Rivlin, T. J., The Chebyshev Polynomials, Wiley-Interscience, New York, 1974. Home/Search Document Not in Database Summary Related Articles Check |
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Generalizations of Müntz's Theorem via a Remez-Type.. - Borwein, Erdelyi (Correct)
.... The answer is given in terms of the Chebyshev polynomials. The extremal polynomials for the above problem are the Chebyshev polynomials Tn (x) cos(n arccos h(x) where h is a linear function which scales [0,s]or [1 s, 1] onto [ 1, 1] For various proofs, extensions, and applications, see [13, 14, 15, 22, 23]. We generalize the Remez inequality in the following way. Let . That is, Mn (#) is the collection of Muntz polynomials #R. We seek to find . These two problems are no longer equivalent as they are in the polynomial case (since x x does not preserve membership in Mn ....
T. J. Rivlin, Chebyshev Polynomials, 2nd ed., Wiley, New York, 1990.
Small Polynomials With Integer Coefficients - Pritsker (Correct)
....and of their zeros. Let P n (C) and P n (Z) be the classes of algebraic polynomials of degree at most n, respectively with complex and with integer coe#cients. The problem of minimizing the uniform norm on E by monic polynomials from P n (C) is well known as the Chebyshev problem (see [4] [31], 43] 16] etc. In the classical case E = 1, 1] the explicit solution of this problem is given by the monic Chebyshev polynomial of degree n: T n (x) 2 1 n cos(n arccos x) n # N. Using a change of variable, we can immediately extend this to an arbitrary interval [a, b] # R, so ....
T. J. Rivlin, Chebyshev Polynomials, John Wiley & Sons, New York, 1990.
Majorization in Lattice Path Enumeration And Creating Order - Tamm (Correct)
....An (#) 6.7) 1 # (# B2n 1 (#) 2 = # B 2 2n 1 (# 2 ) 1 # (# B 2n 1 (#) 2 = # B 2 2n 1 (# 2 ) 6.8) Proof: By definition, 6. 4) follows from the fact that B n = U n (0) Using the following well known properties of the Chebyshev polynomials (cf. 88] and [86], p. 9) t n (x) 2x t n 1 (x) t n 2 (x) u n (x) 2x u n 1 (x) u n 2 (x) 2 t n (x) u n (x) u n 2 (x) 6.5) follows via #Bn (#) # # Bn 1 (#) 2 # Bn 2 (#) # u n 1 ( # 2 ) 2 u n 2 ( # 2 ) u n ( # 2 ) u n 2 ( # 2 ) The ....
T. J. Rivlin, The Chebyshev Polynomials, Wiley, 1974.
Small Polynomials With Integer Coefficients - Pritsker (2000) (Correct)
....constant, integer trans nite diameter, zeros, multiple factors, asymptotics, potentials, weighted polynomials. Research supported in part by the National Science Foundation grant DMS 9996410. 1 2 IGOR E. PRITSKER monic polynomials from Pn (C ) is well known as the Chebyshev problem (see [4] [31], 43] 16] etc. In the classical case E = 1; 1] the explicit solution of this problem is given by the monic Chebyshev polynomial of degree n: Tn (x) 2 1 n cos(n arccos x) n 2 N: Using a change of variable, we can immediately extend this to an arbitrary interval [a; b] R, so that t ....
T. J. Rivlin, Chebyshev Polynomials, John Wiley & Sons, New York, 1990.
Chebyshev Polynomials With Integer Coefficients - Pritsker (Correct)
....31A05, 31A15. Key words and phrases. Chebyshev polynomials; Integer Chebyshev constant; Integer transfinite diameter; Multiple factors; Asymptotics; Potentials; Weighted polynomials. y Research supported in part by the National Science Foundation grant DMS 9707359. 2 IGOR E. PRITSKER [15]) The case of an arbitrary interval [a; b] ae R can be reduced to that of [ Gamma1; 1] by a change of variable. Thus we immediately obtain that t n (x) b Gamma a 2 n T n 2x Gamma a Gamma b b Gamma a is a monic polynomial with the smallest uniform norm on [a; b] among all ....
....2 N; 1.1) and the Chebyshev constant for [a; b] is given by cheb( a; b] lim n 1 kt n k 1=n [a;b] b Gamma a 4 : 1.2) Chebyshev polynomials and Chebyshev constant represent very classical topics in analysis. These ideas have applications in many areas of mathematics, see [2] 12] and [15]. We remark that the Chebyshev constant of a compact set in C is equal to its transfinite diameter and to its logarithmic capacity (cf. 18, pp. 71 75] for the general definitions and a discussion) A corresponding minimization problem in the class of polynomials with integer coefficients P n (Z) ....
T. J. Rivlin, Chebyshev Polynomials, John Wiley & Sons, New York, 1990.
Weierstrass and Approximation Theory - Pinkus (Correct)
....thus, jg( Gamma t e ( j for all 2 [ Gamma; In other words, since g is even we may assume that t is even. Let t( n X m=0 am cos m : Each cos m is a polynomial of exact degree m in cos . In fact cos m = Tm (cos ) where the Tm are the Chebyshev polynomials (see e.g. Rivlin [1974]) Setting p(x) n X m=0 amTm (x) we have jf(x) Gamma p(x)j for all x 2 [0; 1] Proposition 4. If algebraic polynomials are dense in C[a; b] then trigonometric polynomials are dense in e C[0; 2] Proof. The first proof of this fact was the one given by Weierstrass in Section 3. To ....
....for some known constant a. is monic, while Tn is normalized to have norm one. In this case we show how we can further refine formula (5.3) for the h i . The polynomial satisfies the second order differential equation (1 Gamma x 2 ) 00 (x) Gamma x 0 (x) n 2 (x) 0 (see e.g. Rivlin [1974, p. 31] At the points x i we have (x i ) 0 and therefore (1 Gamma x 2 i ) 00 (x i ) x i 0 (x i ) and 00 (x i ) 0 (x i ) x i (1 Gamma x 2 i ) 5:4) Furthermore it is easily verified from the formula Tn (x) cos(n arccos x) that 0 (x i ) aT 0 n (x i ) ....
Rivlin, T. J. [1974] "The Chebyshev Polynomial", John Wiley, New York.
On The Zeros Of Various Kinds Of Orthogonal Polynomials - Brezinski, Redivo-Zaglia (1997) (1 citation) Self-citation (Rivlin) (Correct)
....result of the theorem by taking the lowest upper bound on p. This result can be improved as for Theorem 7. The possibility of having pairs of multiple complex zeros can be studied similarly. 7 3. Generalizing Chebyshev polynomials The Chebyshev polynomials T k can be defined in several ways [18]. They can be introduced 1. in closed form by the formula T k (x) cos(k arccos x) 2. by their recurrence relationship T k 1 (x) 2xT k (x) Gamma T k Gamma1 (x) with T 0 (x) 1 and T 1 (x) x or, equivalently as proved in [3] by the Christoffel Darboux formula, 3. by their orthogonality ....
T.J. Rivlin, Chebyshev Polynomials, 2nd ed., Wiley, New York, 1990.
Best Approximation and Cyclic Variation Diminishing Kernels - Davydov, Pinkus (Correct)
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Rivlin, T. J., The Chebyshev Polynomials, Wiley-Interscience, New York, 1974.
Chebyshev interpolation for DMT modems - Cuypers, Ysebaert, Moonen, Pisoni (2004) (Correct)
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Theodore J. Rivlin, The Chebyshev polynomials, pp. 12--, Wiley, 1974.
Hankel Matrices in Coding Theory and Combinatorics - Tamm (2000) (Correct)
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T. J. Rivlin, The Chebyshev Polynomials, Wiley, 1974.
Chebyshev Approximation of Discrete Polynomials and Splines - Park (1999) (Correct)
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Rivlin, T. J., "Chebyshev Polynomials", John Wiley & Sons, INC., New York, 1990.
On the Evaluation of Polynomial Coefficients - Calvetti, Reichel (Correct)
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T. J. Rivlin, Chebyshev Polynomials, 2nd ed., Wiley, New York, 1990.
Müntz Spaces and Remez Inequalities - Borwein, Erdelyi (Correct)
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T. J. Rivlin, Chebyshev Polynomials, 2nd ed., Wiley, New York, 1990.
The Full M - Untz Theorem In (Correct)
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T. J. Rivlin, Chebyshev Polynomials, 2nd ed., Wiley, New York, 1990.
Barycentric Lagrange Interpolation - Berrut, Trefethen (1 citation) (Correct)
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T. J. Rivlin, The Chebyshev Polynomials, John Wiley, New York, 1974.
Iterative Solution Methods for Large Linear Discrete.. - Calvetti, Reichel, Zhang (1998) (2 citations) (Correct)
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T. J. Rivlin, Chebyshev Polynomials, 2nd ed., Wiley, New York, 1990.
D-Optimal Designs for Trigonometric Regression Models on a.. - Dette, Melas, al. (2001) (Correct)
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T.J. Rivlin (1974). Chebyshev polynomials. Wiley, New York.
Optimal designs for estimating individual coefficients in.. - Dette, Melas (2000) (Correct)
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T.J. Rivlin (1974). Chebyshev polynomials. Wiley, New York.
Approximation Theory - A Short Course - Carothers (1998) (Correct)
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Rivlin, T. J., The Chebyshev Polynomials, Wiley, 1974.
Efficient Computation of Chebyshev Polynomials in Computer Algebra - Koepf (1997) (Correct)
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Rivlin, Theodore J.: The Chebyshev Polynomials. Pure & Applied Mathematics. John Wiley & Sons, New York--London--Sydney--Toronto, 1974.
Lebesgue Functions for Polynomial Interpolation - a Survey - Brutman (1997) (2 citations) (Correct)
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T.J. Rivlin, Chebyshev Polynomials, 2nd Edition. John Wiley & Sons, New York, 1990.
Computing Probabilistic Bounds For Extreme.. - van Dorsselaer.. (2000) (Correct)
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T. J. Rivlin, Chebyshev Polynomials, 2nd ed., John Wiley, New York, 1990.
On Upper Bounds for Minimum Distance and Covering Radius of.. - Laihonen, Litsyn (1998) (Correct)
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T. Rivlin, The Chebyshev polynomials, New York: John Wiley & Sons (1974). 10
Optimal Oscillation Points For Polynomials Of.. - Lawrence Harris.. (Correct)
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T. J. Rivlin, Chebyshev Polynomials, 2nd ed. Wiley,NewYork, 1990.
A Note on Chebyshev's Inequality - Li (Correct)
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T.J. Rivlin, "Chebyshev Polynomials", 2nd edn., John Wiley & Sons, New York, 1990.
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