The Daugavet equation for polynomials

Yun Sung Choi; Domingo García; Manuel Maestre; Miguel Martín

Displaying similar documents to “The Daugavet equation for polynomials”

Extendibility of polynomials and analytic functions on $ℓ_{p}$

Daniel Carando (2001)

Studia Mathematica

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We prove that extendible 2-homogeneous polynomials on spaces with cotype 2 are integral. This allows us to find examples of approximable non-extendible polynomials on $ℓ_{p}$ (1 ≤ p < ∞ ) of any degree. We also exhibit non-nuclear extendible polynomials for 4 < p < ∞. We study the extendibility of analytic functions on Banach spaces and show the existence of functions of infinite radius of convergence whose coefficients are finite type polynomials but which fail to be extendible. ...

Upper bounds for the $L_{q}$ norm of Fekete polynomials on subarcs

(2012)

Acta Arithmetica

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A formula for Jack polynomials of the second order

Francisco J. Caro-Lopera, José A. Díaz-García, Graciela González-Farías (2007)

Applicationes Mathematicae

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This work solves the partial differential equation for Jack polynomials $C_{κ}^{α}$ of the second order. When the parameter α of the solution takes the values 1/2, 1 and 2 we get explicit formulas for the quaternionic, complex and real zonal polynomials of the second order, respectively.

Cotype and absolutely summing homogeneous polynomials in $ℒ_{p}$ spaces

Daniel Pellegrino (2003)

Studia Mathematica

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We lift to homogeneous polynomials and multilinear mappings a linear result due to Lindenstrauss and Pełczyński for absolutely summing operators. We explore the notion of cotype to obtain stronger results and provide various examples of situations in which the space of absolutely summing homogeneous polynomials is different from the whole space of homogeneous polynomials. Among other consequences, these results enable us to obtain answers to some open questions about absolutely summing...

Deformed Heisenberg algebra with reflection and $d$ -orthogonal polynomials

Fethi Bouzeffour, Hanen Ben Mansour, Ali Zaghouani (2017)

Czechoslovak Mathematical Journal

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This paper is devoted to the study of matrix elements of irreducible representations of the enveloping deformed Heisenberg algebra with reflection, motivated by recurrence relations satisfied by hypergeometric functions. It is shown that the matrix elements of a suitable operator given as a product of exponential functions are expressed in terms of $d$ -orthogonal polynomials, which are reduced to the orthogonal Meixner polynomials when $d = 1$ . The underlying algebraic framework allowed a systematic...

Extension and lifting of weakly continuous polynomials

Raffaella Cilia, Joaquín M. Gutiérrez (2005)

Studia Mathematica

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We show that a Banach space X is an ℒ₁-space (respectively, an $ℒ_{\infty}$ -space) if and only if it has the lifting (respectively, the extension) property for polynomials which are weakly continuous on bounded sets. We also prove that X is an ℒ₁-space if and only if the space $_{w b} (^{m} X)$ of m-homogeneous scalar-valued polynomials on X which are weakly continuous on bounded sets is an $ℒ_{\infty}$ -space.

Quadratic polynomials, period polynomials, and Hecke operators

Marie Jameson, Wissam Raji (2013)

Acta Arithmetica

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For any non-square 1 < D ≡ 0,1 (mod 4), Zagier defined $F_{k} (D; x) : = \sum_{\begin{matrix} a, b, c \in ℤ, a < 0 \\ b^{2} - 4 a c = D \end{matrix}} m a x (0, {(a x^{2} + b x + c)}^{k - 1})$ . Here we use the theory of periods to give identities and congruences which relate various values of $F_{k} (D; x)$ .

Existence and reduction of generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials

Luis M. Navas, Francisco J. Ruiz, Juan L. Varona (2019)

Archivum Mathematicum

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One can find in the mathematical literature many recent papers studying the generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials, defined by means of generating functions. In this article we clarify the range of parameters in which these definitions are valid and when they provide essentially different families of polynomials. In particular, we show that, up to multiplicative constants, it is enough to take as the “main family” those given by ${(\frac{2}{λ e^{t} + 1})}^{α} e^{x t} = \sum_{n = 0}^{\infty} ℰ_{n}^{(α)} (x; λ) \frac{t^{n}}{n!}, λ \in ℂ ∖ {- 1},$ and as an “exceptional...

Thom polynomials and Schur functions: the singularities $I I I_{2, 3} (-)$

Özer Öztürk (2010)

Annales Polonici Mathematici

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We give a closed formula for the Thom polynomials of the singularities $I I I_{2, 3} (-)$ in terms of Schur functions. Our computations combine the characterization of the Thom polynomials via the “method of restriction equations” of Rimányi et al. with the techniques of Schur functions.

Unconditionality for m-homogeneous polynomials on $ℓ ⁿ_{\infty}$

Andreas Defant, Pablo Sevilla-Peris (2016)

Studia Mathematica

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Let χ(m,n) be the unconditional basis constant of the monomial basis $z^{α}$ , α ∈ ℕ₀ⁿ with |α| = m, of the Banach space of all m-homogeneous polynomials in n complex variables, endowed with the supremum norm on the n-dimensional unit polydisc ⁿ. We prove that the quotient of $s u p_{m} \sqrt[m]{s u p_{m} χ (m, n)}$ and √(n/log n) tends to 1 as n → ∞. This reflects a quite precise dependence of χ(m,n) on the degree m of the polynomials and their number n of variables. Moreover, we give an analogous formula for m-linear forms, a...

Comparison of L¹- and $L^{\infty}$ -norms of squares of polynomials

A. Schinzel, W. M. Schmidt (2002)

Acta Arithmetica

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An Inequality for Trigonometric Polynomials

N. K. Govil, Mohammed A. Qazi, Qazi I. Rahman (2012)

Bulletin of the Polish Academy of Sciences. Mathematics

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The main result says in particular that if $t (ζ) : = \sum_{ν = - n} ⁿ c_{ν} e^{i ν ζ}$ is a trigonometric polynomial of degree n having all its zeros in the open upper half-plane such that |t(ξ)| ≥ μ on the real axis and cₙ ≠ 0, then |t’(ξ)| ≥ μn for all real ξ.

Lower bounds for norms of products of polynomials on $L_{p}$ spaces

Daniel Carando, Damián Pinasco, Jorge Tomás Rodríguez (2013)

Studia Mathematica

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For 1 < p < 2 we obtain sharp lower bounds for the uniform norm of products of homogeneous polynomials on $L_{p} (μ)$ , whenever the number of factors is no greater than the dimension of these Banach spaces (a condition readily satisfied in infinite-dimensional settings). The result also holds for the Schatten classes $_{p}$ . For p > 2 we present some estimates on the constants involved.

The image of multilinear polynomials evaluated on $3 \times 3$ upper triangular matrices

Thiago Castilho de Mello (2021)

Communications in Mathematics

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We describe the images of multilinear polynomials of arbitrary degree evaluated on the $3 \times 3$ upper triangular matrix algebra over an infinite field.

A note on $q$ -partial difference equations and some applications to generating functions and $q$ -integrals

Da-Wei Niu, Jian Cao (2019)

Czechoslovak Mathematical Journal

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We study the condition on expanding an analytic several variables function in terms of products of the homogeneous generalized Al-Salam-Carlitz polynomials. As applications, we deduce bilinear generating functions for the homogeneous generalized Al-Salam-Carlitz polynomials. We also gain multilinear generating functions for the homogeneous generalized Al-Salam-Carlitz polynomials. Moreover, we obtain generalizations of Andrews-Askey integrals and Ramanujan $q$ -beta integrals. At last,...

Recurrences for the coefficients of series expansions with respect to classical orthogonal polynomials

Stanislaw Lewanowicz (2002)

Applicationes Mathematicae

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Let $P_{k}$ be any sequence of classical orthogonal polynomials. Further, let f be a function satisfying a linear differential equation with polynomial coefficients. We give an algorithm to construct, in a compact form, a recurrence relation satisfied by the coefficients $a_{k}$ in $f = \sum_{k} a_{k} P_{k}$ . A systematic use of the basic properties (including some nonstandard ones) of the polynomials $P_{k}$ results in obtaining a low order of the recurrence.

Symmetric identity for polynomial sequences satisfying $A_{n + 1}^{'} (x) = (n + 1) A_{n} (x)$

Farid Bencherif, Rachid Boumahdi, Tarek Garici (2021)

Communications in Mathematics

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Using umbral calculus, we establish a symmetric identity for any sequence of polynomials satisfying $A_{n + 1}^{'} (x) = (n + 1) A_{n} (x)$ with $A_{0} (x)$ a constant polynomial. This identity allows us to obtain in a simple way some known relations involving Apostol-Bernoulli polynomials, ApostolEuler polynomials and generalized Bernoulli polynomials attached to a primitive Dirichlet character.

M-ideals of homogeneous polynomials

Verónica Dimant (2011)

Studia Mathematica

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We study the problem of whether $_{w} (ⁿ E)$ , the space of n-homogeneous polynomials which are weakly continuous on bounded sets, is an M-ideal in the space (ⁿE) of continuous n-homogeneous polynomials. We obtain conditions that ensure this fact and present some examples. We prove that if $_{w} (ⁿ E)$ is an M-ideal in (ⁿE), then $_{w} (ⁿ E)$ coincides with $_{w 0} (ⁿ E)$ (n-homogeneous polynomials that are weakly continuous on bounded sets at 0). We introduce a polynomial version of property (M) and derive that if $_{w} (ⁿ E) =_{w 0} (ⁿ E)$ and (E) is an...

One-sided approximation by trigonometric polynomials in $L_{p}$ -norm, 0

D. P. Dryanov (1989)

Banach Center Publications

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On certain expansions in series of polynomials of incomplete $β$ - functions.

В. Романовский ([unknown])

Matematiceskij sbornik

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Linearization of Arbitrary products of classical orthogonal polynomials

Mahouton Hounkonnou, Said Belmehdi, André Ronveaux (2000)

Applicationes Mathematicae

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A procedure is proposed in order to expand $w = \prod_{j = 1}^{N} P_{i_{j}} (x) = \sum_{k = 0}^{M} L_{k} P_{k} (x)$ where $P_{i} (x)$ belongs to aclassical orthogonal polynomial sequence (Jacobi, Bessel, Laguerre and Hermite) ( $M = \sum_{j = 1}^{N} i_{j}$ ). We first derive a linear differential equation of order $2^{N}$ satisfied by w, fromwhich we deduce a recurrence relation in k for the linearizationcoefficients $L_{k}$ . We develop in detail the two cases ${[P_{i} (x)]}^{N}$ , $P_{i} (x) P_{j} (x) P_{k} (x)$ and give the recurrencerelation in some cases (N=3,4), when the polynomials $P_{i} (x)$ are monic Hermite orthogonal polynomials.