Smoothness of the Green function for a special domain

Serkan Celik; Alexander Goncharov

Displaying similar documents to “Smoothness of the Green function for a special domain”

Moments of measure orthogonalizing the 2-dimensional Chebyshev polynomials

Łukasz Jan Wojakowski (2006)

Banach Center Publications

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We calculate the moments $m_{k, 0}$ of the measure orthogonalizing the 2-dimensional Chebyshev polynomials introduced by Koornwinder.

Chebyshev polynomials and Pell equations over finite fields

Boaz Cohen (2021)

Czechoslovak Mathematical Journal

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We shall describe how to construct a fundamental solution for the Pell equation $x^{2} - m y^{2} = 1$ over finite fields of characteristic $p \neq 2$ . Especially, a complete description of the structure of these fundamental solutions will be given using Chebyshev polynomials. Furthermore, we shall describe the structure of the solutions of the general Pell equation $x^{2} - m y^{2} = n$ .

A non-Chebyshev finite-dimensional subspace in $H_{1}$

S. B. Stechkin (1989)

Banach Center Publications

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Sign changes in $π_{q, a} (x) - π_{q, b} (x)$

Kevin Ford, Richard H. Hudson (2001)

Acta Arithmetica

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Chebyshev Distance

Roland Coghetto (2016)

Formalized Mathematics

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In [21], Marco Riccardi formalized that ℝN-basis n is a basis (in the algebraic sense defined in [26]) of [...] ℰTn $ℰ_{T}^{n}$ and in [20] he has formalized that [...] ℰTn $ℰ_{T}^{n}$ is second-countable, we build (in the topological sense defined in [23]) a denumerable base of [...] ℰTn $ℰ_{T}^{n}$ . Then we introduce the n-dimensional intervals (interval in n-dimensional Euclidean space, pavé (borné) de ℝn [16], semi-intervalle (borné) de ℝn [22]). We conclude with the definition of Chebyshev distance [11]. ...

Optimality of Chebyshev bounds for Beurling generalized numbers

Harold G. Diamond, Wen-Bin Zhang (2013)

Acta Arithmetica

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If the counting function N(x) of integers of a Beurling generalized number system satisfies both $\int_{1}^{\infty} x^{- 2} | N (x) - A x | d x < \infty$ and $x^{- 1} (l o g x) (N (x) - A x) = O (1)$ , then the counting function π(x) of the primes of this system is known to satisfy the Chebyshev bound π(x) ≪ x/logx. Let f(x) increase to infinity arbitrarily slowly. We give a construction showing that $\int_{1}^{\infty} | N (x) - A x | x^{- 2} d x < \infty$ and $x^{- 1} (l o g x) (N (x) - A x) = O (f (x))$ do not imply the Chebyshev bound.

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.

Transfinite diameter, Chebyshev constants, and capacities in ℂⁿ

Vyacheslav Zakharyuta (2012)

Annales Polonici Mathematici

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The famous result of geometric complex analysis, due to Fekete and Szegö, states that the transfinite diameter d(K), characterizing the asymptotic size of K, the Chebyshev constant τ(K), characterizing the minimal uniform deviation of a monic polynomial on K, and the capacity c(K), describing the asymptotic behavior of the Green function $g_{K} (z)$ at infinity, coincide. In this paper we give a survey of results on multidimensional notions of transfinite diameter, Chebyshev constants and capacities,...

Chebyshev series expansions of the functions $J_{v} (k x) / {(k x)}^{v}$ and $I_{v} (k x) / {(k x)}^{v}$

Z. Cylkowski (1966)

Applicationes Mathematicae

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Explicit extension maps in intersections of non-quasi-analytic classes

Jean Schmets, Manuel Valdivia (2005)

Annales Polonici Mathematici

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We deal with projective limits of classes of functions and prove that: (a) the Chebyshev polynomials constitute an absolute Schauder basis of the nuclear Fréchet spaces $_{()} ({[- 1, 1]}^{r})$ ; (b) there is no continuous linear extension map from $Λ_{()}^{(r)}$ into $_{()} (ℝ^{r})$ ; (c) under some additional assumption on , there is an explicit extension map from $_{()} ({[- 1, 1]}^{r})$ into $_{()} ({[- 2, 2]}^{r})$ by use of a modification of the Chebyshev polynomials. These results extend the corresponding ones obtained by Beaugendre in [1] and [2].

Chebyshev bounds for Beurling numbers

Harold G. Diamond, Wen-Bin Zhang (2013)

Acta Arithmetica

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The first author conjectured that Chebyshev-type prime bounds hold for Beurling generalized numbers provided that the counting function N(x) of the generalized integers satisfies the L¹ condition $\int_{1}^{\infty} | N (x) - A x | d x / x^{2} < \infty$ for some positive constant A. This conjecture was shown false by an example of Kahane. Here we establish the Chebyshev bounds using the L¹ hypothesis and a second integral condition.

The transfinite diameter of the real ball and simplex

T. Bloom, L. Bos, N. Levenberg (2012)

Annales Polonici Mathematici

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We calculate the transfinite diameter for the real unit ball $B_{d} : = x \in ℝ^{d} : | x | \leq 1$ and the real unit simplex $T_{d} : = x \in ℝ_{+}^{d} : \sum_{j = 1}^{d} x_{j} \leq 1 .$

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.

Sign changes of certain arithmetical function at prime powers

Rishabh Agnihotri, Kalyan Chakraborty (2021)

Czechoslovak Mathematical Journal

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We examine an arithmetical function defined by recursion relations on the sequence ${f (p^{k})}_{k \in ℕ}$ and obtain sufficient condition(s) for the sequence to change sign infinitely often. As an application we give criteria for infinitely many sign changes of Chebyshev polynomials and that of sequence formed by the Fourier coefficients of a cusp form.

Kolmogorov problem in $W^{r} H^{ω} [0, 1]$ and extremal Zolotarev ω-splines

Bagdasarov Sergey K.

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AbstractThe main result of the paper, based on the Borsuk Antipodality Theorem, describes extremal functions of the Kolmogorov-Landau problem(*) $f^{(m)} (ξ) \to s u p$ , $f \in W^{r} H^{ω} [ξ, b]$ , ${| | f | |}_{ℂ [a, b]} \leq B$ ,for all 0 < m ≤ r, ξ ≤ a or ξ = (a+b)/2, all B > 0 and concave moduli of continuity ω on ℝ₊. It is shown that any extremal function $=_{B, r, m, ω, ξ}$ of the problem (*) enjoys the following two characteristic properties. First, the function $^{(r)} (\cdot) -^{(r)} (ξ)$ is extremal for the problem(**) $\int_{ξ}^{b} h (t) ψ (t) d t \to s u p$ , $h \in H^{ω} [ξ, b]$ , h(ξ) = 0,for an appropriate choice of the kernel ψ with a finite...

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...

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.

An Elementary Proof of the Exponential Conditioning of Real Vandermonde Matrices

Stefano Serra Capizzano (2007)

Bollettino dell'Unione Matematica Italiana

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We provide and discuss an elementary proof of the exponential con- ditioning of real Vandermonde matrices which can be easily given in undergraduate courses: we exclusively use the definition of conditioning and the sup-norm formula on $[-1,1]$ for Chebyshev polynomials of first kind. The same proof idea works virtually unchanged for the famous Hilbert matrix.

Nilakantha's accelerated series for π

David Brink (2015)

Acta Arithmetica

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We show how the idea behind a formula for π discovered by the Indian mathematician and astronomer Nilakantha (1445-1545) can be developed into a general series acceleration technique which, when applied to the Gregory-Leibniz series, gives the formula $π = \sum_{n = 0}^{\infty} ((5 n + 3) n! (2 n)!) / (2^{n - 1} (3 n + 2)!)$ with convergence as $13 . 5^{- n}$ , in much the same way as the Euler transformation gives $π = \sum_{n = 0}^{\infty} (2^{n + 1} n! n!) / (2 n + 1)!$ with convergence as $2^{- n}$ . Similar transformations lead to other accelerated series for π, including three “BBP-like” formulas, all of which are collected in...

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 $F = C \cap Fix (σ) \cap Const (δ)$ .

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.

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...

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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