Upper bounds for the $L_q$ norm of Fekete polynomials on subarcs

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Displaying similar documents to “Upper bounds for the $L_{q}$ norm of Fekete polynomials on subarcs”

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.

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. ...

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...

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...

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

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.

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.

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.

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...

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...

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

D. P. Dryanov (1989)

Banach Center Publications

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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.

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

В. Романовский ([unknown])

Matematiceskij sbornik

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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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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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Rademacher-Carlitz polynomials

Matthias Beck, Florian Kohl (2014)

Acta Arithmetica

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We introduce and study the Rademacher-Carlitz polynomial $R (u, v, s, t, a, b) : = \sum_{k = ⌈ s ⌉}^{⌈ s ⌉ + b - 1} u^{⌊ (k a + t) ⌋} / b_{v^{k}}$ where $a, b \in ℤ_{> 0}$ , s,t ∈ ℝ, and u and v are variables. These polynomials generalize and unify various Dedekind-like sums and polynomials; most naturally, one may view R(u,v,s,t,a,b) as a polynomial analogue (in the sense of Carlitz) of the Dedekind-Rademacher sum $r_{t} (a, b) : = \sum_{k = 0}^{b - 1} (((k a + t) / b)) ((k / b))$ , which appears in various number-theoretic, combinatorial, geometric, and computational contexts. Our results come in three flavors: we prove a reciprocity theorem for Rademacher-Carlitz...

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...

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].

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...

Fejér–Riesz factorizations and the structure of bivariate polynomials orthogonal on the bi-circle

Jeffrey S. Geronimo, Plamen Iliev (2014)

Journal of the European Mathematical Society

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We give a complete characterization of the positive trigonometric polynomials $Q (θ, ϕ)$ on the bi-circle, which can be factored as $Q (θ, ϕ) = | p (e^{i θ}, e^{i ϕ}) |^{2}$ where $p (z, w)$ is a polynomial nonzero for $| z | = 1$ and $| w | \leq 1$ . The conditions are in terms of recurrence coefficients associated with the polynomials in lexicographical and reverse lexicographical ordering orthogonal with respect to the weight $\frac{1}{4 π^{2} Q (θ, ϕ)}$ on the bi-circle. We use this result to describe how specific factorizations of weights on the bi-circle can be translated into identities...

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...

Displaying similar documents to “Upper bounds for the L q norm of Fekete polynomials on subarcs”

Displaying similar documents to “Upper bounds for the $L_{q}$ norm of Fekete polynomials on subarcs”