A note on the weighted Hilbert's inequality

A finite Hilbert transformation associated with a polynomial is the analogue of a Hilbert transformation associated with an entire function which is a generalization of the classical Hilbert transformation. The weighted Hilbert inequality, which has applications in analytic number theory, is closely related to the finite Hilbert transformation associated with a polynomial. In this note, we study a relation between the finite Hilbert transformation and the weighted Hilbert's inequality. A question is posed about the finite Hilbert transformation, of which an affirmative answer implies the weighted Hilbert inequality.

     

Differential operators on Hilbert modular forms

We investigate differential operators and their compatibility with subgroups of SL₂n(R). In particular, we construct Rankin-Cohen brackets for Hilbert modular forms, and more generally, multilinear differential operators on the space of Hilbert modular forms. As an application, we explicitly determine the Rankin-Cohen bracket of a Hilbert-Eisenstein series and an arbitrary Hilbert modular form. We use this result to compute the Petersson inner product of such a bracket and a Hilbert modular cusp form.     

The maximal coarse Baum–Connes conjecture for spaces which admit a fibred coarse embedding into Hilbert space

We introduce a notion of fibred coarse embedding into Hilbert space for metric spaces, which is a generalization of Gromov?s notion of coarse embedding into Hilbert space. It turns out that a large class of expander graphs admit such an embedding. We show that the maximal coarse Baum–Connes conjecture holds for metric spaces with bounded geometry which admit a fibred coarse embedding into Hilbert space.     

Dilation of Dual Frame Pairs in Hilbert C*-Modules

We investigate the geometric properties for Hilbert C*-modular frames. We show that any dual frame pair in a Hilbert C*-module is an orthogonal compression of a Riesz basis and its canonical dual for some larger Hilbert C*-module. This generalizes the Hilbert space dual frame pair dilation theory due to Casazza, Han and Larson to dual Hilbert C*-modular frame pairs. Additionally, for any Hilbert C*-modular dual frame pair induced by a group of unitary operators, we show that there is a dilated dual pair which inherits the same group structure.     

On additive and multiplicative Hilbert cubes

Given subset E of natural numbers FS(E) is defined as the collection of all sums of elements of finite subsets of E and any translation of FS(E) is said to be Hilbert cube. We can define the multiplicative analog of Hilbert cube as well. E.G. Strauss proved that for every ε>0 there exists a sequence with density >1−ε which does not contain an infinite Hilbert cube. On the other hand, Nathanson showed that any set of density 1 contains an infinite Hilbert cube. In the present note we estimate the density of Hilbert cubes which can be found avoiding sufficiently sparse (in particular, zero density) sequences. As a consequence we derive a result in which we ensure a dense additive Hilbert cube which avoids a multiplicative one.     

Direct limits, multiresolution analyses, and wavelets

A multiresolution analysis for a Hilbert space realizes the Hilbert space as the direct limit of an increasing sequence of closed subspaces. In a previous paper, we showed how, conversely, direct limits could be used to construct Hilbert spaces which have multiresolution analyses with desired properties. In this paper, we use direct limits, and in particular the universal property which characterizes them, to construct wavelet bases in a variety of concrete Hilbert spaces of functions. Our results apply to the classical situation involving dilation matrices on L²(Rn), the wavelets on fractals studied by Dutkay and Jorgensen, and Hilbert spaces of functions on solenoids.     

A polynomial expression for the Hilbert series of the quotient ring of diagonal coinvariants

A special case of Haiman?s identity [M. Haiman, Vanishing theorems and character formulas for the Hilbert scheme of points in the plane, Invent. Math. 149 (2002) 371–407] for the character of the quotient ring of diagonal coinvariants under the diagonal action of the symmetric group yields a formula for the bigraded Hilbert series as a sum of rational functions in q,t. In this paper we show how a summation identity of Garsia and Zabrocki for Macdonald polynomial Pieri coefficients can be used to transform Haiman?s formula for the Hilbert series into an explicit polynomial in q,t with integer coefficients. We also provide an equivalent formula for the Hilbert series as the constant term in a multivariate Laurent series.     

A converse to the Oka principle over certain complex Banach manifolds

First Kajiwara then Leiterer gave geometric or cohomological criteria in the spirit of the Grauert-Oka principle for an open subset D of a Stein manifold M to be itself Stein. We give here criteria analogous to Leiterer's, e.g., for a relatively open subset D of a closed complex Hilbert submanifold M of separable Hilbert space to be itself biholomorphic to a closed complex Hilbert submanifold of separable Hilbert space.     

Scalar and matrix Riemann–Hilbert approach to the strong asymptotics of Padé approximants and complex orthogonal polynomials with varying weight

We describe methods for the derivation of strong asymptotics for the denominator polynomials and the remainder of Padé approximants for a Markov function with a complex and varying weight. Two approaches, both based on a Riemann–Hilbert problem, are presented. The first method uses a scalar Riemann–Hilbert boundary value problem on a two-sheeted Riemann surface, the second approach uses a matrix Riemann–Hilbert problem. The result for a varying weight is not with the most general conditions possible, but the loss of generality is compensated by an easier and transparent proof.     

On the Hilbert function of Artinian local complete intersections of codimension three

In singularity theory or algebraic geometry, it is natural to investigate possible Hilbert functions for special algebras A such as local complete intersections or more generally Gorenstein algebras. The sequences that occur as the Hilbert functions of standard graded complete intersections are well understood classically thanks to Macaulay and Stanley. Very little is known in the local case except in codimension two. In this paper we characterise the Hilbert functions of quadratic Artinian complete intersections of codimension three. Interestingly we prove that a Hilbert function is admissible for such a Gorenstein ring if and only if is admissible for such a complete intersection. We provide an effective construction of a local complete intersection for a given Hilbert function. We prove that the symmetric decomposition of such a complete intersection ideal is determined by its Hilbert function.     

Finite dimensional semisimple Q-algebras

A Q-algebra can be represented as an operator algebra on an infinite dimensional Hilbert space. However we don’t know whether a finite n-dimensional Q-algebra can be represented on a Hilbert space of dimension n except n = 1, 2. It is known that a two dimensional Q-algebra is just a two dimensional commutative operator algebra on a two dimensional Hilbert space. In this paper we study a finite n-dimensional semisimple Q-algebra on a finite n-dimensional Hilbert space. In particular we describe a three dimensional Q-algebra of the disc algebra on a three dimensional Hilbert space. Our studies are related to the Pick interpolation problem for a uniform algebra.     

Symmetric Hilbert spaces arising from species of structures

Symmetric Hilbert spaces such as the bosonic and the fermionic Fock spaces over some lsquo;one particle space’ are formed by certain symmetrization procedures performed on the full Fock space. We investigate alternative ways of symmetrization by building on Joyal's notion of a combinatorial species. Any such species F gives rise to an endofunctor of the category of Hilbert spaces with contractions mapping a Hilbert space to a symmetric Hilbert space with the same symmetry as the species F. A general framework for annihilation and creation operators on these spaces is developed, and compared to the generalised Brownian motions of R. Speicher and M. Bożejko. As a corollary we find that the commutation relation with admits a realization on a symmetric Hilbert space whenever f has a power series with infinite radius of convergence and positive coefficients. Received: 7 April 2000; in final form: 28 November 2000 / Published online: 19 October 2001     

Hilbert schemes and maximal Betti numbers over veronese rings

Macaulay??s Theorem (Macaulay in Proc. Lond Math Soc 26:531?C555, 1927) characterizes the Hilbert functions of graded ideals in a polynomial ring over a field. We characterize the Hilbert functions of graded ideals in a Veronese ring R (the coordinate ring of a Veronese embedding of P ^r-1). We also prove that the Hilbert scheme, which parametrizes all graded ideals in R with a fixed Hilbert function, is connected; this is an analogue of Hartshorne??s Theorem (Hartshorne in Math. IHES 29:5?C48, 1966) that Hilbert schemes over a polynomial ring are connected. Furthermore, we prove that each lex ideal in R has the greatest Betti numbers among all graded ideals with the same Hilbert function.     

Hilbert空间中的g-Riesz框架

g-框架作为Hilbert空间中的推广框架最近被提出,它们有许多和框架类似的性质,但并不是所有的性质都是相似的.Christensen已指出了每个Riesz框架都包含一个Riesz基.本文指出并不是所有的g-Riesz框架都包含一个g-Riesz基,但我们得到了每个g-Riesz框架都包含一个无冗g-框架,同时给出了H...     

Smooth and irreducible multigraded Hilbert schemes

The multigraded Hilbert scheme parametrizes all homogeneous ideals in a polynomial ring graded by an abelian group with a fixed Hilbert function. We prove that any multigraded Hilbert scheme is smooth and irreducible when the polynomial ring is Z[x,y], which establishes a conjecture of Haiman and Sturmfels.     

Representation and duality for Hilbert algebras

In this paper we introduce a special kind of ordered topological spaces, called Hilbert spaces. We prove that the category of Hilbert algebras with semi-homomorphisms is dually equivalent to the category of Hilbert spaces with certain relations. We restrict this result to give a duality for the category of Hilbert algebras with homomorphisms. We apply these results to prove that the lattice of the deductive systems of a Hilbert algebra and the lattice of open subsets of its dual Hilbert space, are isomorphic. We explore how this duality is related to the duality given in [6] for finite Hilbert algebras, and with the topological duality developed in [7] for Tarski algebras.      

Hilbert algebras as implicative partial semilattices

The infimum of elements a and b of a Hilbert algebra are said to be the compatible meet of a and b, if the elements a and b are compatible in a certain strict sense. The subject of the paper will be Hilbert algebras equipped with the compatible meet operation, which normally is partial. A partial lower semilattice is shown to be a reduct of such an expanded Hilbert algebra i ?both algebras have the same ?lters.An expanded Hilbert algebra is actually an implicative partial semilattice (i.e., a relative subalgebra of an implicative semilattice),and conversely.The implication in an implicative partial semilattice is characterised in terms of ?lters of the underlying partial semilattice.     

A Fractional Hilbert Transform for 2D Signals

The Hilbert transform is an important tool in image processing and optics. The Hilbert transform can be generalized to a fractional Hilbert transform. The generalization is driven by optics and image processing. We will generalize the fractional Hilbert transform into 2 dimensions by rotating the Hilbert transform in \({\mathbb{R}^{3}}\) . The definition of the Hilbert transform as well as of the rotations will be done by quaternions.     

Reduced arithmetically Gorenstein schemes and simplicial polytopes with maximal Betti numbers

An SI-sequence is a finite sequence of positive integers which is symmetric, unimodal and satisfies a certain growth condition. These are known to correspond precisely to the possible Hilbert functions of graded Artinian Gorenstein algebras with the weak Lefschetz property, a property shared by a nonempty open set of the family of all graded Artinian Gorenstein algebras having a fixed Hilbert function that is an SI sequence. Starting with an arbitrary SI-sequence, we construct a reduced, arithmetically Gorenstein configuration G of linear varieties of arbitrary dimension whose Artinian reduction has the given SI-sequence as Hilbert function and has the weak Lefschetz property. Furthermore, we show that G has maximal graded Betti numbers among all arithmetically Gorenstein subschemes of projective space whose Artinian reduction has the weak Lefschetz property and the given Hilbert function. As an application we show that over a field of characteristic zero every set of simplicial polytopes with fixed h-vector contains a polytope with maximal graded Betti numbers.