An Inequality for Trigonometric Polynomials

N. K. Govil; Mohammed A. Qazi; Qazi I. Rahman

Displaying similar documents to “An Inequality for Trigonometric Polynomials”

On realizability of sign patterns by real polynomials

Vladimir Kostov (2018)

Czechoslovak Mathematical Journal

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The classical Descartes’ rule of signs limits the number of positive roots of a real polynomial in one variable by the number of sign changes in the sequence of its coefficients. One can ask the question which pairs of nonnegative integers $(p, n)$ , chosen in accordance with this rule and with some other natural conditions, can be the pairs of numbers of positive and negative roots of a real polynomial with prescribed signs of the coefficients. The paper solves this problem for degree $8$ polynomials. ...

A note on the number of zeros of polynomials in an annulus

Xiangdong Yang, Caifeng Yi, Jin Tu (2011)

Annales Polonici Mathematici

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Let p(z) be a polynomial of the form $p (z) = \sum_{j = 0}^{n} a_{j} z^{j}$ , $a_{j} \in - 1, 1$ . We discuss a sufficient condition for the existence of zeros of p(z) in an annulus z ∈ ℂ: 1 - c < |z| < 1 + c, where c > 0 is an absolute constant. This condition is a combination of Carleman’s formula and Jensen’s formula, which is a new approach in the study of zeros of polynomials.

Inequalities concerning polar derivative of polynomials

Arty Ahuja, K. K. Dewan, Sunil Hans (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In this paper we obtain certain results for the polar derivative of a polynomial $p (z) = c_{n} z^{n} + \sum_{j = μ}^{n} c_{n - j} z^{n - j}$ , $1 \leq μ \leq n$ , having all its zeros on $| z | = k$ , $k \leq 1$ , which generalizes the results due to Dewan and Mir, Dewan and Hans. We also obtain certain new inequalities concerning the maximum modulus of a polynomial with restricted zeros. [Editor’s note: There are flaws in the paper, see M. A. Qazi, Remarks on some recent results about polynomials with restricted zeros, Ann. Univ. Mariae Curie-Skłodowska Sect. A 67 (2), (2013),...

Symmetric identity for polynomial sequences satisfying $A_{n + 1}^{'} (x) = (n + 1) A_{n} (x)$

Farid Bencherif, Rachid Boumahdi, Tarek Garici (2021)

Communications in Mathematics

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Using umbral calculus, we establish a symmetric identity for any sequence of polynomials satisfying $A_{n + 1}^{'} (x) = (n + 1) A_{n} (x)$ with $A_{0} (x)$ a constant polynomial. This identity allows us to obtain in a simple way some known relations involving Apostol-Bernoulli polynomials, ApostolEuler polynomials and generalized Bernoulli polynomials attached to a primitive Dirichlet character.

A formula for Jack polynomials of the second order

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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.

The algebra of polynomials on the space of ultradifferentiable functions

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Banach Center Publications

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We consider the space $^{ℳ}$ of ultradifferentiable functions with compact supports and the space of polynomials on $^{ℳ}$ . A description of the space $(^{ℳ})$ 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 $f = \sum_{k} a_{k} P_{k}$ . 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.

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...

Criterion of the reality of zeros in a polynomial sequence satisfying a three-term recurrence relation

Innocent Ndikubwayo (2020)

Czechoslovak Mathematical Journal

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This paper establishes the necessary and sufficient conditions for the reality of all the zeros in a polynomial sequence ${P_{i}}_{i = 1}^{\infty}$ generated by a three-term recurrence relation $P_{i} (x) + Q_{1} (x) P_{i - 1} (x) + Q_{2} (x) P_{i - 2} (x) = 0$ with the standard initial conditions $P_{0} (x) = 1, P_{- 1} (x) = 0,$ where $Q_{1} (x)$ and $Q_{2} (x)$ are arbitrary real polynomials.

Uniqueness results for differential polynomials sharing a set

Soniya Sultana, Pulak Sahoo (2025)

Mathematica Bohemica

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We investigate the uniqueness results of meromorphic functions if differential polynomials of the form ${(Q (f))}^{(k)}$ and ${(Q (g))}^{(k)}$ share a set counting multiplicities or ignoring multiplicities, where $Q$ is a polynomial of one variable. We give suitable conditions on the degree of $Q$ and on the number of zeros and the multiplicities of the zeros of $Q^{'}$ . The results of the paper generalize some results due to T. T. H. An and N. V. Phuong (2017) and that of N. V. Phuong (2021).

Upper bounds for the $L_{q}$ norm of Fekete polynomials on subarcs

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Acta Arithmetica

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

Fejér–Riesz factorizations and the structure of bivariate polynomials orthogonal on the bi-circle

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 $Q (θ, ϕ)$ on the bi-circle, which can be factored as $Q (θ, ϕ) = | p (e^{i θ}, e^{i ϕ}) |^{2}$ where $p (z, w)$ is a polynomial nonzero for $| z | = 1$ and $| w | \leq 1$ . The conditions are in terms of recurrence coefficients associated with the polynomials in lexicographical and reverse lexicographical ordering orthogonal with respect to the weight $\frac{1}{4 π^{2} Q (θ, ϕ)}$ on the bi-circle. We use this result to describe how specific factorizations of weights on the bi-circle can be translated into identities...

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

Didier D&#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 ${| \nabla f | \geq C | f |}^{ϱ}$ in a neighbourhood of the origin. We prove that the smallest such exponent ϱ is not greater than $1 - R {(n, d)}^{- 1}$ with $R (n, d) = d {(3 d - 3)}^{n - 1}$ .

Jumps of ternary cyclotomic coefficients

Bartłomiej Bzdęga (2014)

Acta Arithmetica

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It is known that two consecutive coefficients of a ternary cyclotomic polynomial $Φ_{p q r} (x) = \sum_{k} a_{p q r} (k) x^{k}$ differ by at most one. We characterize all k such that $| a_{p q r} (k) - a_{p q r} (k - 1) | = 1$ . We use this to prove that the number of nonzero coefficients of the nth ternary cyclotomic polynomial is greater than $n^{1 / 3}$ .

On Bernstein inequalities for multivariate trigonometric polynomials in $L_{p}$ , $0 \leq p \leq \infty$

Laiyi Zhu, Xingjun Zhao (2022)

Czechoslovak Mathematical Journal

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Let $𝕋_{n}$ be the space of all trigonometric polynomials of degree not greater than $n$ with complex coefficients. Arestov extended the result of Bernstein and others and proved that $∥ (1 / n) T_{n}^{'} ∥_{p} \leq {∥ T_{n} ∥}_{p}$ for $0 \leq p \leq \infty$ and $T_{n} \in 𝕋_{n}$ . We derive the multivariate version of the result of Golitschek and Lorentz ${∥{|T_{n} cos α + \frac{1}{n} \nabla T_{n} sin α|}_{l_{\infty}^{(m)}}∥}_{p} \leq {∥ T_{n} ∥}_{p}, 0 \leq p \leq \infty$ for all trigonometric polynomials (with complex coeffcients) in $m$ variables of degree at most $n$ .

Mixed norm condition numbers for the univariate Bernstein basis

Tom Lyche, Karl Scherer (2006)

Banach Center Publications

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We study mixed norm condition numbers for the univariate Bernstein basis for polynomials of degree n, that is, we measure the stability of the coefficients of the basis in the $l_{q}$ -sequence norm whereas the polynomials to be represented are measured in the $L_{p}$ -function norm. The resulting condition numbers differ from earlier results obtained for p = q.

On prime values of reducible quadratic polynomials

W. Narkiewicz, T. Pezda (2002)

Colloquium Mathematicae

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It is shown that Dickson’s Conjecture about primes in linear polynomials implies that if f is a reducible quadratic polynomial with integral coefficients and non-zero discriminant then for every r there exists an integer $N_{r}$ such that the polynomial $f (X) / N_{r}$ represents at least r distinct primes.

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.

The image of multilinear polynomials evaluated on $3 \times 3$ upper triangular matrices

Thiago Castilho de Mello (2021)

Communications in Mathematics

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We describe the images of multilinear polynomials of arbitrary degree evaluated on the $3 \times 3$ upper triangular matrix algebra over an infinite field.

The zeros of functions of finite order in $C^{n}$

Lawrence Gruman (1983)

Annales Polonici Mathematici

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Rademacher-Carlitz polynomials

Matthias Beck, Florian Kohl (2014)

Acta Arithmetica

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We introduce and study the Rademacher-Carlitz polynomial $R (u, v, s, t, a, b) : = \sum_{k = ⌈ s ⌉}^{⌈ s ⌉ + b - 1} u^{⌊ (k a + t) ⌋} / b_{v^{k}}$ where $a, b \in ℤ_{> 0}$ , s,t ∈ ℝ, and u and v are variables. These polynomials generalize and unify various Dedekind-like sums and polynomials; most naturally, one may view R(u,v,s,t,a,b) as a polynomial analogue (in the sense of Carlitz) of the Dedekind-Rademacher sum $r_{t} (a, b) : = \sum_{k = 0}^{b - 1} (((k a + t) / b)) ((k / b))$ , which appears in various number-theoretic, combinatorial, geometric, and computational contexts. Our results come in three flavors: we prove a reciprocity theorem for Rademacher-Carlitz...

Sum of squares and the Łojasiewicz exponent at infinity

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Annales Polonici Mathematici

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Let V ⊂ ℝⁿ, n ≥ 2, be an unbounded algebraic set defined by a system of polynomial equations $h ₁ (x) = \dots = h_{r} (x) = 0$ and let f: ℝⁿ→ ℝ be a polynomial. It is known that if f is positive on V then ${f |}_{V}$ extends to a positive polynomial on the ambient space ℝⁿ, provided V is a variety. We give a constructive proof of this fact for an arbitrary algebraic set V. Precisely, if f is positive on V then there exists a polynomial $h (x) = \sum_{i = 1}^{r} h ²_{i} (x) σ_{i} (x)$ , where $σ_{i}$ are sums of squares of polynomials of degree at most p, such that f(x) + h(x) >...

Reduction and specialization of polynomials

Pierre Dèbes (2016)

Acta Arithmetica

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We show explicit forms of the Bertini-Noether reduction theorem and of the Hilbert irreducibility theorem. Our approach recasts in a polynomial context the geometric Grothendieck good reduction criterion and the congruence approach to HIT for covers of the line. A notion of “bad primes” of a polynomial P ∈ ℚ[T,Y] irreducible over ℚ̅ is introduced, which plays a central and unifying role. For such a polynomial P, we deduce a new bound for the least integer t₀ ≥ 0 such that P(t₀,Y) is...