequalities for the gamma function

We prove the following two theorems:

(i) Let be the th power mean of and . The inequality

holds for all if and only if , where denotes Euler's constant. This refines results established by W. Gautschi (1974) and the author (1997).

(ii) The inequalities

are valid for all if and only if and , while holds for all if and only if and . These bounds for improve those given by G. D. Anderson an S.-L. Qiu (1997).

     

Long time heat diffusion on homogeneous trees

Let be a homogeneous tree of degree , , the Laplace operator of and the fundamental solution of the heat equation on . We show that the heat kernel is asymptotically concentrated in an annulus moving to infinity with finite speed . Asymptotic concentration of heat in the norm is also investigated.

     

On conditions for polyconvexity

We give an example of a smooth function , which is not polyconvex and which has the property that its restriction to any ball of radius one can be extended to a smooth polyconvex function . In particular, it implies that there exists no `local condition' which is necessary and sufficient for polyconvexity of functions , where , . We also briefly discuss connections with quasiconvexity.

     

On the Gelfand-Kirillov conjecture for quantum algebras

Let be a complex not a root of unity and be a semi-simple Lie -algebra. Let be the quantized enveloping algebra of Drinfeld and Jimbo, be its triangular decomposition, and the associated quantum group. We describe explicitly and as a quantum Weyl field. We use for this a quantum analogue of the Taylor lemma.

     

Extremal points of a functional on the set of convex functions

We investigate the extremal points of a functional , for a convex or concave function . The admissible functions are convex themselves and satisfy a condition . We show that the extremal points are exactly and if these functions are convex and coincide on the boundary . No explicit regularity condition is imposed on , , or . Subsequently we discuss a number of extensions, such as the case when or are non-convex or do not coincide on the boundary, when the function also depends on , etc.

     

Universal -lattices of minimal rank

Let be the minimal rank of -universal -lattices, by which we mean positive definite -lattices which represent all positive -lattices of rank . It is a well known fact that for . In this paper, we determine and find all -universal lattices of rank for .

     

The globally irreducible representations of symmetric groups

Let be an algebraic number field and be the ring of integers of . Let be a finite group and be a finitely generated torsion free -module. We say that is a globally irreducible -module if, for every maximal ideal of , the -module is irreducible, where stands for the residue field .

Answering a question of Pham Huu Tiep, we prove that the symmetric group does not have non-trivial globally irreducible modules. More precisely we establish that if is a globally irreducible -module, then is an -module of rank with the trivial or sign action of .