Note on the multidimensional Gebelein inequality
Marek Beśka, Agnieszka Wałachowska (2013)
Applicationes Mathematicae
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We generalize the Gebelein inequality for Gaussian random vectors in .
Marek Beśka, Agnieszka Wałachowska (2013)
Applicationes Mathematicae
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We generalize the Gebelein inequality for Gaussian random vectors in .
Z. Sadlok, Z. Tyc (1977)
Annales Polonici Mathematici
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L. Carlitz, H. M. Srivastava (1976)
Matematički Vesnik
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L. Carlitz, H. M. Srivastava (1976)
Matematički Vesnik
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Stanislaw Lewanowicz (2002)
Applicationes Mathematicae
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Let 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 in . A systematic use of the basic properties (including some nonstandard ones) of the polynomials results in obtaining a low order of the recurrence.
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 -orthogonal polynomials, which are reduced to the orthogonal Meixner polynomials when . The underlying algebraic framework allowed a systematic...
Joe Callaghan (2007)
Annales Polonici Mathematici
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Let K be any subset of . We define a pluricomplex Green’s function for θ-incomplete polynomials. We establish properties of analogous to those of the weighted pluricomplex Green’s function. When K is a regular compact subset of , we show that every continuous function that can be approximated uniformly on K by θ-incomplete polynomials, must vanish on . We prove a version of Siciak’s theorem and a comparison theorem for θ-incomplete polynomials. We compute when K is a compact...
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 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.
Marek Bożejko (2007)
Banach Center Publications
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In this paper we give a construction of operators satisfying q-CCR relations for q > 1: and also q-CAR relations for q < -1: , where N is the number operator on a suitable Fock space acting as Nx₁ ⊗ ⋯ ⊗ xₙ = nx₁ ⊗ ⋯ ⊗xₙ. Some applications to combinatorial problems are also given.
Stephan Ruscheweyh, Magdalena Wołoszkiewicz (2011)
Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica
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Let be the uniform norm in the unit disk. We study the quantities where the infimum is taken over all polynomials of degree with and . In a recent paper by Fournier, Letac and Ruscheweyh (Math. Nachrichten 283 (2010), 193-199) it was shown that . We find the exact values of and determine corresponding extremal polynomials. The method applied uses known cases of maximal ranges of polynomials.
Katarzyna Grasela (2010)
Banach Center Publications
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We consider the space of ultradifferentiable functions with compact supports and the space of polynomials on . A description of the space of polynomial ultradistributions as a locally convex direct sum is given.
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 on the bi-circle, which can be factored as where is a polynomial nonzero for and . The conditions are in terms of recurrence coefficients associated with the polynomials in lexicographical and reverse lexicographical ordering orthogonal with respect to the weight on the bi-circle. We use this result to describe how specific factorizations of weights on the bi-circle can be translated into identities...
Marek Bożejko, Gero Fendler (2006)
Banach Center Publications
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We show that for the t-deformed semicircle measure, where 1/2 < t ≤ 1, the expansions of functions with respect to the associated orthonormal polynomials converge in norm when 3/2 < p < 3 and do not converge when 1 ≤ p < 3/2 or 3 < p. From this we conclude that natural expansions in the non-commutative spaces of free group factors and of free commutation relations do not converge for 1 ≤ p < 3/2 or 3 < p.
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 in the special case where is a monic quadratic polynomial with negative discriminant. We also mention similar results in the case that is monic and linear.
Özer Öztürk (2010)
Annales Polonici Mathematici
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We give a closed formula for the Thom polynomials of the singularities 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.
Maritza M. Branker (2005)
Annales Polonici Mathematici
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We apply pluripotential theory to establish results in 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 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...
Wojciech Banaszczyk, Artur Lipnicki (2015)
Annales Polonici Mathematici
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The paper deals with the approximation by polynomials with integer coefficients in , 1 ≤ p ≤ ∞. Let be the space of polynomials of degree ≤ n which are divisible by the polynomial , r ≥ 0, and let be the set of polynomials with integer coefficients. Let be the maximal distance of elements of from in . We give rather precise quantitative estimates of for n ≳ 6r. Then we obtain similar, somewhat less precise, estimates of for p ≠ 2. It follows that as n → ∞. The results...
Michael Voit (2011)
Colloquium Mathematicae
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Let p,q be positive integers. The groups and act on the Heisenberg group canonically as groups of automorphisms, where is the vector space of all complex p × q matrices. The associated orbit spaces may be identified with and respectively, being the cone of positive semidefinite matrices and the Weyl chamber . In this paper we compute the associated convolutions on and explicitly, depending on p. Moreover, we extend these convolutions by analytic continuation to series...
Giacomo Gigante (2010)
Colloquium Mathematicae
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Let be a sequence of arbitrary complex numbers, let α,β > -1, let Pₙα,βn=0+∞ , . Then, for any non-negative integer n, . When , this formula reduces to Bateman’s expansion for Bessel functions. For particular values of y and n one obtains generalizations of several formulas already known for Bessel functions, like Sonine’s first and second finite integrals and certain Neumann series expansions. Particular choices of allow one to write all these type of formulas...
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 Radon projections, consisting of k parallel X-ray beams in each direction , k = 0, ..., n, determines uniquely algebraic polynomials of degree n in two variables.
Pradipto Banerjee, Michael Filaseta, Carrie E. Finch, J. Russell Leidy (2013)
Journal de Théorie des Nombres de Bordeaux
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We show that the discriminant of the generalized Laguerre polynomial is a non-zero square for some integer pair , with , if and only if belongs to one of explicitly given infinite sets of pairs or to an additional finite set of pairs. As a consequence, we obtain new information on when the Galois group of over is the alternating group . For example, we establish that for all but finitely many positive integers , the only for which the Galois group of over is is...
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 , 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 . For p > 2 we present some estimates on the constants involved.
Laura G. DeMarco, Curtis T. McMullen (2008)
Annales scientifiques de l'École Normale Supérieure
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In this paper we study branched coverings of metrized, simplicial trees which arise from polynomial maps with disconnected Julia sets. We show that the collection of all such trees, up to scale, forms a contractible space compactifying the moduli space of polynomials of degree ; that records the asymptotic behavior of the multipliers of ; and that any meromorphic family of polynomials over can be completed by a unique tree at its central fiber. In the cubic case we give a...
Claudia Garetto (2010)
Banach Center Publications
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We provide a deep investigation of the notions of - and -hypoellipticity for partial differential operators with constant Colombeau coefficients. This involves generalized polynomials and fundamental solutions in the dual of a Colombeau algebra. Sufficient conditions and necessary conditions for - and -hypoellipticity are given
Iwona Naraniecka, Jan Szynal, Anna Tatarczak (2013)
Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica
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The extremal functions realizing the maxima of some functionals (e.g. , and ) within the so-called universal linearly invariant family (in the sense of Pommerenke [10]) have such a form that looks similar to generating function for Meixner-Pollaczek (MP) polynomials [2], [8]. This fact gives motivation for the definition and study of the generalized Meixner-Pollaczek (GMP) polynomials of a real variable as coefficients of where the parameters , , satisfy the conditions:...