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Displaying similar documents to “Discrepancy estimates for some linear generalized monomials”

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.

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.

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...

Piatetski-Shapiro sequences via Beatty sequences

Lukas Spiegelhofer (2014)

Acta Arithmetica

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Integer sequences of the form n c , where 1 < c < 2, can be locally approximated by sequences of the form ⌊nα+β⌋ in a very good way. Following this approach, we are led to an estimate of the difference n x φ ( n c ) - 1 / c n x c φ ( n ) n 1 / c - 1 , which measures the deviation of the mean value of φ on the subsequence n c from the expected value, by an expression involving exponential sums. As an application we prove that for 1 < c ≤ 1.42 the subsequence of the Thue-Morse sequence indexed by n c attains both of its values with...

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...

Some results on derangement polynomials

Mehdi Hassani, Hossein Moshtagh, Mohammad Ghorbani (2022)

Commentationes Mathematicae Universitatis Carolinae

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We study moments of the difference D n ( x ) - x n n ! e - 1 / x concerning derangement polynomials D n ( x ) . For the first moment, we obtain an explicit formula in terms of the exponential integral function and we show that it is always negative for x > 0 . For the higher moments, we obtain a multiple integral representation of the order of the moment under computation.

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.

Explicit bounds for the Łojasiewicz exponent in the gradient inequality for polynomials

Didier D&amp;#039;Acunto, Krzysztof Kurdyka (2005)

Annales Polonici Mathematici

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Let f: ℝⁿ → ℝ be a polynomial function of degree d with f(0) = 0 and ∇f(0) = 0. Łojasiewicz’s gradient inequality states that there exist C > 0 and ϱ ∈ (0,1) such that | f | C | f | ϱ in a neighbourhood of the origin. We prove that the smallest such exponent ϱ is not greater than 1 - R ( n , d ) - 1 with R ( n , d ) = d ( 3 d - 3 ) n - 1 .

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 ∈...

On the structure of sequences with forbidden zero-sum subsequences

W. D. Gao, R. Thangadurai (2003)

Colloquium Mathematicae

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We study the structure of longest sequences in d which have no zero-sum subsequence of length n (or less). We prove, among other results, that for n = 2 a and d arbitrary, or n = 3 a and d = 3, every sequence of c(n,d)(n-1) elements in d which has no zero-sum subsequence of length n consists of c(n,d) distinct elements each appearing n-1 times, where c ( 2 a , d ) = 2 d and c ( 3 a , 3 ) = 9 .

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...

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...

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...

Primefree shifted Lucas sequences

Lenny Jones (2015)

Acta Arithmetica

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We say a sequence = ( s ) n 0 is primefree if |sₙ| is not prime for all n ≥ 0, and to rule out trivial situations, we require that no single prime divides all terms of . In this article, we focus on the particular Lucas sequences of the first kind, a = ( u ) n 0 , defined by u₀ = 0, u₁ = 1, and uₙ = aun-1 + un-2 for n≥2, where a is a fixed integer. More precisely, we show that for any integer a, there exist infinitely many integers k such that both of the shifted sequences a ± k are simultaneously primefree. This...

Explicit construction of normal lattice configurations

Mordechay B. Levin, Meir Smorodinsky (2005)

Colloquium Mathematicae

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We extend Champernowne’s construction of normal numbers to base b to the d case and obtain an explicit construction of a generic point of the d shift transformation of the set 0 , 1 , . . . , b - 1 d .

Characterization of functions whose forward differences are exponential polynomials

J. M. Almira (2017)

Commentationes Mathematicae Universitatis Carolinae

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Given { h 1 , , h t } a finite subset of d , we study the continuous complex valued functions and the Schwartz complex valued distributions f defined on d with the property that the forward differences Δ h k m k f are (in distributional sense) continuous exponential polynomials for some natural numbers m 1 , , m t .