On Runge approximation in a Banach space

We show that on some open sets, more general than balls, Runge approximation is possible in certain Banach spaces, and also in certain complex Banach manifolds. We also show that there is an entire holomorphic curve in Hilbert space on which there is a bounded holomorphic function on the trace of a ball that has no bounded holomorphic extension to even a smaller concentric ball. Using the same technique we also prove that a form of Runge approximation better than an error function is not always possible.     

Weight enumerators,intersection enumerators,and Jacobi polynomials

The intersection enumerator and the Jacobi polynomial in an arbitrary genus for a binary code are introduced. Adding the weight enumerator into our discussion, we give the explicit relations among them and give some of their basic properties.     

On Jacobi and continuous Hahn polynomials

Jacobi polynomials are mapped onto the continuous Hahn polynomials by the Fourier transform, and the orthogonality relations for the continuous Hahn polynomials then follow from the orthogonality relations for the Jacobi polynomials and the Parseval formula. In a special case this relation dates back to work by Bateman in 1933 and we follow a part of the historical development for these polynomials. Some applications of this relation are given.

     

Bases in spaces of homogeneous polynomials on Banach spaces

In this work we present some conditions of equivalence for the existence of a monomial basis in spaces of homogeneous polynomials on Banach spaces.     

On a new geometric property for Banach spaces

In this paper we study a geometric property for Banach spaces called condition (*), introduced by de Reynaet al in [3], A Banach space has this property if for any weakly null sequencex _n of unit vectors inX, ifx _* ⁿ is any sequence of unit vectors inX ^* that attain their norm at x_n’s, then . We show that a Banach space satisfies condition (*) for all equivalent norms iff the space has the Schur property. We also study two related geometric conditions, one of which is useful in calculating the essential norm of an operator.     

A Banach space with a countable infinite number of complex structures

We give examples of real Banach spaces with exactly infinite countably many complex structures and with _ω1${\omega }_1$ many complex structures.     

On the summability of weighted Lagrange interpolation on the roots of Jacobi polynomials

The aim of this paper is to investigate the summability of weighted Lagrange interpolation on the roots of Jacobi polynomials. Starting from the Lagrange interpolation polynomials, we shall construct a wide class of discrete processes which are uniformly convergent in a suitable weighted space of continuous functions. Error estimates for the approximation will also be considered. This revised version was published online in June 2006 with corrections to the Cover Date.     

On conjectured inequalities for zeros of Jacobi polynomials

Inequalities for the largest zero of Jacobi polynomials, conjectured recently by us and in joint work with P. Leopardi, are here extended to all zeros of Jacobi polynomials, and new relevant conjectures are formulated based on extensive computation.      

On a conjectured inequality of Gautschi and Leopardi for Jacobi polynomials

Motivated by work on positive cubature formulae over the spherical surface, Gautschi and Leopardi conjectured that the inequality holds for α,β > − 1 and n ≥ 1, θ ∈ (0, π), where are the Jacobi polynomials of degree n and parameters (α, β). We settle this conjecture in the special cases where .      

Ideals of polynomials between Banach spaces revisited

Ideals of polynomials and multilinear operators between Banach spaces have been exhaustively investigated in the last decades. In this paper, we introduce a unified (and more general) approach and propose some lines of investigation in this new framework. Among other results, we prove a Bohnenblust–Hille inequality in this more general setting.     

A class of matrix-valued polynomials generalizing Jacobi polynomials

A hierarchy of matrix-valued polynomials which generalize the Jacobi polynomials is found. Defined by a Rodrigues formula, they are also products of a sequence of differential operators. Each class of polynomials is complete, satisfies a three term recurrence relation, integral inter-relations, and weak orthogonality relations.     

On non-fragmentability of Banach spaces

In this paper, we use the game characterization of Kenderov and Moors [11] to construct an example of a non-fragmentable Banach space. More precisely, we will show that ifX is the tree-complete Banach algebra of Haydon and Zizler [3], (X/c ₀, weak) is not fragmentable by any metric. In particular, this shows thatX/c ₀ cannot be equivalently renormed to be rotund.     

Jacobi polynomials in Bernstein form

The paper describes a method to compute a basis of mutually orthogonal polynomials with respect to an arbitrary Jacobi weight on the simplex. This construction takes place entirely in terms of the coefficients with respect to the so-called Bernstein–Bézier form of a polynomial.     

On Uniformly Finitely Extensible Banach spaces

We continue the study of Uniformly Finitely Extensible Banach spaces (in short, UFO) initiated in Moreno and Plichko (2009) [39] and Castillo and Plichko (2010) [18]. We show that they have the Uniform Approximation Property of Pe?czyński and Rosenthal and are compactly extensible. We will also consider their connection with the automorphic space problem of Lindenstrauss and Rosenthal – do there exist automorphic spaces other than _c0(I)$c_0(I)$ and _?2(I)${?}_2(I)$? – showing that a space all whose subspaces are UFO must be automorphic when it is Hereditarily Indecomposable (HI), and a Hilbert space when it is either locally minimal or isomorphic to its square. We will finally show that most HI – among them, the super-reflexive HI space constructed by Ferenczi – and asymptotically _?2${?}_2$ spaces in the literature cannot be automorphic.     

Some generating functions of Jacobi polynomials

In this paper, Weisner’s group-theoretic method of obtaining generating functions is utilized in the study of Jacobi polynomialsP> _n ^(a,ß)(x) by giving suitable interpretations to the index (n) and the parameter (β) to find out the elements for constructing a six-dimensional Lie algebra.     

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.