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

A. Schinzel; W. M. Schmidt

Displaying similar documents to “Comparison of L¹- and $L^{\infty}$ -norms of squares of polynomials”

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.

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.

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...

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.

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.

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 sets of polynomials whose difference set contains no squares

Thái Hoàng Lê, Yu-Ru Liu (2013)

Acta Arithmetica

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Let $_{q} [t]$ be the polynomial ring over the finite field $_{q}$ , and let $_{N}$ be the subset of $_{q} [t]$ containing all polynomials of degree strictly less than N. Define D(N) to be the maximal cardinality of a set $A \subseteq_{N}$ for which A-A contains no squares of polynomials. By combining the polynomial Hardy-Littlewood circle method with the density increment technology developed by Pintz, Steiger and Szemerédi, we prove that $D (N) ≪ q^{N} {(l o g N)}^{7} / N$ .

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.

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...

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.

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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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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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.

Factorization of irreducible polynomials over a finite field with the substitution $x^{(} q^{r}) - x$ for x

Andrew Long (1973)

Acta Arithmetica

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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].

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.

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.

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...

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

Sum of squares and the Łojasiewicz exponent at infinity

Krzysztof Kurdyka, Beata Osińska-Ulrych, Grzegorz Skalski, Stanisław Spodzieja (2014)

Annales Polonici Mathematici

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Let V ⊂ ℝⁿ, n ≥ 2, be an unbounded algebraic set defined by a system of polynomial equations $h ₁ (x) = \dots = h_{r} (x) = 0$ and let f: ℝⁿ→ ℝ be a polynomial. It is known that if f is positive on V then ${f |}_{V}$ extends to a positive polynomial on the ambient space ℝⁿ, provided V is a variety. We give a constructive proof of this fact for an arbitrary algebraic set V. Precisely, if f is positive on V then there exists a polynomial $h (x) = \sum_{i = 1}^{r} h ²_{i} (x) σ_{i} (x)$ , where $σ_{i}$ are sums of squares of polynomials of degree at most p, such that f(x) + h(x) >...

The $V_{a}$ -deformation of the classical convolution

Anna Dorota Krystek (2007)

Banach Center Publications

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We study deformations of the classical convolution. For every invertible transformation T:μ ↦ Tμ, we are able to define a new associative convolution of measures by $μ *_{T} ν = T^{- 1} (T μ * T ν)$ . We deal with the $V_{a}$ -deformation of the classical convolution. We prove the analogue of the classical Lévy-Khintchine formula. We also show the central limit measure, which turns out to be the standard Gaussian measure. Moreover, we calculate the Poisson measure in the $V_{a}$ -deformed classical convolution and give the orthogonal...

On a theorem of Saeki concerning convolution squares of singular measures

Thomas Körner (2008)

Bulletin de la Société Mathématique de France

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If $1 > α > 1 / 2$ , then there exists a probability measure $μ$ such that the Hausdorff dimension of the support of $μ$ is $α$ and $μ * μ$ is a Lipschitz function of class $α - \frac{1}{2}$ .

Displaying similar documents to “Comparison of L¹- and L ∞ -norms of squares of polynomials”

Displaying similar documents to “Comparison of L¹- and $L^{\infty}$ -norms of squares of polynomials”