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

Daniel Carando; Damián Pinasco; Jorge Tomás Rodríguez

Displaying similar documents to “Lower bounds for norms of products of polynomials on $L_{p}$ spaces”

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...

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.

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,...

Estimates for polynomials in the unit disk with varying constant terms

Stephan Ruscheweyh, Magdalena Wołoszkiewicz (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let $∥ \cdot ∥$ be the uniform norm in the unit disk. We study the quantities $M_{n} (α) : = inf (∥ z P (z) + α ∥ - α)$ where the infimum is taken over all polynomials $P$ of degree $n - 1$ with $∥ P (z) ∥ = 1$ and $α > 0$ . In a recent paper by Fournier, Letac and Ruscheweyh (Math. Nachrichten 283 (2010), 193-199) it was shown that ${inf}_{α > 0} M_{n} (α) = 1 / n$ . We find the exact values of $M_{n} (α)$ and determine corresponding extremal polynomials. The method applied uses known cases of maximal ranges of polynomials.

A Green's function for θ-incomplete polynomials

Joe Callaghan (2007)

Annales Polonici Mathematici

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Let K be any subset of $ℂ^{N}$ . We define a pluricomplex Green’s function $V_{K, θ}$ for θ-incomplete polynomials. We establish properties of $V_{K, θ}$ analogous to those of the weighted pluricomplex Green’s function. When K is a regular compact subset of $ℝ^{N}$ , we show that every continuous function that can be approximated uniformly on K by θ-incomplete polynomials, must vanish on $K ∖ s u p p {(d d^{c} V_{K, θ})}^{N}$ . We prove a version of Siciak’s theorem and a comparison theorem for θ-incomplete polynomials. We compute $s u p p {(d d^{c} V_{K, θ})}^{N}$ when K is a compact...

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...

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.

Some results on derangement polynomials

Mehdi Hassani, Hossein Moshtagh, Mohammad Ghorbani (2022)

Commentationes Mathematicae Universitatis Carolinae

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We study moments of the difference $D_{n} (x) - x^{n} n! e^{- 1 / x}$ concerning derangement polynomials $D_{n} (x)$ . For the first moment, we obtain an explicit formula in terms of the exponential integral function and we show that it is always negative for $x > 0$ . For the higher moments, we obtain a multiple integral representation of the order of the moment under computation.

Inequalities for two sine polynomials

Horst Alzer, Stamatis Koumandos (2006)

Colloquium Mathematicae

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We prove: (I) For all integers n ≥ 2 and real numbers x ∈ (0,π) we have $α \leq \sum_{j = 1}^{n - 1} 1 / (n ² - j ²) s i n (j x) \leq β$ , with the best possible constant bounds α = (15-√2073)/10240 √(1998-10√2073) = -0.1171..., β = 1/3. (II) The inequality $0 < \sum_{j = 1}^{n - 1} (n ² - j ²) s i n (j x)$ holds for all even integers n ≥ 2 and x ∈ (0,π), and also for all odd integers n ≥ 3 and x ∈ (0,π - π/n].

The algebra of polynomials on the space of ultradifferentiable functions

Katarzyna Grasela (2010)

Banach Center Publications

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We consider the space $^{ℳ}$ of ultradifferentiable functions with compact supports and the space of polynomials on $^{ℳ}$ . A description of the space $(^{ℳ})$ of polynomial ultradistributions as a locally convex direct sum is given.

A generalisation of Amitsur's A-polynomials

Adam Owen, Susanne Pumplün (2021)

Communications in Mathematics

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We find examples of polynomials $f \in D [t; σ, δ]$ whose eigenring $ℰ (f)$ is a central simple algebra over the field $F = C \cap Fix (σ) \cap Const (δ)$ .

On the lattice of polynomials with integer coefficients: the covering radius in $L_{p} (0, 1)$

Wojciech Banaszczyk, Artur Lipnicki (2015)

Annales Polonici Mathematici

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The paper deals with the approximation by polynomials with integer coefficients in $L_{p} (0, 1)$ , 1 ≤ p ≤ ∞. Let $P_{n, r}$ be the space of polynomials of degree ≤ n which are divisible by the polynomial $x^{r} {(1 - x)}^{r}$ , r ≥ 0, and let $P_{n, r}^{ℤ} \subset P_{n, r}$ be the set of polynomials with integer coefficients. Let $μ (P_{n, r}^{ℤ}; L_{p})$ be the maximal distance of elements of $P_{n, r}$ from $P_{n, r}^{ℤ}$ in $L_{p} (0, 1)$ . We give rather precise quantitative estimates of $μ (P_{n, r}^{ℤ}; L ₂)$ for n ≳ 6r. Then we obtain similar, somewhat less precise, estimates of $μ (P_{n, r}^{ℤ}; L_{p})$ for p ≠ 2. It follows that $μ (P_{n, r}^{ℤ}; L_{p}) ≍ n^{- 2 r - 2 / p}$ as n → ∞. The results...

On the value set of small families of polynomials over a finite field, II

Guillermo Matera, Mariana Pérez, Melina Privitelli (2014)

Acta Arithmetica

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We obtain an estimate on the average cardinality (d,s,a) of the value set of any family of monic polynomials in $_{q} [T]$ of degree d for which s consecutive coefficients $a = (a_{d - 1}, . . ., a_{d - s})$ are fixed. Our estimate asserts that $(d, s, a) = μ_{d} q + (q^{1 / 2})$ , where $μ_{d} : = \sum_{r = 1}^{d} ({(- 1)}^{r - 1}) / (r!)$ . We also prove that $₂ (d, s, a) = μ ²_{d} q ² + (q^{3 / 2})$ , where ₂(d,s,a) is the average second moment of the value set cardinalities for any family of monic polynomials of $_{q} [T]$ of degree d with s consecutive coefficients fixed as above. Finally, we show that $₂ (d, 0) = μ ²_{d} q ² + (q)$ , where ₂(d,0) denotes the average second moment for...

The factorization of $f (x) x^{n} + g (x)$ with $f (x)$ monic and of degree $\leq 2$ .

Joshua Harrington, Andrew Vincent, Daniel White (2013)

Journal de Théorie des Nombres de Bordeaux

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In this paper we investigate the factorization of the polynomials $f (x) x^{n} + g (x) \in ℤ [x]$ in the special case where $f (x)$ is a monic quadratic polynomial with negative discriminant. We also mention similar results in the case that $f (x)$ is monic and linear.

Some hypergeometric polynomials associated with the Lauricella function $F_{D}$ of several variables. II

L. Carlitz, H. M. Srivastava (1976)

Matematički Vesnik

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Some hypergeometric polynomials associated with the Lauricella function $F_{D}$ of several variables. I

L. Carlitz, H. M. Srivastava (1976)

Matematički Vesnik

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The multiplicity of the zero at 1 of polynomials with constrained coefficients

Peter Borwein, Tamás Erdélyi, Géza Kós (2013)

Acta Arithmetica

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For n ∈ ℕ, L > 0, and p ≥ 1 let $κ_{p} (n, L)$ be the largest possible value of k for which there is a polynomial P ≠ 0 of the form $P (x) = \sum_{j = 0}^{n} a_{j} x^{j}$ , $| a_{0} | \geq L (\sum_{j = 1}^{n} | a_{j} |^{p}$ 1/p $,$ aj ∈ ℂ $,$ such that ${(x - 1)}^{k}$ divides P(x). For n ∈ ℕ and L > 0 let $κ_{\infty} (n, L)$ be the largest possible value of k for which there is a polynomial P ≠ 0 of the form $P (x) = \sum_{j = 0}^{n} a_{j} x^{j}$ , $| a_{0} | \geq L m a x_{1 \leq j \leq n} | a_{j} |$ , $a_{j} \in ℂ$ , such that ${(x - 1)}^{k}$ divides P(x). We prove that there are absolute constants c₁ > 0 and c₂ > 0 such that $c_{1} \sqrt (n / L) - 1 \leq κ_{\infty} (n, L) \leq c_{2} \sqrt (n / L)$ for every L ≥ 1. This complements an earlier result of the authors valid for every n ∈ ℕ and L ∈...

The norm of the polynomial truncation operator on the unit disk and on [-1,1]

Tamás Erdélyi (2001)

Colloquium Mathematicae

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Let D and ∂D denote the open unit disk and the unit circle of the complex plane, respectively. We denote by ₙ (resp. $ₙ^{c}$ ) the set of all polynomials of degree at most n with real (resp. complex) coefficients. We define the truncation operators Sₙ for polynomials $P ₙ \in ₙ^{c}$ of the form $P ₙ (z) : = \sum_{j = 0}^{n} a_{j} z_{j}$ , $a_{j} \in C$ , by $S ₙ (P ₙ) (z) : = \sum_{j = 0}^{n} a ̃_{j} z_{j}$ , $a ̃_{j} : = a_{j} | a_{j} | m i n | a_{j} |, 1$ (here 0/0 is interpreted as 1). We define the norms of the truncation operators by $∥ S ₙ ∥_{\infty, \partial D}^{r e a l} : = s u p_{P ₙ \in ₙ} (m a x_{z \in \partial D} | S ₙ (P ₙ) (z) |) / (m a x_{z \in \partial D} | P ₙ (z) |)$ , $∥ S ₙ ∥_{\infty, \partial D}^{c o m p} : = s u p_{P ₙ \in ₙ^{c}} (m a x_{z \in \partial D} | S ₙ (P ₙ) (z) |) / (m a x_{z \in \partial D} | P ₙ (z) |$ . Our main theorem establishes the right order of magnitude of the above norms: there is an absolute constant c₁...

Approximation by weighted polynomials in $ℝ^{k}$

Maritza M. Branker (2005)

Annales Polonici Mathematici

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We apply pluripotential theory to establish results in $ℝ^{k}$ concerning uniform approximation by functions of the form wⁿPₙ where w denotes a continuous nonnegative function and Pₙ is a polynomial of degree at most n. Then we use our work to show that on the intersection of compact sections $Σ \subset ℝ^{k}$ a continuous function on Σ is uniformly approximable by θ-incomplete polynomials (for a fixed θ, 0 < θ < 1) iff f vanishes on θ²Σ. The class of sets Σ expressible as the intersection of compact...

Explicit bounds for the Łojasiewicz exponent in the gradient inequality for polynomials

Didier D&#039;Acunto, Krzysztof Kurdyka (2005)

Annales Polonici Mathematici

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Let f: ℝⁿ → ℝ be a polynomial function of degree d with f(0) = 0 and ∇f(0) = 0. Łojasiewicz’s gradient inequality states that there exist C > 0 and ϱ ∈ (0,1) such that ${| \nabla f | \geq C | f |}^{ϱ}$ in a neighbourhood of the origin. We prove that the smallest such exponent ϱ is not greater than $1 - R {(n, d)}^{- 1}$ with $R (n, d) = d {(3 d - 3)}^{n - 1}$ .

Type and cotype of operator spaces

Hun Hee Lee (2008)

Studia Mathematica

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We consider two operator space versions of type and cotype, namely $S_{p}$ -type, $S_{q}$ -cotype and type (p,H), cotype (q,H) for a homogeneous Hilbertian operator space H and 1 ≤ p ≤ 2 ≤ q ≤ ∞, generalizing “OH-cotype 2” of G. Pisier. We compute type and cotype of some Hilbertian operator spaces and $L_{p}$ spaces, and we investigate the relationship between a homogeneous Hilbertian space H and operator spaces with cotype (2,H). As applications we consider operator space versions of generalized little...

Characterization of functions whose forward differences are exponential polynomials

J. M. Almira (2017)

Commentationes Mathematicae Universitatis Carolinae

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Given ${h_{1}, \dots, h_{t}}$ a finite subset of $ℝ^{d}$ , we study the continuous complex valued functions and the Schwartz complex valued distributions $f$ defined on $ℝ^{d}$ with the property that the forward differences $Δ_{h_{k}}^{m_{k}} f$ are (in distributional sense) continuous exponential polynomials for some natural numbers $m_{1}, \dots, m_{t}$ .

On some properties of Chebyshev polynomials

Hacène Belbachir, Farid Bencherif (2008)

Discussiones Mathematicae - General Algebra and Applications

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Letting $T_{n}$ (resp. $U_{n}$ ) be the n-th Chebyshev polynomials of the first (resp. second) kind, we prove that the sequences ${(X^{k} T_{n - k})}_{k}$ and ${(X^{k} U_{n - k})}_{k}$ for n - 2⎣n/2⎦ ≤ k ≤ n - ⎣n/2⎦ are two basis of the ℚ-vectorial space $_{n} [X]$ formed by the polynomials of ℚ[X] having the same parity as n and of degree ≤ n. Also $T_{n}$ and $U_{n}$ admit remarkableness integer coordinates on each of the two basis.

Representations of the general linear group over symmetry classes of polynomials

Yousef Zamani, Mahin Ranjbari (2018)

Czechoslovak Mathematical Journal

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Let $V$ be the complex vector space of homogeneous linear polynomials in the variables $x_{1}, ..., x_{m}$ . Suppose $G$ is a subgroup of $S_{m}$ , and $χ$ is an irreducible character of $G$ . Let $H_{d} (G, χ)$ be the symmetry class of polynomials of degree $d$ with respect to $G$ and $χ$ . For any linear operator $T$ acting on $V$ , there is a (unique) induced operator $K_{χ} (T) \in End (H_{d} (G, χ))$ acting on symmetrized decomposable polynomials by $K_{χ} (T) (f_{1} * f_{2} * ... * f_{d}) = T f_{1} * T f_{2} * ... * T f_{d} .$ In this paper, we show that the representation $T \mapsto K_{χ} (T)$ of the general linear group $G L (V)$ is equivalent to the direct sum of $χ (1)$ copies...

Displaying similar documents to “Lower bounds for norms of products of polynomials on L p spaces”

Displaying similar documents to “Lower bounds for norms of products of polynomials on $L_{p}$ spaces”