D. Hilbert, Ein Beitrag zur Theorie des Legendreschen Polynoms,ActaMath.18 (1894), 155--159.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....= 1 is a candidate for achieving the infimum in (1.4) and since in (1.3) 2( b a) 4) 1ifb a#4, we observe that on intervals [a, b] of length greater than or equal to 4 we have n [a, b] # [a, b] 1. We will thus from now on restrict our attention to intervals of length at most 4. Hilbert [12] showed that there exists an absolute constant c so that L2 [a,b] cn and Fekete [9] showed that 1 2 n 1 n 2 . For refinements of their inequalities, see Kashin [13] From the above it follows that (1.8) 4 # Recall that b 4. There is a pretty argument ....

D. Hilbert, Ein Beitrag zur Theorie des Legendreschen Polynoms,ActaMath.18 (1894), 155--159.

....readily observe that if E = a, b] and b a # 4, then Q n (x) # 1, n # N, by (1.1) and (1.6) so that (1.7) t Z ( a, b] 1, b a # 4. On the other hand, we obtain directly from the definition and (1.2) that (1.8) b a 4 = t C ( a, b] # t Z ( a, b] b a 4. Hilbert [19] proved an important upper bound (1.9) t Z ( a, b] # # b a 4 , by using Legendre polynomials and Minkowski theorem on the integer lattice points in a convex body. Actually, he worked with L 2 norm on [a, b] but this gives the same n th root behavior as for L# norm in (1.6) With the ....

....and # 3 # 0.0037, and R n # P n (Z) n # N. This gives a good indication of what might be the asymptotic structure of the integer Chebyshev polynomials on [0, 1] and other sets. Thus Amoroso [1] considered intervals with rational endpoints, and applied a refinement of Hilbert s approach in [19] to the polynomials vanishing with high multiplicities at the endpoints, to improve upon (1.9) Essentially the same ideas were used by Kashin [20] for dealing with the symmetric intervals [ a, a] in which case one should consider polynomials with factors x k . Borwein and Erdelyi [5] used ....

D. Hilbert, Ein Beitrag zur Theorie des Legendreschen Polynoms, Acta Math. 18 (1894), 155-159.

....the reader to the subject, illustrate some of the main ideas, and provide pointers to the literature. The subject of numerical stability instability of fast algorithms is confused for several reasons: 1. Structured matrices are often very ill conditioned. For example, the Hilbert matrix [50], defined by a i;j = 1= i j Gamma 1) is often used as an example of a poorly To appear as a chapter in Stability of Fast Methods for Linear Systems with Structure (editors, Ali H. Sayed and Thomas Kailath) SIAM, Philadelphia. y Copyright c fl 1997, R. P. Brent rpb177tr 1 2 R. P. BRENT ....

D. Hilbert, Ein Beitrag zur Theorie des Legendre'schen Polynoms, Acta Math. 18 (1894), 155--160.

....loss of generality. One may readily observe that if E = a; b] and b a 4, then Qn (x) 1; n 2 N; by (1.1) and (1.6) so that (1.7) t Z ( a; b] 1; b a 4: On the other hand, we obtain directly from the de nition and (1.2) that (1. 8) b a 4 = t C ( a; b] t Z ( a; b] b a 4: Hilbert [19] proved an important upper bound (1.9) t Z ( a; b] r b a 4 ; by using Legendre polynomials and Minkowski theorem on the integer lattice points in a convex body. Actually, he worked with L 2 norm on [a; b] but this gives the same n th root behavior as for L1 norm in (1.6) With the help of ....

.... 0:0166 and 3 0:0037; and Rn 2 Pn (Z) n 2 N. This gives a good indication of what might be the asymptotic structure of the integer Chebyshev polynomials on [0; 1] and other sets. Thus Amoroso [1] considered intervals with rational endpoints, and applied a re nement of Hilbert s approach in [19] to the polynomials vanishing with high multiplicities at the endpoints, to improve upon (1.9) Essentially the same ideas were used by Kashin [20] for dealing with the symmetric intervals [ a; a] in which case one should consider polynomials with factors x k : Borwein and Erd elyi [5] used ....

D. Hilbert, Ein Beitrag zur Theorie des Legendreschen Polynoms, Acta Math. 18 (1894), 155-159.

....to see that the above limit exists (cf. 14, Ch. 10] or [3] Observe from (1.1) and (1.3) that if b Gamma a 4 then q n (x) j 1 for any n 2 N and inch( a; b] 1. However, if b Gamma a 4 then b Gamma a 4 = cheb( a; b] inch( a; b] 1. 5) On the other hand, the results of Hilbert [10] and Fekete [5] imply that inch( a; b] s b Gamma a 4 (1.6) see [3] The exact value of the integer Chebyshev constant and an explicit (or even asymptotic) form of the integer Chebyshev polynomials is not known for any [a; b] with b Gammaa 4. Perhaps the most studied case, due to the ....

D. Hilbert, Ein Beitrag zur Theorie des Legendreschen Polynoms, Acta Math. 18 (1894), 155-159.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC