Chebyshev series expansions of the functions and
Z. Cylkowski (1966)
Applicationes Mathematicae
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Z. Cylkowski (1966)
Applicationes Mathematicae
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Hacène Belbachir, Farid Bencherif (2008)
Discussiones Mathematicae - General Algebra and Applications
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Letting (resp. ) be the n-th Chebyshev polynomials of the first (resp. second) kind, we prove that the sequences and for n - 2⎣n/2⎦ ≤ k ≤ n - ⎣n/2⎦ are two basis of the ℚ-vectorial space formed by the polynomials of ℚ[X] having the same parity as n and of degree ≤ n. Also and admit remarkableness integer coordinates on each of the two basis.
T. Alden Gassert (2014)
Journal de Théorie des Nombres de Bordeaux
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Let be any integer and fix an odd prime . Let denote the -fold composition of the Chebyshev polynomial of degree shifted by . If this polynomial is irreducible, let , where is a root of . We use a theorem of Dedekind in conjunction with previous results of the author to give conditions on that ensure is monogenic. For other values of , we apply a result of Guàrdia, Montes, and Nart to obtain a formula for the discriminant of and compute an integral basis for the ring...
Łukasz Piasecki (2014)
Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica
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The aim of this paper is to show that for every Banach space containing asymptotically isometric copy of the space there is a bounded, closed and convex set with the Chebyshev radius such that for every there exists a -contractive mapping with for any .
Jon Eivind Vatne (2017)
Applications of Mathematics
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Acute triangles are defined by having all angles less than , and are characterized as the triangles containing their circumcenter in the interior. For simplices of dimension , acuteness is defined by demanding that all dihedral angles between -dimensional faces are smaller than . However, there are, in a practical sense, too few acute simplices in general. This is unfortunate, since the acuteness property provides good qualitative features for finite element methods. The property...
Jiří Matoušek, Martin Tancer, Uli Wagner (2011)
Journal of the European Mathematical Society
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Let be the following algorithmic problem: Given a finite simplicial complex of dimension at most , does there exist a (piecewise linear) embedding of into ? Known results easily imply polynomiality of (; the case is graph planarity) and of for all . We show that the celebrated result of Novikov on the algorithmic unsolvability of recognizing the 5-sphere implies that and are undecidable for each . Our main result is NP-hardness of and, more generally, of for all...
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 ; (b) there is no continuous linear extension map from into ; (c) under some additional assumption on , there is an explicit extension map from into by use of a modification of the Chebyshev polynomials. These results extend the corresponding ones obtained by Beaugendre in [1] and [2].
Dušan Pokorný, Luděk Zajíček (2022)
Czechoslovak Mathematical Journal
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We give a complete characterization of closed sets whose distance function is DC (i.e., is the difference of two convex functions on ). Using this characterization, a number of properties of such sets is proved.
Brandts, Jan, Cihangir, Apo
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This paper is about -triangles, which are the simplest nontrivial examples of -polytopes: convex hulls of a subset of vertices of the unit -cube . We consider the subclasses of right -triangles, and acute -triangles, which only have acute angles. They can be explicitly counted and enumerated, also modulo the symmetries of .
Th. Friedrich (1974)
Colloquium Mathematicae
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Dalík, Josef
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The general method of averaging for the superapproximation of an arbitrary partial derivative of a smooth function in a vertex of a simplicial triangulation of a bounded polytopic domain in for any is described and its complexity is analysed.
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. ) the set of all polynomials of degree at most n with real (resp. complex) coefficients. We define the truncation operators Sₙ for polynomials of the form , , by , (here 0/0 is interpreted as 1). We define the norms of the truncation operators by , . Our main theorem establishes the right order of magnitude of the above norms: there is an absolute constant c₁...
Małgorzata Michalska, Maria Nowak, Paweł Sobolewski (2011)
Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica
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We give new characterizations of the analytic Besov spaces on the unit ball of in terms of oscillations and integral means over some Euclidian balls contained in .
Christian Samuel (2010)
Colloquium Mathematicae
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We show that every operator from to is compact when 1 ≤ p,q < s and that every operator from to is compact when 1/p + 1/q > 1 + 1/s.
Albert Baernstein II, Robert C. Culverhouse (2002)
Studia Mathematica
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Let , , where the are independent random vectors, each uniformly distributed on the unit sphere in ℝⁿ, and are real constants. We prove that if is majorized by in the sense of Hardy-Littlewood-Pólya, and if Φ: ℝⁿ → ℝ is continuous and bisubharmonic, then EΦ(X) ≤ EΦ(Y). Consequences include most of the known sharp Khinchin inequalities for sums of the form X. For radial Φ, bisubharmonicity is necessary as well as sufficient for the majorization inequality to always hold. Counterparts...
Krzysztof Kołodziejczyk (1987)
Colloquium Mathematicae
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Marco Cicalese, Gian Paolo Leonardi (2013)
Journal of the European Mathematical Society
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We prove some results in the context of isoperimetric inequalities with quantitative terms. In the -dimensional case, our main contribution is a method for determining the optimal coefficients in the inequality , valid for each Borel set with positive and finite area, with and being, respectively, the and the of . In dimensions, besides proving existence and regularity properties of minimizers for a wide class of including the lower semicontinuous extension of , we...
Kevin Ford, Richard H. Hudson (2001)
Acta Arithmetica
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Sandro Manfredini, Simona Settepanella (2014)
Annales de la faculté des sciences de Toulouse Mathématiques
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Let be the -th ordered configuration space of all distinct points in the Grassmannian of -dimensional subspaces of , whose sum is a subspace of dimension . We prove that is (when non empty) a complex submanifold of of dimension and its fundamental group is trivial if , and and equal to the braid group of the sphere if . Eventually we compute the fundamental group in the special case of hyperplane arrangements, i.e. .
Finbarr Holland, David Walsh (1995)
Studia Mathematica
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Let 1 < p < ∞, q = p/(p-1) and for define , x > 0. Moser’s Inequality states that there is a constant such that where is the unit ball of . Moreover, the value a = 1 is sharp. We observe that f where the integral operator has a simple kernel K. We consider the question of for what kernels K(t,x), 0 ≤ t, x < ∞, this result can be extended, and proceed to discuss this when K is non-negative and homogeneous of degree -1. A sufficient condition on K is found for...
Kuzman Adzievski (2006)
Annales Polonici Mathematici
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We study questions related to exceptional sets of pluri-Green potentials in the unit ball B of ℂⁿ in terms of non-isotropic Hausdorff capacity. For suitable measures μ on the ball B, the pluri-Green potentials are defined by , where for a fixed z ∈ B, denotes the holomorphic automorphism of B satisfying , and for every w ∈ B. If dμ(w) = f(w)dλ(w), where f is a non-negative measurable function of B, and λ is the measure on B, invariant under all holomorphic automorphisms of...
Gennadiy Averkov, Horst Martini (2009)
Colloquium Mathematicae
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Let be a d-dimensional normed space with norm ||·|| and let B be the unit ball in . Let us fix a Lebesgue measure in with . This measure will play the role of the volume in . We consider an arbitrary simplex T in with prescribed edge lengths. For the case d = 2, sharp upper and lower bounds of are determined. For d ≥ 3 it is noticed that the tight lower bound of is zero.
Binlong Li, Bo Ning (2014)
Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica
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Let and be two given graphs. The Ramsey number is the least integer such that for every graph on vertices, either contains a or contains a . Parsons gave a recursive formula to determine the values of , where is a path on vertices and is a star on vertices. In this note, we study the Ramsey numbers , where is a linear forest on vertices. We determine the exact values of for the cases and , and for the case that has no odd component. Moreover, we...
Jean Saint Raymond (2007)
Fundamenta Mathematicae
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Let X be a Polish space, and let C₀ and C₁ be disjoint coanalytic subsets of X. The pair (C₀,C₁) is said to be complete if for every pair (D₀,D₁) of disjoint coanalytic subsets of there exists a continuous function such that and . We give several explicit examples of complete pairs of coanalytic sets.
Hans Triebel (1994)
Studia Mathematica
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Let , where the sum is taken over the lattice of all points k in having integer-valued components, j∈ℕ and . Let be either or (s ∈ ℝ, 0 < p < ∞, 0 < q ≤ ∞) on The aim of the paper is to clarify under what conditions is equivalent to .