Polynomials and multilinear mappings in topological modules

Criteria for the equicontinuity of sets of multilinear mappings between topological modules are studied, as well as topological modules of continuous multilinear mappings. As a consequence, criteria for the equicontinuity of sets of homogeneous polynomials between topological modules are also studied, as well as topological modules of continuous homogeneous polynomials.     

Bounded and unbounded polynomials and multilinear forms: Characterizing continuity

In this paper we prove a characterization of continuity for polynomials on a normed space. Namely, we prove that a polynomial is continuous if and only if it maps compact sets into compact sets. We also provide a partial answer to the question as to whether a polynomial is continuous if and only if it transforms connected sets into connected sets. These results motivate the natural question as to how many non-continuous polynomials there are on an infinite dimensional normed space. A problem on the lineability of the sets of non-continuous polynomials and multilinear mappings on infinite dimensional normed spaces is answered.     

On factorization of Hilbert-Schmidt mappings

We present some results on factorization of Hilbert-Schmidt multilinear mappings and polynomials through infinite dimensional Banach spaces, L₁ and L_∞ spaces. We conclude this work with a result on factorization of holomorphic mappings of Hilbert-Schmidt type.     

Summability and estimates for polynomials and multilinear mappings

In this paper we extend and generalize several known estimates for homogeneous polynomials and multilinear mappings on Banach spaces. Applying the theory of absolutely summing nonlinear mappings, we prove that estimates which are known for mappings on ?_p spaces in fact hold true for mappings on arbitrary Banach spaces.     

Almost summing mappings

We introduce a general definition of almost p-summing mappings and give several concrete examples of such mappings. Some known results are considerably generalized and we present various situations in which the space of almost p-summing multilinear mappings coincides with the whole space of continuous multilinear mappings. Received: 17 June 2002     

The Laguerre polynomials in several variables

In this paper, we give some relations between multivariable Laguerre polynomials and other well-known multivariable polynomials. We get various families of multilinear and multilateral generating functions for these polynomials. Some special cases are also presented.     

Multilinear differential operators on modular forms

We construct multilinear differential operators on modular forms and prove that they are essentially unique. We also discuss certain homogeneous polynomials associated to such differential operators as well as some related multilinear differential operators that do not produce modular forms.

     

On the convergence of random polynomials and multilinear forms

We consider different kinds of convergence of homogeneous polynomials and multilinear forms in random variables. We show that for a variety of complex random variables, the almost sure convergence of the polynomial is equivalent to that of the multilinear form, and to the square summability of the coefficients. Also, we present polynomial Khintchine inequalities for complex gaussian and Steinhaus variables. All these results have no analogues in the real case. Moreover, we study the Lp-convergence of random polynomials and derive certain decoupling inequalities without the usual tetrahedral hypothesis. We also consider convergence on “full subspaces” in the sense of Sjögren, both for real and complex random variables, and relate it to domination properties of the polynomial or the multilinear form, establishing a link with the theory of homogeneous polynomials on Banach spaces.     

On a multivariable extension for the extended Jacobi polynomials

The main object of this paper is to construct a systematic investigation of a multivariable extension of the extended Jacobi polynomials and give some relations for these polynomials. We derive various families of multilinear and multilateral generating functions. We also obtain relations between the polynomials extended Jacobi polynomials and some other well-known polynomials. Other miscellaneous properties of these general families of multivariable polynomials are also discussed. Furthermore, some special cases of the results are presented in this study.     

Multilinear extensions of absolutely (p;q;r)-summing operators

In this paper we introduce and study a new class containing the class of absolutely summing multilinear mappings, which we call absolutely (p;q ₁,…,q _m ;r)-summing multilinear mappings. We investigate some interesting properties concerning the absolutely (p;q ₁,…,q _m ;r)-summing m-linear mappings defined on Banach spaces. In particular, we prove a kind of Pietsch’s Domination Theorem and a multilinear version of the Factorization Theorem.     

On a multivariable extension of the Humbert polynomials

In this paper, we present a multivariable extension of the Humbert polynomials, which is motivated by the Chan-Chyan-Srivastava multivariable polynomials, the multivariable extension of the familiar Lagrange-Hermite polynomials and Erkus-Srivastava multivariable polynomials. We derive various families of multilinear and mixed multilateral generating functions for these polynomials. Other miscellaneous properties of these multivariable polynomials are also discussed. Some special cases of the results presented in this study are also indicated.     

Diagonal multilinear operators on Köthe sequence spaces

We analyse the interplay between maximal/minimal/adjoint ideals of multilinear operators (between sequence spaces) and their associated Köthe sequence spaces. We establish relationships with spaces of multipliers and apply these results to describe diagonal multilinear operators from Lorentz sequence spaces. We also define and study some properties of the ideal of (E, p)-summing multilinear mappings, a natural extension of the linear ideal of absolutely (E, p)-summing operators.     

Summability of multilinear mappings: Littlewood,Orlicz and beyond

In this paper we prove new results concerning summability properties of multilinear mappings between Banach spaces, such as an extension of Littlewood’s 4/3 Theorem. The role of the Littlewood–Orlicz property in the theory is established, especially in the question of determining when multilinear mappings are (1; 2, . . . , 2)-summing.     

Factorization of injective ideals by interpolation

We construct a factorization of certain multilinear mappings through linear operators belonging to closed, injective operator ideals using interpolation technique. An extension of the duality theorem for interpolation spaces is also obtained.     

Summability of multilinear mappings: Littlewood, Orlicz and beyond

In this paper we prove new results concerning summability properties of multilinear mappings between Banach spaces, such as an extension of Littlewood’s 4/3 Theorem. The role of the Littlewood–Orlicz property in the theory is established, especially in the question of determining when multilinear mappings are (1; 2, . . . , 2)-summing.     

A nonlinear Pietsch domination theorem

Pietsch’s domination theorem, which is known for linear, multilinear and polynomial mappings, is extended to a larger class of nonlinear mappings.     

Bornologically isomorphic representations of distributions on manifolds

Distributional tensor fields can be regarded as multilinear mappings on smooth tensor fields with distributional values or as (classical) tensor fields with distributional coefficients. We show that the corresponding isomorphisms hold also in the bornological setting.     

Nicodemi sequences of operators between spaces of multilinear mappings

We obtain results on three aspects of Nicodemi extensions of multilinear mappings between Banach spaces: (i) subspace invariance, (ii) the norms of the extension operators, (iii) when Aron–Berner extensions are Nicodemi extensions.     

Norm or Numerical Radius Attaining Multilinear Mappings and Polynomials

We study the denseness or norm of numerical radius attainingmultilinear mappings and polynomials between Banach spaces,and examine the relations between norms and numerical radiiof such mappings.     

Critical points and values of complex polynomials

This paper is primarily concerned with complex polynomials which have critical points which are also fixed points. We show that certain perturbations of a critical fixed point satisfy an inequality. This inequality permits us to prove a local version of Smale's mean value conjecture. We also use Thurston's topological characterization of critically finite rational mappings to enumerate explicitly as branched mappings the set of complex polynomials which have all their critical points fixed.