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V. P. Kondrat'ev 《Mathematical Notes》1977,22(3):696-698
For n=8 an upper bound is given for the functional $$V_n = \mathop {\inf }\limits_{t_n } \frac{{\alpha _1 + \alpha _2 + \cdots + \alpha _n }}{{\left( {\sqrt {\alpha _1 } - \sqrt {\alpha _0 } } \right)^2 }}$$ , which is defined on the class of even, nonnegative, trigonometric polynomials \(t_n (\phi ) = \sum\nolimits_{k = 0}^n {\alpha _k } cos k\phi \) , such that α k ? 0 (k=0, ...,n) α1>α0 :V s ? 34.54461566. 相似文献
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Eduard Belinsky 《Journal of Mathematical Analysis and Applications》2003,286(2):675-681
Three extremal problems for trigonometric polynomials are studied in this paper. The first was initiated by Maiorov. It relates to the trigonometric polynomials with n nonzero harmonics. The Lp-norm of the Weyl derivative is compared with the Lq-norm of the polynomial. The other two problems have appeared in the recent paper by Oswald. They deal with the polynomials of degree n. The minimum of Lp-norm with respect to the choice of phases is compared with lq-norm of its coefficients. 相似文献
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S. B. Stechkin 《Mathematical Notes》1994,55(2):195-203
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Using a particular way of normalizing the orthogonal polynomials, which is most commonly encountered in the synthesis of filtering networks in communication and electronic engineering, two theorems concerning the extremal properties of orthogonal polynomials are first proved. The results are then applied to find the minimum value and the minimizing function for various definite integrals involving weight functions of classical orthogonal polynomials. 相似文献
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J. Marshall Ash Michael Ganzburg 《Proceedings of the American Mathematical Society》1999,127(1):211-216
Let be a trigonometric polynomial of degree The problem of finding the largest value for in the inequality is studied. We find exactly provided is the conjugate of an even integer and For general we get an interval estimate for where the interval length tends to as tends to
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1. Introduction and Main ResultsDenote by Pn the set of algebraic polynomials of degree not exceeding n. LetX = {X = (xl,xz,...,x.). 1 = xl > xz >' > xtL--l > xtL = --l}, n 2 2and let for X E X.Erd6s in [21 raised the question of determining Y E X such thatReceived July 21, 1999.also conjectured that Y = Z satisfyingw,,(Z, x) = c1' l" Pt.--1 (x) dx,j-- 1 Pt.-- 1 (x) dx,1 .2>where P,--1 stands for the (n -- 1)th Legendre polynomial normalized by Pn--1 (1) = l. Thiscolljecture was di… 相似文献
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The refinements of some well-known Bemstein-type inequalities for trigonometric polynomials are obtained.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 3, pp. 428–430, March, 1993. 相似文献
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É. S. Belinskii 《Mathematical Notes》1991,49(1):10-14
Translated from Matematicheskie Zametki, Vol. 49, No. 1, pp. 12–18, January, 1991. 相似文献
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A. V. Moskovskii 《Mathematical Notes》1995,58(6):1355-1356
Translated fromMatematicheskie Zametki, Vol. 58, No. 6, pp. 940–941, December, 1995. 相似文献
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L. V. Taikov 《Mathematical Notes》1990,48(4):1044-1046
Translated from Matematicheskie Zametki, Vol. 48, No. 4, pp. 110–114, October, 1990. 相似文献
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B. P. Osilenker 《Russian Mathematics (Iz VUZ)》2010,54(2):46-56
In a loaded Jacobi space with the inner product
$
\left\langle {f,g} \right\rangle = \frac{{\Gamma (\alpha + \beta + 2)}}{{2^{\alpha + \beta + 1} \Gamma (\alpha + 1)\Gamma (\beta + 1)}}\smallint _{ - 1}^1 fg(1 - x)^\alpha (1 + x)^\beta dx + Lf(1)g(1) + Mf( - 1)g( - 1)(L,M \ge 0)
$
\left\langle {f,g} \right\rangle = \frac{{\Gamma (\alpha + \beta + 2)}}{{2^{\alpha + \beta + 1} \Gamma (\alpha + 1)\Gamma (\beta + 1)}}\smallint _{ - 1}^1 fg(1 - x)^\alpha (1 + x)^\beta dx + Lf(1)g(1) + Mf( - 1)g( - 1)(L,M \ge 0)
相似文献
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T. V. Radoslavova 《Mathematical Notes》1975,18(4):903-910
Suppose Φp, E (p>0 an integer, E ?[0, 2π]) is a family of positive nondecreasing functions? x(t) (t>0, x∈ E) such that? x(nt)≤nP ? x(t) (n=0,1,...), tn is a trigonometric polynomial of order at most n, and Δ h l (f, x) (l>0 an integer) is the finite difference of orderl with step h of the functionf.THEOREM. Supposef (x) is a function which is measurable, finite almost everywhere on [0, 2π], and integrable in some neighborhood of each point xε E,? X εΦp,E and $$\overline {\mathop {\lim }\limits_{\delta \to \infty } } |(2\delta )^{ - 1} \smallint _{ - \delta }^\delta \Delta _u^l (f,x)du|\varphi _x^{ - 1} (\delta ) \leqslant C(x)< \infty (x \in E).$$ . Then there exists a sequence {t n } n=1 ∞ which converges tof (x) almost everywhere, such that for x ε E $$\overline {\mathop {\lim }\limits_{n \to \infty } } |f(x) - l_n (x)|\varphi _x^{ - 1} (l/n) \leqslant AC(x),$$ where A depends on p andl. 相似文献
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If p(z) is a polynomial of degree n having all its zeros on |z| = k, k ≤ 1, then it is proved[5] that max |z|=1 |p′(z)| ≤ kn1n + kn m|z|=ax1 |p(z)|. In this paper, we generalize the above inequality by extending it to the polar derivative of a polynomial of the type p(z) = cnzn + ∑n j=μ cn jzn j, 1 ≤μ≤ n. We also obtain certain new inequalities concerning the maximum modulus of a polynomial with restricted zeros. 相似文献
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I. V. Belyakov 《Mathematical Notes》2006,80(3-4):339-344
For a wide class of symmetric trigonometric polynomials, the minimal deviation property is established. As a corollary, the extremal property is proved for the components of the Chebyshev polynomial mappings corresponding to symmetric algebras A α. 相似文献
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