Exact Kronecker constants of Hadamard sets

Kathryn E. Hare; L. Thomas Ramsey

Displaying similar documents to “Exact Kronecker constants of Hadamard sets”

Lagrange approximation in Banach spaces

Lisa Nilsson, Damián Pinasco, Ignacio M. Zalduendo (2015)

Czechoslovak Mathematical Journal

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Starting from Lagrange interpolation of the exponential function $e^{z}$ in the complex plane, and using an integral representation formula for holomorphic functions on Banach spaces, we obtain Lagrange interpolating polynomials for representable functions defined on a Banach space $E$ . Given such a representable entire funtion $f : E \to ℂ$ , in order to study the approximation problem and the uniform convergence of these polynomials to $f$ on bounded sets of $E$ , we present a sufficient growth condition on...

Interpolation of Cesàro sequence and function spaces

Sergey V. Astashkin, Lech Maligranda (2013)

Studia Mathematica

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The interpolation properties of Cesàro sequence and function spaces are investigated. It is shown that $C e s_{p} (I)$ is an interpolation space between $C e s_{p ₀} (I)$ and $C e s_{p ₁} (I)$ for 1 < p₀ < p₁ ≤ ∞ and 1/p = (1 - θ)/p₀ + θ/p₁ with 0 < θ < 1, where I = [0,∞) or [0,1]. The same result is true for Cesàro sequence spaces. On the other hand, $C e s_{p} [0, 1]$ is not an interpolation space between Ces₁[0,1] and $C e s_{\infty} [0, 1]$ .

Interpolation of quasicontinuous functions

Joan Cerdà, Joaquim Martín, Pilar Silvestre (2011)

Banach Center Publications

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If C is a capacity on a measurable space, we prove that the restriction of the K-functional $K (t, f; L^{p} (C), L^{\infty} (C))$ to quasicontinuous functions f ∈ QC is equivalent to $K (t, f; L^{p} (C) \cap Q C, L^{\infty} (C) \cap Q C)$ . We apply this result to identify the interpolation space ${(L^{p ₀, q ₀} (C) \cap Q C, L^{p ₁, q ₁} (C) \cap Q C)}_{θ, q}$ .

The Lizorkin-Freitag formula for several weighted $L_{p}$ spaces and vector-valued interpolation

Irina Asekritova, Natan Krugljak, Ludmila Nikolova (2005)

Studia Mathematica

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A complete description of the real interpolation space $L = {(L_{p ₀} (ω ₀), . . ., L_{p ₙ} (ω ₙ))}_{θ ⃗, q}$ is given. An interesting feature of the result is that the whole measure space (Ω,μ) can be divided into disjoint pieces $Ω_{i}$ (i ∈ I) such that L is an $l_{q}$ sum of the restrictions of L to $Ω_{i}$ , and L on each $Ω_{i}$ is a result of interpolation of just two weighted $L_{p}$ spaces. The proof is based on a generalization of some recent results of the first two authors concerning real interpolation of vector-valued spaces.

Modulus of dentability in $L ¹ + L^{\infty}$

Adam Bohonos, Ryszard Płuciennik (2008)

Banach Center Publications

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We introduce the notion of the modulus of dentability defined for any point of the unit sphere S(X) of a Banach space X. We calculate effectively this modulus for denting points of the unit ball of the classical interpolation space $L ¹ + L^{\infty} .$ Moreover, a criterion for denting points of the unit ball in this space is given. We also show that none of denting points of the unit ball of $L ¹ + L^{\infty}$ is a LUR-point. Consequently, the set of LUR-points of the unit ball of $L ¹ + L^{\infty}$ is empty.

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...

Real method of interpolation on subcouples of codimension one

S. V. Astashkin, P. Sunehag (2008)

Studia Mathematica

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We find necessary and sufficient conditions under which the norms of the interpolation spaces ${(N ₀, N ₁)}_{θ, q}$ and ${(X ₀, X ₁)}_{θ, q}$ are equivalent on N, where N is the kernel of a nonzero functional ψ ∈ (X₀ ∩ X₁)* and $N_{i}$ is the normed space N with the norm inherited from $X_{i}$ (i = 0,1). Our proof is based on reducing the problem to its partial case studied by Ivanov and Kalton, where ψ is bounded on one of the endpoint spaces. As an application we completely resolve the problem of when the range of the operator $T_{θ} = S - 2^{θ} I$ (S...

Complex interpolation of function spaces with general weights

Douadi Drihem (2023)

Commentationes Mathematicae Universitatis Carolinae

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We present the complex interpolation of Besov and Triebel–Lizorkin spaces with generalized smoothness. In some particular cases these function spaces are just weighted Besov and Triebel–Lizorkin spaces. As a corollary of our results, we obtain the complex interpolation between the weighted Triebel–Lizorkin spaces ${\dot{F}}_{p_{0}, q_{0}}^{s_{0}} (ω_{0})$ and ${\dot{F}}_{\infty, q_{1}}^{s_{1}} (ω_{1})$ with suitable assumptions on the parameters $s_{0}, s_{1}, p_{0}, q_{0}$ and $q_{1}$ , and the pair of weights $(ω_{0}, ω_{1})$ .

Convergence of greedy approximation II. The trigonometric system

S. V. Konyagin, V. N. Temlyakov (2003)

Studia Mathematica

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We study the following nonlinear method of approximation by trigonometric polynomials. For a periodic function f we take as an approximant a trigonometric polynomial of the form $G ₘ (f) : = \sum_{k \in Λ} f ̂ (k) e^{i (k, x)}$ , where $Λ \subset ℤ^{d}$ is a set of cardinality m containing the indices of the m largest (in absolute value) Fourier coefficients f̂(k) of the function f. Note that Gₘ(f) gives the best m-term approximant in the L₂-norm, and therefore, for each f ∈ L₂, ||f-Gₘ(f)||₂ → 0 as m → ∞. It is known from previous results that in...

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].

Embeddings of Besov spaces of logarithmic smoothness

Fernando Cobos, Óscar Domínguez (2014)

Studia Mathematica

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This paper deals with Besov spaces of logarithmic smoothness $B_{p, r}^{0, b}$ formed by periodic functions. We study embeddings of $B_{p, r}^{0, b}$ into Lorentz-Zygmund spaces $L_{p, q} {(l o g L)}_{β}$ . Our techniques rely on the approximation structure of $B_{p, r}^{0, b}$ , Nikol’skiĭ type inequalities, extrapolation properties of $L_{p, q} {(l o g L)}_{β}$ and interpolation.

A recursive interpolation method in $R^{n}$

M. Ajzerle (1992)

Matematički Vesnik

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Several notes on the circumradius condition

Václav Kučera (2016)

Applications of Mathematics

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Recently, the so-called circumradius condition (or estimate) was derived, which is a new estimate of the $W^{1, p}$ -error of linear Lagrange interpolation on triangles in terms of their circumradius. The published proofs of the estimate are rather technical and do not allow clear, simple insight into the results. In this paper, we give a simple direct proof of the $p = \infty$ case. This allows us to make several observations such as on the optimality of the circumradius estimate. Furthermore, we show how...

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.

$H^{\infty}$ functional calculus in real interpolation spaces

Giovanni Dore (1999)

Studia Mathematica

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Let A be a linear closed densely defined operator in a complex Banach space X. If A is of type ω (i.e. the spectrum of A is contained in a sector of angle 2ω, symmetric around the real positive axis, and $∥ λ {(λ I - A)}^{- 1} ∥$ is bounded outside every larger sector) and has a bounded inverse, then A has a bounded $H^{\infty}$ functional calculus in the real interpolation spaces between X and the domain of the operator itself.

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...

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...

Interpolation theorem for the p-harmonic transform

Luigi D&#039;Onofrio, Tadeusz Iwaniec (2003)

Studia Mathematica

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We establish an interpolation theorem for a class of nonlinear operators in the Lebesgue spaces $ℒ^{s} (ℝ ⁿ)$ arising naturally in the study of elliptic PDEs. The prototype of those PDEs is the second order p-harmonic equation ${d i v | \nabla u |}^{p - 2 \nabla} u = d i v$ . In this example the p-harmonic transform is essentially inverse to $d i v (| \nabla |^{p - 2} \nabla)$ . To every vector field $\in ℒ^{q} (ℝ ⁿ, ℝ ⁿ)$ our operator $ℋ_{p}$ assigns the gradient of the solution, $ℋ_{p} = \nabla u \in ℒ^{p} (ℝ ⁿ, ℝ ⁿ)$ . The core of the matter is that we go beyond the natural domain of definition of this operator. Because of nonlinearity our...

Approximation of sets defined by polynomials with holomorphic coefficients

Marcin Bilski (2012)

Annales Polonici Mathematici

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Let X be an analytic set defined by polynomials whose coefficients $a ₁, . . ., a_{s}$ are holomorphic functions. We formulate conditions on sequences $a_{1, ν}, . . ., a_{s, ν}$ of holomorphic functions converging locally uniformly to $a ₁, . . ., a_{s}$ , respectively, such that the sequence $X_{ν}$ of sets obtained by replacing $a_{j}$ ’s by $a_{j, ν}$ ’s in the polynomials converges to X.

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.

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

D. P. Dryanov (1989)

Banach Center Publications

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Interpolation and extension of Lipschitz-Hölder maps on $C_{p}$ spaces

Charles E. C Cleaver (1973)

Colloquium Mathematicae

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$L_{\infty}$ -Convergence of Finite Element Approximation

J. A. Nitsche (1975)

Publications mathématiques et informatique de Rennes

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Structure of Cesàro function spaces: a survey

Sergey V. Astashkin, Lech Maligranda (2014)

Banach Center Publications

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Geometric structure of Cesàro function spaces $C e s_{p} (I)$ , where I = [0,1] and [0,∞), is investigated. Among other matters we present a description of their dual spaces, characterize the sets of all q ∈ [1,∞] such that $C e s_{p} [0, 1]$ contains isomorphic and complemented copies of $l_{q}$ -spaces, show that Cesàro function spaces fail the fixed point property, give a description of subspaces generated by Rademacher functions in spaces $C e s_{p} [0, 1]$ .

On some properties of generalized Marcinkiewicz spaces

Evgeniy Pustylnik (2001)

Studia Mathematica

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We give a full solution of the following problems concerning the spaces $M_{φ} (X ⃗)$ : (i) to what extent two functions φ and ψ should be different in order to ensure that $M_{φ} (X ⃗) \neq M_{ψ} (X ⃗)$ for any nontrivial Banach couple X⃗; (ii) when an embedding $M_{φ} (X ⃗) ⊊ M_{ψ} (X ⃗)$ can (or cannot) be dense; (iii) which Banach space can be regarded as an $M_{φ} (X ⃗)$ -space for some (unknown beforehand) Banach couple X⃗.

Interpolation by bivariate polynomials based on Radon projections

B. Bojanov, I. K. Georgieva (2004)

Studia Mathematica

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For any given set of angles θ₀ < ... < θₙ in [0,π), we show that a set of $(\binom{n + 2}{2})$ Radon projections, consisting of k parallel X-ray beams in each direction $θ_{k}$ , k = 0, ..., n, determines uniquely algebraic polynomials of degree n in two variables.

$H^{\infty}$ functional calculus in real interpolation spaces, II

Giovanni Dore (2001)

Studia Mathematica

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Let A be a linear closed one-to-one operator in a complex Banach space X, having dense domain and dense range. If A is of type ω (i.e.the spectrum of A is contained in a sector of angle 2ω, symmetric about the real positive axis, and ${| | λ (λ I - A)}^{- 1} | |$ is bounded outside every larger sector), then A has a bounded $H^{\infty}$ functional calculus in the real interpolation spaces between X and the intersection of the domain and the range of the operator itself.

Correction to “Walsh equiconvergence for best $l_{2}$ approximates” Studia Math.. 77 (1984), 523-528

A. Sharma, Z. Ziegler (1987)

Studia Mathematica

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