tailieunhanh - numerical mathematics and scientific computation volume 1 Episode 9

Tham khảo tài liệu 'numerical mathematics and scientific computation volume 1 episode 9', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | . Acceleration of Convergence 277 Comments The only if parts of these iff statements are deep results mainly due to Hausdorff and Bernstein and we omit the rather technical proofs. The relatively simple proofs of the if parts of the first three statements will be sketched since they provide some useful insight. 1. Assume that un is a moment sequence 3 0 0 3 is continuous at t 0 and non-decreasing for t 0. Note that multiplication by E or A outside the integral sign in corresponds to multiplication by t or t 1 inside. Then for j n o 1 2 . f-lVAJ z f-iv t - ýtndfí tẦ fl tVtndíỉ t4 0. V -L AA U Ị V 7 V 7 1 U Py1- I b b bbfjyb u. Jo Jo hence un is . 2. Assume that u s satisfies . It is rather easy to legitimate the differentiation under the integral sign in this equation. Differentiation j times with respect to s yields for j 1 2 3 . l Ju s 1 J í xý e sx da x i Xp sx da x 0 Jo Jo hence u s is . 3. Assume that un u n jp e nxda x . Define t e x 3 0 0 3 t 3 e x u 0 a x and note that t 1 o X 0 t 0 o X 00 and that u 0 lima a x . It follows that 3 t is non-negative and nondecreasing since X decreases as t increases. Note that 3 t ị 3 0 as t ị 0. Then un i tnd 3 t i tnd 3 t Ji Jo hence tzn is a minimal . 4. The distinction is illustrated for a x eax u s s a -1 for a real a. u is analytic for a and bounded only for a for any a a. The basic formula for the application of complete monotonicity to the summation of power series reads 00 . 00 f1 . . r1 00 . . T S z yy UịZi yy zitid t yy iCd 3 t 1 rt -1d 3 t . i 0 0 Jo Jo 0 Jo The inversion of the summation and integration is legitimate when 1. Note that the last integral exists for more general z a classical principle of Complex Analysis then yields the following interesting result. Lemma . If the sequence uj is . then the last integral of formula provides the unique single-valued analytic continuation of S z to the whole complex plane save for a cut along the real .