L. Chen and M. E. H. Ismail, On asymptotics of Jacobi polynomials,SIAMJ.Math.Anal.22 (1991), 1442--1449.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....# # # # # # 2#(n 1) ##(n 2# 1) n # for n =0,1, and # where w(x) 1 x . For nonsymmetric Jacobi weights much less is known. In 1988, L. Gatteschi [10] extended Bernstein s results to Jacobi polynomials with . For instance, he proved that if and # # 0then ##[0, 2 ] # #(sin # 2) cos # 2) n (cos #) # #(# 1) n n # for n =0,1, Again, in terms of the the orthonormal Jacobi polynomials, this can be stated as x#[0,1] # # # # # # 2# 1 #(n # # 1)#(n # 1) ##(n 1)#(n # 1) 2n # # 1) # for n ....

L. Chen and M. E. H. Ismail, On asymptotics of Jacobi polynomials,SIAMJ.Math.Anal.22 (1991), 1442--1449.

....Hermite polynomials. We will consider the cases where N = an # 1 2 (a ##. For these special parameters N satisfying (1. 8) the strong asymptotics will be performed on the basis of known Darboux type formulae for Jacobi polynomials with varying weights (Sections 3 and 4) which we take from [2, 3, 10] and by the use of a RiemannHilbert approach (Section 5) The central results are formulae of Plancherel Rotach type for n including Airy asymptotics. 2 Jacobi polynomials with varying weights In this section we collect some known Darboux type formulae which after some modifications we take ....

Li-chen Chen and M. E. H. Ismail, On asymptotics of Jacobi polynomials, SIAM J. Math. Anal. 22 (1991), 1442--1449.

.... t nJ The Jacobi polynomial representation immediately gives the following symmetries, which allow us to consider values of n with one sign only. THEOREM 6. We have 2, t) t n We then find asymptotic approximations to the amplitudes via large parameter asymptotics for Jacobi polynomials [7]. This uses the Darboux method, starting from a generating function for J. The second approach we consider is a Fourier analysis of the Hadamard walk. It is a counterpart of the SchrSdinger approach in quantum mechanics. The basic result is the following lemma. LEMMA T. We have b(n,t) dk ie ....

.... t T. 1 1(1 1) 2 oy 1 and for t Tu. 2uJ . 1 Then bR(n,t) is [ 2 2 1Y (1 ) if 0 n t; kt nj t n 1 ( 1) n) 2 X 9n 2 1 [ 1, n ) 2 if t n 0. From these representations of the wave function we ob tain Lemma 5 and Theorem 6. Chen and Ismail [7] have analyzed the asymptotics for values of Jacobi polynomials whose parameters are linear functions of the degree. Using their ideas, we obtain, after some work, Theorems 1 and 2. The results as stated in [7] have some minor errors. In particular, to obtain Theorem 1 one must take to in (2.17) ....

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L.-D. Chen and M. E. H. Ismall, On asymptotics of Jacobi polynomials, SIAM J. Math. Anal., 22:1442-1449, 1991.

.... 2) 4.2) Thus the Jacobi polynomials have the arcsine measure [ 1;1] as asymptotic zero distribution. More interesting limiting behavior can be obtained if the parameters = N and = N depend on N . Jacobi polynomials with varying parameters were considered before in a number of papers [1, 4, 8, 11, 15, 22]. The corresponding recurrence coecients are denoted here by a n;N and b n;N . Suppose the following limits exist lim N 1 N N = A 0; lim N 1 N N = B 0: Then it is easy to get from (4.1) 4.2) that for t 0, lim n=N t a n;N = 2 q t(t A B) t A) t B) 2t A B) 2 ; 4.3) ....

.... (t) 1 x 2 ; x 2 [ t) t) In Figure 2 we have plotted this density for A = 1, B = 2 and various values of t. This limiting density for the zeros of Jacobi polynomials was obtained earlier by Sa , Ullman and Varga [31] in their work on incomplete polynomials, see also Chen and Ismail [1] and Gawronski and Shawyer [11] for strong asymptotics. If at least one of A and B is di erent from 0, then the limiting relations (4.3) and (4.4) are also valid for t = 0. Then Theorem 1.10 gives us the limits of the smallest and largest zeros, as obtained earlier by Moak, Sa and Varga [22] ....

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L-C. Chen and M.E.H. Ismail, On asymptotics of Jacobi polynomials, SIAM J. Math. Anal. 22 (1991), 1442-1449.

....paper to derive and to work out the details of strong asymptotics for the relativistic Hermite polynomials. For special parameters N satisfying (1. 8) this will be performed on the basis of known Darboux type formulae for Jacobi polynomials with varying weights (sections 3 and 4) which we take from [2, 3, 10] and by the use of a Riemann Hilbert approach (section 5) The central results are formulae of Plancherel Rotach type for H N n including Airy asymptotics. 2 Jacobi polynomials with varying weights In this section we collect some known Darboux type formulae which after some modifications we take ....

Li-chen Chen and M. E. H. Ismail, On asymptotics of Jacobi polynomials, SIAM J. Math. Anal. 22 (1991), 1442--1449.

....gives the following symmetries, which allow us to consider values of n with one sign only. Theorem 6. We have pL( n; t) pL(n 2; t) pR( n; t) t n t n 2 pR(n; t) We then nd asymptotic approximations to the amplitudes via large parameter asymptotics for Jacobi polynomials [7]. This uses the Darboux method, starting from a generating function for J . The second approach we consider is a Fourier analysis of the Hadamard walk. It is a counterpart of the Schr odinger approach in quantum mechanics. The basic result is the following lemma. Lemma 7. We have L(n; t) Z ....

.... T ; 1 2 J (1; 1) Then R(n; t) is ( 1) t n) 2 8 : t n t n 2 n=2 1 J (1;n) t n 2 1 ; if 0 n t; 2 n=2 1 J (1; n) t n 2 1 ; if t n 0: From these representations of the wave function we obtain Lemma 5 and Theorem 6. Chen and Ismail [7] have analyzed the asymptotics for values of Jacobi polynomials whose parameters are linear functions of the degree. Using their ideas, we obtain, after some work, Theorems 1 and 2. The results as stated in [7] have some minor errors. In particular, to obtain Theorem 1 one must take t0 in (2.17) ....

[Article contains additional citation context not shown here]

L.-D. Chen and M. E. H. Ismail, On asymptotics of Jacobi polynomials, SIAM J. Math. Anal., 22:1442-1449, 1991.

.... bounds on the distance distribution of codes and other invariants we need asymptotic formulas for orthogonal polynomials involved in inequalities (7) 9) These problems have been studied more or less independently in coding theory [34] 24] 29] 25] 32] 1] 4] and analysis [35] [12], 21] 16] 15] 28] We quote results from the coding theory side since they are in the form better suited to our needs. Asymptotics of extremal zeros found in [34] 24] were used in these papers to derive the bounds ffi (lp) R) and d (kl) R) respectively. However, to derive bounds on ....

L.-C. Chen and M. E. H. Ismail, On asymptotics of Jacobi polynomials, SIAM J. Math. Anal. 22 (1991), no. 5, 1442--1449.

....subsets that in the asymptotics possess distance invariance properties similar to those of linear codes in the Hamming space, namely that the distance spectrum with respect to any given vector in the subset is one and the same. 1 After this paper was submitted, we learned of related results [10]. 7 The remaining part of the proof of Theorem 1, given in Section 5, is geometrically much more intuitive. It is accomplished by an inclusion exclusion argument first employed for the Hamming case in [25] A few remarks on the asymptotic notation. Since in the paper we are interested only ....

L.-C. Chen and M. E. H. Ismail, On asymptotics of Jacobi polynomials, SIAM J. Math. Anal. 22 (1991), no. 5, 1442--1449.

....Thus the Jacobi polynomials have the arcsine measure [ Gamma1;1] as asymptotic zero distribution. More interesting limiting behavior can be obtained if the parameters ff = ff N and fi = fi N depend on N . Jacobi polynomials with varying parameters were considered before in a number of papers [2, 4, 8, 11, 15, 22]. The corresponding recurrence coefficients are denoted here by a n;N and b n;N . Suppose the following limits exist lim N 1 ff N N = A 0; lim N 1 fi N N = B 0: Then it is easy to get from (4.1) 4.2) that for t 0, lim n=N t a n;N = 2 q t(t A B) t A) t B) 2t A B) 2 ; 4.3) ....

....of t. t=4 t=2 t=1 t=1 2 0 0.2 0.4 0.6 0.8 1 1 0.5 0. 5 1 Figure 2: Some densities for the asymptotic zero distribution This limiting density for the zeros of Jacobi polynomials was obtained earlier by Saff, Ullman and Varga [31] in their work on incomplete polynomials, see also Chen and Ismail [2] and Gawronski and Shawyer [11] for strong asymptotics. If at least one of A and B is different from 0, then the limiting relations (4.3) and (4.4) are also valid for t = 0. Then Theorem 1.10 gives us the limits of the smallest and largest zeros, as obtained earlier by Moak, Saff and Varga [22] ....

[Article contains additional citation context not shown here]

L.-C. Chen and M.E.H. Ismail, On asymptotics of Jacobi polynomials, SIAM J. Math. Anal. 22 (1991), 1442--1449.

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