Results 181 to 190 of about 2,041 (214)
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Polynomials and Trigonometric Polynomials
1976Setting cos ϑ = x, the expressions $$ T_n \left( x \right) = \cos n\vartheta {\text{ }}U_n \left( x \right) = \frac{1} {{n + 1}}T'_{n + 1} \left( x \right) = \frac{{\sin \left( {n + 1} \right)\vartheta }} {{\sin \vartheta }}'{\text{ }}n = 0,1,2,...
George Pólya, Gabor Szegö
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Minima of Trigonometric Polynomials
Bulletin of the London Mathematical Society, 1998Let \(00\) such that \[ -\min_{x\in (0,2\pi]} \sum^N_{k= 1} (\cos n_kx+ \sin n_kx)\geq c{N^{1/2}\over\log N}. \]
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On the trigonometric polynomials of Fejér and Young
Periodica Mathematica Hungarica, 2011Let \[ S_{n}(x)=\sum_{k=1}^{n} \frac{\sin(kx)}{k} \;\;\text{and}\;\; C_{n}(x)=1+\sum_{k=1}^{n}\frac{\cos(kx)}{k} \] be the trigonometric sums of Fejér and Young, respectively. The authors prove the following result: For all natural numbers \(n\geq 2\) and real numbers \(x\in (0,\,\pi)\) one has \[ \frac{C_{n}(x)}{S_{n}(x)}\geq \frac{1}{9}\,\sqrt{15}\,.\
Horst Alzer, Qinghe Yin
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Positivity of trigonometric polynomials
42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475), 2004The paper introduces a modification of the well-known sum-of-squares relaxation scheme for semi-algebraic programming by Shor based on replacing the ordinary polynomials by their trigonometric counterparts. It is shown that the new scheme has certain theoretical advantages over the classical one: in particular, a trigonometric polynomial is positive if
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Analysis and Mathematical Physics, 2023
The paper investigates two types of real trigonometric polynomial equations: \[ A(\theta)y'=B_1(\theta)+B_n(\theta)y^n \] and \[ A(\theta)y^{n-1}y'=B_1(\theta)+B_n(\theta)y^n \] The authors focus on the first equation and demonstrate that when $n\geq 4$, it has a maximum of 3 real trigonometric polynomial solutions if $n$ is even and 5 real ...
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The paper investigates two types of real trigonometric polynomial equations: \[ A(\theta)y'=B_1(\theta)+B_n(\theta)y^n \] and \[ A(\theta)y^{n-1}y'=B_1(\theta)+B_n(\theta)y^n \] The authors focus on the first equation and demonstrate that when $n\geq 4$, it has a maximum of 3 real trigonometric polynomial solutions if $n$ is even and 5 real ...
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A Property of the Ratio of Trigonometric Polynomials
Journal of the Society for Industrial and Applied Mathematics Series B Numerical Analysis, 1964The purpose of this note is to give a simple and relatively brief proof that it is possible to "factor out" zeros common to the numerator and de? nominator of a ratio of trigonometric polynomials. This result is needed in the paper of Cheney and Loeb [1, Lemma 9] where a different and longer proof is given.
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Integral Norms of Trigonometric Polynomials
Mathematical Notes, 2001zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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Trigonometric polynomials with simple roots
Information Processing Letters, 1991zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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Ukrainian Mathematical Journal, 2009
Summary: We study inequalities of the Turán type for trigonometric polynomials and conjugate trigonometric polynomials in the quasi-norm of \(L_0\) and derivatives of any order. We present expressions for constants in these inequalities and obtain double-sided estimates for them.
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Summary: We study inequalities of the Turán type for trigonometric polynomials and conjugate trigonometric polynomials in the quasi-norm of \(L_0\) and derivatives of any order. We present expressions for constants in these inequalities and obtain double-sided estimates for them.
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Turan's Inequalities for Trigonometric Polynomials
Journal of the London Mathematical Society, 1996We present a technique for establishing inequalities of the form \[ c |f |_\infty \leq \int^{2 \pi}_0 \varphi \biggl (\bigl |f^{(k)} (t) \bigr |\biggr) dt \leq M |f |_\infty \] in the set of all trigonometric polynomials of order \(n\) which have only real zeros. The function \(\varphi\) is assumed to be convex and increasing on \([0, \infty)\).
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