Rivlin, T. J., The Chebyshev Polynomials, Wiley-Interscience, New York, 1974.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... 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.

....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.

....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.

....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.

....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.

....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.

....That is, H(#) # C(A) and every p # Hn(#) having at least n 1 (distinct) zeros in A is identically 0 on A. In fact, cosh spaces Hn (#) are simple examples for Chebyshev spaces, hence they share the following well known properties of general Chebyshev spaces (see, for example, 2] 14] and [21]) Theorem 4.1 (Existence of Chebyshev Polynomials) Let A be a compact subset of [0, #) containing at least n 1points. Then there exists a unique (extended) Chebyshev polynomial Tn : Tn # 1 ,# 2, # n;A for Hn(#) on A defined by Tn (x) c # # (cosh(# n t) 1) # # a 0 n 1 ....

Rivlin, T.J., Chebyshev Polynomials, 2nd ed., Wiley, New York, 1990.

....: 0 0 0 1 2x 1 : 0 0 0 . 0 0 0 : 1 2x 1 0 0 0 : 0 1 2x 1 1 C C C C C C A ; 17) Observe that the transition matrix AM = VM (1) occurs for the special value x = 1. The following well known properties of the Chebyshev polynomials (cf. 8] and [9]) will be used later on. t n (x) 2x Delta t n Gamma1 (x) Gamma t n Gamma2 (x) u n (x) 2x Delta u n Gamma1 (x) Gamma u n Gamma2 (x) 18) t n (x) u n (x) Gamma u n Gamma2 (x) 19) 4 Delta sin i 2n 1 ; i = 1; n are the eigenvalues of V n (x) 20) Proof of Theorem 1: i) ....

T. J. Rivlin, The Chebyshev polynomials, Wiley, 1974.

....Table 1 shows the byte sizes of T n (x) in input form. The Chebyshev polynomials have the nice property T n (1) 1. This can be used to check the accuracy of the numerical computations. For further details about these (and other families of orthogonal) polynomials we refer the reader to [2] x22, [6], 7] and [8] All timings are given in CPU seconds truncated to three digits, and were calculated on a SUN Sparc 10 with 85 MByte memory under SunOS 4.1.3 with the versions Maple V.3, Mathematica 2.2, REDUCE 3.6 1 and MuPAD 1.2.2. We issued the statements in separate sessions to avoid the ....

Rivlin, Th. J.: The Chebyshev Polynomials. Pure & Applied Mathematics. John Wiley & Sons, New York--London--Sydney--Toronto, 1974.

....respectively. It is easy to see that F [f ] 1 2 P p (T ) and that F [f ] 1 (x; 0) f(x) Lemma 4.1 Let f 2 P p (I 1 ) and let F [f ] 1 be defined by (4.1) Then, jjF [f ] 1 jj H s 1=2 (T ) Cjjf jj H s (I 1 ) 4. 2) The following is the classical Markov inequality; see, e.g. Rivlin [57]. 44 Figure 4.1: Extension of boundary values I 3 I 1 I 2 y x x x y y Lemma 4.2 Let f 2 Q p ( 0; 1] Then, max [0;1] fi fi fi fi df dx (x) fi fi fi fi 2p 2 max [0;1] jf(x)j: The following result is Theorem 6.2 in Babuska, Craig, Mandel, and Pitkaranta [9] Lemma 4.3 Let u 2 Q p ....

Theodore J. Rivlin. The Chebyshev Polynomials. Wiley Interscience, 1990.

....r , then we will still have a constant bound. The proof of this theorem can be found in Pavarino [13] and is based on a series of technical results concerning the decomposition of discrete harmonic polynomials and on Theorem 3.1. The main tools used in the proof are Markov s theorem (see Rivlin [15]) and a p version analog of the decomposition lemma 3.2 in Widlund [17] A two dimensional extension theorem for polynomial finite elements (see Babuska, Craig, Mandel and Pitkaranta [1] is needed to prove the logarithmic bound, while the constant bound can be obtained without it. ....

T. J. Rivlin, Chebyshev Polynomials, Wiley Interscience, 1990.

....in terms of the Chebyshev polynomials. The extremal polynomials for the above problem are the Chebyshev polynomials SigmaT n (x) Sigma cos(n arccosh(x) where h is a linear function which scales [0; s] or [1 Gamma s; 1] onto [ Gamma1; 1] For various proofs, extensions, and applications see [13, 14, 15, 20, 21]. We announce the following bounded Remez type inequality for M ( whose proof, which is quite difficult, will appear elsewhere. Theorem 2.1. Suppose P 1 i=1 1= i 1. Let s 0. Then there exists a constant c depending only on : f i g 1 i=0 and s (and not on , A, or the length of p) so ....

T. J. Rivlin, Chebyshev Polynomials, 2nd ed., Wiley, New York, 1990.

....1) containing at least n 1 points. That is, M ( ae C[A] and every p 2 Mn ( having at least n 1 (distinct) zeros in A is identically 0. In fact, Muntz spaces are the canonical examples for Chebyshev spaces and the following properties of Muntz spaces Mn ( are well known (see, for example, [10, 12, 21]) FULL M UNTZ THEOREM IN C[0; 1] AND L 1 [0; 1] 5 Theorem 3.1 (Existence of Chebyshev Polynomials) Let A be a compact subset of [0; 1) containing at least n 1 points. Then there exists a unique (extended) Chebyshev polynomial Tn : Tn f 0 ; 1 ; n ; Ag for Mn ( on A defined by Tn ....

T. J. Rivlin, Chebyshev Polynomials, 2nd ed., Wiley, New York, 1990.

....terms of the Chebyshev polynomials. The extremal polynomials for the above problem are the Chebyshev polynomials SigmaT n (x) Sigma cos(n arccosh(x) where h is a linear function which scales [0; s] or [1 Gamma s; 1] onto [ Gamma1; 1] For various proofs, extensions, and applications, see [13, 14, 15, 22, 23]. We generalize the Remez inequality in the following way. Let Mn ( spanfx 0 ; x 1 ; x n g: That is, Mn ( is the collection of Muntz polynomials p(x) n X i=0 a i x i ; a i 2 R: We seek to find (1) max ae jp(0)j kpkA : 0 6= p 2 Mn ( A ae [0; 1] m(A) s oe and ....

T. J. Rivlin, Chebyshev Polynomials, 2nd ed., Wiley, New York, 1990.

....1 The Chebyshev polynomials have the nice property that T n (1) 1. This can be used to check the accuracy of the numerical computations. For further details about these (and other families of orthogonal) polynomials including the algorithms of this article we refer the reader to [2] x22, 5] [6], 7] and [8] We think that the user of a computer algebra system is mostly interested in good timings. The memory management is not of such a large interest to him besides the fact that large memory usage might influence the timings, or may even crash the system. By this reason we just compared ....

Rivlin, Th. J.: The Chebyshev Polynomials. Pure & Applied Mathematics. John Wiley & Sons, New York--London--Sydney--Toronto, 1974.

....X s=0 fl s N(L; s; b) Hence N(L; s; b) 6= 0 for at least one s (s = 0; 1; r) and so R r. 4 We should now find a polynomial of a low degree such that jf(i)j is small compared to f(0) when i 6= 0 and fi i (b) 6= 0. The Chebyshev polynomial of the first kind and degree r is defined in [14], p.5 by T r (x) 1 2 ii x p x 2 Gamma 1 j r i x Gamma p x 2 Gamma 1 j r j : So clearly, x 1, T r (x) 1 2 ( x p x 2 Gamma 1) r 1) 6) Assume that 0 a b. Among the polynomials p r (x) of degree at most r such that p r (0) 1 the one defined by t r (x) ....

T. Rivlin, The Chebyshev polynomials, New York: John Wiley & Sons (1974).

....are independent of the diameter H of the elements. We have chosen not to show how the constants of our auxiliary estimates depend on H: 5.1. Technical tools. We will now give a series of lemmas that are needed in the proof of Theorem 3.1. We begin with the classical Markov inequality; cf. Rivlin [46]. Lemma 5.1. Let f be a polynomial of degree p defined on [ Gamma1; 1] Then max [ Gamma1;1] jf 0 (x)j p 2 max [ Gamma1;1] jf(x)j: The following result is a discrete Sobolev inequality for polynomials; see Theorem 6.2 in Babuska, Craig, Mandel, and Pitkaranta [1] Lemma 5.2. Let F = ....

Theodore J. Rivlin. The Chebyshev Polynomials. Wiley Interscience, 1990.

....kf(x) 0 g(x)k for all x 2 J . Such density results are now known for many different classes of functions and norms; see, e.g. 49] 5 Lower bounds on rate of approximation Neural networks are one of several branches of the larger enterprise of approximation theory; see, e.g. 45] 1] [58] and [63] We review the classical theory of functional approximation by polynomials and its modern generalization to the nonlinear case. Chebyshev and Weierstrass showed that all continuous functions on a closed bounded subset J of a Euclidean space R d can be uniformly approximated by linear ....

Rivlin, T. J. (1990). Chebyshev Polynomials. New York: Wiley.

....terms of the Chebyshev polynomials. The extremal polynomials for the above problem are the Chebyshev polynomials SigmaT n (x) Sigma cos(n arccos h(x) where h is a linear function that scales [0; s] or [1 Gamma s; 1] onto [ Gamma1; 1] For various proofs, extensions, and applications, see [13, 14, 15, 22, 23]. Our bounded Remez type inequality of Theorem 3.1 states the following. If ( i ) 1 i=0 is a sequence of distinct real numbers satisfying 1 X i=0 i 6=0 1 j i j 1; then there is a constant c depending only on ( i ) 1 i=0 , A, ff, and fi (and not on the number of terms in p) so that ....

T. J. Rivlin, Chebyshev Polynomials, 2nd ed., Wiley, New York, 1990.

....0 (x)j p 2 max I jv(x)j ; with equality only for v = SigmaT p ; the Chebyshev polynomial of degree p. This inequality is not scale invariant: on the interval I H = GammaH; H ] max IH jv 0 (x)j p 2 H max IH jv(x)j : A proof and many useful generalizations can be found in Rivlin [21]. The following result is Lemma 2.2 in Bramble and Xu [7] Lemma 1. If D is a bounded Lipschitz domain in R 2 , then kwk L 1 (D) C(j log fflj 1=2 kwk H 1 (D) fflkwk W 1;1 (D) for every function w 2 W 1;1 (D) and any ffl 2 (0; 1) With this Lemma, we can prove a Sobolev like ....

Rivlin, T. (1990): The Chebyshev Polynomials. Wiley Interscience

....I jv 0 (x)j p 2 max I jv(x)j; with equality only for v = SigmaT p ; the Chebyshev polynomial of degree p. This inequality is not scale invariant: on the interval I= GammaH; H ] max I jv 0 (x)j p 2 H max I jv(x)j; A proof and many useful generalizations can be found in Rivlin [72]. The following result is stated and proved as Lemma 2.2 in Bramble and Xu [22] Lemma 4.1 If D is a bounded domain in R 2 and D is Lipschitz continuous, then kwk L 1 (D) C(j log fflj 1=2 kwk H 1 (D) fflkwk W 1;1 (D) for every function w 2 W 1;1 (D) and any ffl 2 (0; 1) With ....

T. J. Rivlin. Chebyshev Polynomials. Wiley Interscience, 1990.

....of how to use polynomials to increase efficiency when computing a large number of eigenpairs. A brief summary is provided at the end. 5. 2 Characteristics of the polynomials Polynomials we will use in this study are: the Chebyshev polynomials [51, 116] the Chebyshev polynomials of the second kind [101], the least squares polynomials [111] and the Kernel polynomials [1, 23, 33, 141] This section will not give detailed algorithms for these polynomials since they are widely available. The main purpose of this section is to familiarize readers with some general characteristics of these polynomials ....

....2.7.4] If the extreme eigenvalues are clustered, this property indicates that using the Chebyshev polynomial to transform the spectrum is most effective in increasing the separations. The Chebyshev polynomial we referred to in the above discussion is the Chebyshev polynomial of the first kind [101]. The above reference to the derivative leads us to consider the Chebyshev polynomial of the second kind which is the derivative of the Chebyshev polynomial of the first kind. Let U k (i) denote kth order Chebyshev polynomial of the second kind. The first two U k s are: U 0 = 0, U 1 = 2i. The ....

Theodore J. Rivlin. Chebyshev Polynomials. Wiley, New York, 2nd edition, 1990.

.... : with deg(f i (z) i, it is true that f i (f j (z) f j (f i (z) for any i; j 0; then, either f i (z) Gamma1 (z) i ) for all i, or, f i (z) T i ( Gamma1 (z) for all i where (z) az b and Gamma1 (z) z Gamma a) b for some constants a; b (see, for instance, Rivlin (1974)) 3 . The algorithm can be adapted easily to interpolate polynomials that are sparse in such generalized power bases or Chebyshev bases. One can consider several possible natural bases and polynomials that are sparse in those bases. For example, suppose we know that the polynomial given by a ....

Rivlin, T. (1974), " The Chebyshev polynomials," New York, Wiley.

....this section by studying the conditioning in the case ff = fi = Gamma1=2; similar results are valid for ff = fi = 1=2. Therefore, let w h be the vector whose coordinates are the zeros of T h 1 , the Chebyshev polynomial of the first kind. The orthogonality properties of the T h imply that [14] l m (w) 2 N 1 N X h=0 0 T h (wNm )T h (w) so that, if T h (w) h X k=0 t hk w k ; then 6 l mn = 2 N 1 N X h=n t hn T h (wNm ) 19) Thus, N X m;n=0 jl mn j 2 N 1 N X m;n=0 N X h=n jt hn j = 2 N X h=0 h X n=0 jt hn j : 6 P h=n means P 0 h=n ....

Rivlin T. (1990): Chebyshev Polynomials. Wiley Interscience,

....direct interpolation seems impractical but the function f(z) i z 2 j Gamman Yn (z) Gamma 2 ln(z)J n (z) 1 n Gamma1 X r=0 (n Gamma r Gamma 1) r i z 2 j 2r Gamman ; is a regular function of z and so can be handled. For the Chebyshev coefficients of (1) Rivlin [7] showed that ja 2r j 2M( 2r ; for any 1, where M( sup jzj= jf(z)j. Also jT 2r (j)j 1 2 i 1 p 2 j 2r i 1 Gamma p 2 j 2r ; for jjj 1, with equality if and only if j = Sigmai, hence ja 2r T 2r i z j j 2M( Gamma 1 p 2 Delta 2r 2r : It ....

T.J. Rivlin, Chebyshev Polynomials, Wiley, New York, 1974.

....2 4 6 8 10 12 14 16 18 20 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 degree Figure 1: Results with Horner( and Simplex method ( 3.2 Chebyshev polynomials 3.2.1 Chebyshev constraints In this section, we show how the use of Chebyshev polynomials can lead to an improvement of our method. It is known[4] that, if P n Gamma1 denotes the set of polynomials defined on [ 1,1] whose degree do not exceed n Gamma 1, the polynomial in P n Gamma1 closest to the function f(x) x n , where the distance is given by the infinite norm is the polynomial: 1 The subdistributivity of interval multiplication ....

Rivlin, T.J. Chebyshev Polynomials. John Wiley & Sons. (1990).

....Z 1 Gamma1 w f p k : The case = 0 corresponds to expansion in Chebyshev polynomials, and the classical result of M. Riesz [6] shows that S n : L 0 p L 0 p is bounded for 1 p 1. Furthermore, it is well known that kS n k 0 1 = kS n k 0 1 C log n as n 1; see e.g. Rivlin [7]. In two papers, 3, 4] Pollard establishes the boundedness of kS n k p for (2 1) 1) p (2 1) 0, and shows that S n is unbounded if either p (2 1) or p (2 1) 1) Furthermore, it has been shown by Ragozin [5] that for kS n k 0 = jS n k 1 i n as n 1. The ....

T. J. Rivlin, The Chebyshev Polynomials, Wiley--Interscience, New York, 1974.

....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.

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Rivlin, T. J., The Chebyshev Polynomials, Wiley-Interscience, New York, 1974.

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Theodore J. Rivlin, The Chebyshev polynomials, pp. 12--, Wiley, 1974.

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T. J. Rivlin, The Chebyshev Polynomials, Wiley, 1974.

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Rivlin, T. J., "Chebyshev Polynomials", John Wiley & Sons, INC., New York, 1990.

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T. J. Rivlin, Chebyshev Polynomials, 2nd ed., Wiley, New York, 1990.

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T. J. Rivlin, Chebyshev Polynomials, 2nd ed., Wiley, New York, 1990.

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T. J. Rivlin, Chebyshev Polynomials, 2nd ed., Wiley, New York, 1990.

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T. J. Rivlin, The Chebyshev Polynomials, John Wiley, New York, 1974.

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T. J. Rivlin, Chebyshev Polynomials, 2nd ed., Wiley, New York, 1990.

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T.J. Rivlin (1974). Chebyshev polynomials. Wiley, New York.

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T.J. Rivlin (1974). Chebyshev polynomials. Wiley, New York.

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Rivlin, T. J., The Chebyshev Polynomials, Wiley, 1974.

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Rivlin, Theodore J.: The Chebyshev Polynomials. Pure & Applied Mathematics. John Wiley & Sons, New York--London--Sydney--Toronto, 1974.

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T.J. Rivlin, Chebyshev Polynomials, 2nd Edition. John Wiley & Sons, New York, 1990.

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T. J. Rivlin, Chebyshev Polynomials, 2nd ed., John Wiley, New York, 1990.

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T. Rivlin, The Chebyshev polynomials, New York: John Wiley & Sons (1974). 10

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T. J. Rivlin, Chebyshev Polynomials, 2nd ed. Wiley,NewYork, 1990.

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T.J. Rivlin, "Chebyshev Polynomials", 2nd edn., John Wiley & Sons, New York, 1990.

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T.J. Rivlin, The Chebyshev Polynomials, 2nd Ed., John Wiley & Sons, New York, 1990.

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T.J. Rivlin. The Chebyshev polynomials. John Wiley and Sons, 1974.

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T. J. Rivlin, Chebyshev Polynomials, 2nd ed. Wiley, New York, 1990.

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Theodore J. Rivlin. The Chebyshev Polynomials. John Wiley & Sons, 1974.

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