Displaying similar documents to “Correction to “Walsh equiconvergence for best l 2 approximates” Studia Math.. 77 (1984), 523-528”

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 · 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.

A Green's function for θ-incomplete polynomials

Joe Callaghan (2007)

Annales Polonici Mathematici

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Let K be any subset of 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 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...

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 P n , r be the set of polynomials with integer coefficients. Let μ ( P n , r ; L p ) be the maximal distance of elements of P n , r from P n , r in L p ( 0 , 1 ) . We give rather precise quantitative estimates of μ ( P n , r ; L ) for n ≳ 6r. Then we obtain similar, somewhat less precise, estimates of μ ( P n , r ; L p ) for p ≠ 2. It follows that μ ( P n , r ; L p ) n - 2 r - 2 / p as n → ∞. The results...

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 = 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.

The norm of the polynomial truncation operator on the unit disk and on [-1,1]

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. c ) the set of all polynomials of degree at most n with real (resp. complex) coefficients. We define the truncation operators Sₙ for polynomials P c of the form P ( z ) : = j = 0 n a j z j , a j C , by S ( P ) ( z ) : = j = 0 n a ̃ j z j , a ̃ j : = a j | a j | m i n | a j | , 1 (here 0/0 is interpreted as 1). We define the norms of the truncation operators by S , D r e a l : = s u p P ( m a x z D | S ( P ) ( z ) | ) / ( m a x z D | P ( z ) | ) , S , D c o m p : = s u p P c ( m a x z D | S ( P ) ( z ) | ) / ( m a x z D | P ( z ) | . Our main theorem establishes the right order of magnitude of the above norms: there is an absolute constant c₁...

The multiplicity of the zero at 1 of polynomials with constrained coefficients

Peter Borwein, Tamás Erdélyi, Géza Kós (2013)

Acta Arithmetica

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For n ∈ ℕ, L > 0, and p ≥ 1 let κ p ( n , L ) be the largest possible value of k for which there is a polynomial P ≠ 0 of the form P ( x ) = j = 0 n a j x j , | a 0 | L ( j = 1 n | a j | p 1/p , aj ∈ ℂ , such that ( x - 1 ) k divides P(x). For n ∈ ℕ and L > 0 let κ ( n , L ) be the largest possible value of k for which there is a polynomial P ≠ 0 of the form P ( x ) = j = 0 n a j x j , | a 0 | L m a x 1 j n | a j | , a j , such that ( x - 1 ) k divides P(x). We prove that there are absolute constants c₁ > 0 and c₂ > 0 such that c 1 ( n / L ) - 1 κ ( n , L ) c 2 ( n / L ) for every L ≥ 1. This complements an earlier result of the authors valid for every n ∈ ℕ and L ∈...

The factorization of f ( x ) x n + g ( x ) with f ( x ) monic and of degree 2 .

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 f ( x ) x n + g ( x ) [ x ] in the special case where f ( x ) is a monic quadratic polynomial with negative discriminant. We also mention similar results in the case that f ( x ) is monic and linear.

The algebra of polynomials on the space of ultradifferentiable functions

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.

On the value set of small families of polynomials over a finite field, II

Guillermo Matera, Mariana Pérez, Melina Privitelli (2014)

Acta Arithmetica

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We obtain an estimate on the average cardinality (d,s,a) of the value set of any family of monic polynomials in q [ T ] of degree d for which s consecutive coefficients a = ( a d - 1 , . . . , a d - s ) are fixed. Our estimate asserts that ( d , s , a ) = μ d q + ( q 1 / 2 ) , where μ d : = r = 1 d ( ( - 1 ) r - 1 ) / ( r ! ) . We also prove that ( d , s , a ) = μ ² d q ² + ( q 3 / 2 ) , where ₂(d,s,a) is the average second moment of the value set cardinalities for any family of monic polynomials of q [ T ] of degree d with s consecutive coefficients fixed as above. Finally, we show that ( d , 0 ) = μ ² d q ² + ( q ) , where ₂(d,0) denotes the average second moment for...

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 ( f ) : = k Λ f ̂ ( k ) e i ( k , x ) , where Λ 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...

Approximation by weighted polynomials in k

Maritza M. Branker (2005)

Annales Polonici Mathematici

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We apply pluripotential theory to establish results in 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 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...

Coppersmith-Rivlin type inequalities and the order of vanishing of polynomials at 1

(2016)

Acta Arithmetica

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For n ∈ ℕ, L > 0, and p ≥ 1 let κ p ( n , L ) be the largest possible value of k for which there is a polynomial P ≢ 0 of the form P ( x ) = j = 0 n a j x j , | a 0 | L ( j = 1 n | a j | p ) 1 / p , a j , such that ( x - 1 ) k divides P(x). For n ∈ ℕ, L > 0, and q ≥ 1 let μ q ( n , L ) be the smallest value of k for which there is a polynomial Q of degree k with complex coefficients such that | Q ( 0 ) | > 1 / L ( j = 1 n | Q ( j ) | q ) 1 / q . We find the size of κ p ( n , L ) and μ q ( n , L ) for all n ∈ ℕ, L > 0, and 1 ≤ p,q ≤ ∞. The result about μ ( n , L ) is due to Coppersmith and Rivlin, but our proof is completely different and much shorter even...

On the Gauss-Lucas'lemma in positive characteristic

Umberto Bartocci, Maria Cristina Vipera (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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If f ( x ) is a polynomial with coefficients in the field of complex numbers, of positive degree n , then f ( x ) has at least one root a with the following property: if μ k n , where μ is the multiplicity of α , then f ( k ) ( α ) 0 (such a root is said to be a "free" root of f ( x ) ). This is a consequence of the so-called Gauss-Lucas'lemma. One could conjecture that this property remains true for polynomials (of degree n ) with coefficients in a field of positive characteristic p > n (Sudbery's Conjecture). In this paper it...

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 , . . . , a s are holomorphic functions. We formulate conditions on sequences a 1 , ν , . . . , a s , ν of holomorphic functions converging locally uniformly to a , . . . , 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.

A transplantation theorem for ultraspherical polynomials at critical index

J. J. Guadalupe, V. I. Kolyada (2001)

Studia Mathematica

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We investigate the behaviour of Fourier coefficients with respect to the system of ultraspherical polynomials. This leads us to the study of the “boundary” Lorentz space λ corresponding to the left endpoint of the mean convergence interval. The ultraspherical coefficients c ( λ ) ( f ) of λ -functions turn out to behave like the Fourier coefficients of functions in the real Hardy space ReH¹. Namely, we prove that for any f λ the series n = 1 c ( λ ) ( f ) c o s n θ is the Fourier series of some function φ ∈ ReH¹ with | | φ | | R e H ¹ c | | f | | λ . ...

Calculation of the greatest common divisor of perturbed polynomials

Zítko, Jan, Eliaš, Ján

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The coefficients of the greatest common divisor of two polynomials f and g (GCD ( f , g ) ) can be obtained from the Sylvester subresultant matrix S j ( f , g ) transformed to lower triangular form, where 1 j d and d = deg(GCD ( f , g ) ) needs to be computed. Firstly, it is supposed that the coefficients of polynomials are given exactly. Transformations of S j ( f , g ) for an arbitrary allowable j are in details described and an algorithm for the calculation of the GCD ( f , g ) is formulated. If inexact polynomials are given, then an approximate...

On sets of polynomials whose difference set contains no squares

Thái Hoàng Lê, Yu-Ru Liu (2013)

Acta Arithmetica

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Let q [ t ] be the polynomial ring over the finite field q , and let N be the subset of q [ t ] containing all polynomials of degree strictly less than N. Define D(N) to be the maximal cardinality of a set A N for which A-A contains no squares of polynomials. By combining the polynomial Hardy-Littlewood circle method with the density increment technology developed by Pintz, Steiger and Szemerédi, we prove that D ( N ) q N ( l o g N ) 7 / N .

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.

On classifying Laguerre polynomials which have Galois group the alternating group

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 L n ( α ) ( x ) is a non-zero square for some integer pair ( n , α ) , with n 1 , if and only if ( n , α ) belongs to one of 30 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 L n ( α ) ( x ) over is the alternating group A n . For example, we establish that for all but finitely many positive integers n 2 ( mod 4 ) , the only α for which the Galois group of L n ( α ) ( x ) over is A n is...