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

Daniel Carando

Displaying similar documents to “Extendibility of polynomials and analytic functions on $ℓ_{p}$ ”

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.

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

(2012)

Acta Arithmetica

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The Daugavet equation for polynomials

Yun Sung Choi, Domingo García, Manuel Maestre, Miguel Martín (2007)

Studia Mathematica

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We study when the Daugavet equation is satisfied for weakly compact polynomials on a Banach space X, i.e. when the equality ||Id + P|| = 1 + ||P|| is satisfied for all weakly compact polynomials P: X → X. We show that this is the case when X = C(K), the real or complex space of continuous functions on a compact space K without isolated points. We also study the alternative Daugavet equation $m a x_{| ω | = 1} | | I d + ω P | | = 1 + | | P | |$ for polynomials P: X → X. We show that this equation holds for every polynomial on the complex...

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...

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...

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.

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 $F_{k} (D; x)$ .

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,...

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

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

Acta Arithmetica

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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.

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.

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.

Banach spaces of homogeneous polynomials without the approximation property

Seán Dineen, Jorge Mujica (2015)

Czechoslovak Mathematical Journal

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We present simple proofs that spaces of homogeneous polynomials on $L_{p} [0, 1]$ and $ℓ_{p}$ provide plenty of natural examples of Banach spaces without the approximation property. By giving necessary and sufficient conditions, our results bring to completion, at least for an important collection of Banach spaces, a circle of results begun in 1976 by R. Aron and M. Schottenloher (1976).

Nonreciprocal algebraic numbers of small Mahler's measure

Artūras Dubickas, Jonas Jankauskas (2013)

Acta Arithmetica

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We prove that there exist at least cd⁵ monic irreducible nonreciprocal polynomials with integer coefficients of degree at most d whose Mahler measures are smaller than 2, where c is some absolute positive constant. These polynomials are constructed as nonreciprocal divisors of some Newman hexanomials $1 + x^{r ₁} + \dots + x^{r ₅}$ , where the integers 1 ≤ r₁ < ⋯ < r₅ ≤ d satisfy some restrictions including $2 r_{j} < r_{j + 1}$ for j = 1,2,3,4. This result improves the previous lower bound cd³ and seems to be closer to the correct...

On highly nonintegrable functions and homogeneous polynomials

P. Wojtaszczyk (1997)

Annales Polonici Mathematici

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We construct a sequence of homogeneous polynomials on the unit ball $_{d}$ in $ℂ^{d}$ which are big at each point of the unit sphere . As an application we construct a holomorphic function on $_{d}$ which is not integrable with any power on the intersection of $_{d}$ with any complex subspace.

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

D. P. Dryanov (1989)

Banach Center Publications

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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...

Displaying similar documents to “Extendibility of polynomials and analytic functions on ℓ p ”

Displaying similar documents to “Extendibility of polynomials and analytic functions on $ℓ_{p}$ ”