Monotonluğu Koruyan Matrisler. Monotonicity Preserving Matrices

Transcription

1 SAKARYA ÜNİVERSİTESİ FEN BİLİMLERİ ENSTİTÜSÜ DERGİSİ SAKARYA UNIVERSITY JOURNAL OF SCIENCE e-issn: X Degi sayfası: Geliş/Received Kabul/Accepted Doi /saufenbilde Monotonluğu Kouyan Matisle Fatma Aydin Agun *, Billy E. Rhoades, 2 Sameeah Mahmood Mohammad 3 ÖZ Negatif olmayan üçgensel matis dönüşümleinin ço geniş bi ailesi için, monoton azalan dizilein monoton azalan dizilee dönüşebilmesi oşullaını elde etti. Anahta Kelimele: monoton azalan dizile, genelleştiilmiş Hausdoff matislei, Üst üçgen matisle, toplanabilme matislei ABSTRACT Monotonicity Peseving Matices We obtain the conditions fo a lage class of nonnegative tiangula matix tansfomations to map positive monotone deceasing sequences into positive monotone deceasing sequences. Keywods: monotone deceasing sequences, genealized Hausdoff matices, Uppe taingula matices, summability matices. * Yıldız Teni Ünivesitesi, Matemati Mühendisliği. Yıldız Kampüsü - fagun@yildiz.edu.t

2 F. Aydın Agun, B. E. Rhoades, S. M. Mohammad Monotonicity Peseving Matices 1. INTRODUCTION A positive sequence x = {x n } is a monotone deceasing sequence, o simply deceasing, if x n x fo each n 0.A matix A is called tiangula if a n = 0 fo each >n, and a tiangle if it is tiangula and a nn 0 fo each n. We shall say that a matix peseves monotonicity if it maps evey monotone deceasing sequence into a monotone deceasing sequence. Bennett [1] pove the following theoem. Theoem 1. Let A = (a n ) be a matix with nonnegative enties, and conside the associated tansfom x y, given by y n = a n x. Then the following conditions ae equivalent. (i) y 0 y 1 0 wheneve x 0 x 1 x n (ii) a n a,, n, = 0,1,2, This theoem yields a numbe of coollaies fo some well-nown summability matices. Bennett has noted that the positive Hausdoff matices map deceasing sequences into deceasing sequences. As ou fist esult we shall show that the same is tue fo genealized Hausdoff matices. 2. PRELIMINARIES An odinay Hausdoff matix H is a tiangula matix with nonzeo enties ( n ) n μ (1) whee μ n is a eal o complex sequence and is the fowad diffeence opeato defined by μ = μ μ +1 and μ = n μ n μ +1. Thee ae seveal genealizations of Hausdoff matices. One of this is called the H-J matices. A sequence (λ n ) is called acceptable sequence if it satisfies the following popeties: 0 λ 0 < λ 1 < < λ n, with λ n, but slowly enough so that 1 λ n =. n=1 The nonzeo enties of an H-J matix H(μ; λ) ae defined by h(μ; λ) n =λ +1 λ n [μ,, μ n ], 0 n whee μ n is a eal o complex sequence, [.] is the divided diffeence opeato defined by and [μ, μ +1 ] = 1 λ +1 λ [μ μ +1 ] 1 [μ,, μ n ] = ([μ λ n λ,.., μ n 1 ] [μ +1,, μ n ]) and whee it is undestood that λ +1 λ +2 λ n = 1 when = n. Hausdoff [5] defined this genealization fo λ 0 = 0, and, Jaimovsi [6] extended this class fo λ 0 > MAIN THEOREMS Theoem 2. A positive H-J matix, with λ 0 = 0, maps deceasing sequences into deceasing sequences. Poof: Using (ii) of Bennett's theoem h n h, = λ +1 λ n [μ,, μ n ] λ +1 λ [μ,, μ ]. (2) Fom the definition of divided diffeences, (λ λ )[μ, μ ] = [μ, μ n ] [μ +1, μ ] Substituting into (2), we have h n h, = λ +1 λ n (λ λ λ )[μ,, μ ] + λ +1 λ n [μ +1,, μ ] = λ λ n [μ,, μ ] +1 + λ λ n [μ,, μ ] =1 = λ +1 λ n [μ +1,, μ ] λ 0 λ n [μ 0,, μ ]. Since λ 0 = 0 and the matix is positive, Saaya Ünivesitesi Fen Bilimlei Enstitüsü Degisi, vol. 21, no. 3: pp ,

3 F. Aydın Agun, B. E. Rhoades, S. M. Mohammad Monotonicity Peseving Matices h n h, = λ +1 λ n [μ +1,, μ ] 0. (3) The E-J genealized Hausdoff matices, denoted by H α μ = (h (α) n ), wee defined independently by Endl [2] and Jaimovsi [6], with nonzeo enties h (α) n =( n ) n μ (α), 0 n, fo any α 0. Fo α = 0, the E-J matices educe to the odinay Hausdoff matices. Coollay 1. Fo 0 α < 1, a positive E-J genealized Hasudoff matix maps deceasing seuences into deceasing sequences. Poof. h n h, n ) n μ n α ( n + 1 ) μ. Since μ = n μ n μ +1, h n h, n ) { μ + n μ +1 } n α ( n + 1 ) μ n α ) (1 n n + 1 ) μ + ( n ) n μ +1 (4) n + 1 ) μ +1 + ( n + 1 ) μ. =1 Fo 0 α < 1, the binomial fom ( ), will vanish, n + 1 so the above equality will be h n h, = ( n ) n μ Coollay 2. Evey positive Hausdoff matix maps deceasing sequences into deceasing sequences. Poof: Odinay Hausdoff matices ae the special case of H-J matices obtained by setting λ n = n, o by using an E-J matices with α = 0. Let {p n } be a nonnegative sequence with p 0 > 0. A Nölund matix is a tiangula matix B with enties b n = p n, whee n = p n. Coollay 3. Evey Nölund matix, with deceasing sequence {p n }, peseves deceasing sequences. Poof: Using (ii) of Bennett's theoem b n b, = p n p +1 Since {p n } is deceasing, p n p, we can wite the above equation as p n p p n p n = ( 1 1 ) p = p p. Since (p ) is nonnegative sequence, Saaya Ünivesitesi Fen Bilimlei Enstitüsü Degisi, vol. 21, no. 3: pp ,

4 F. Aydın Agun, B. E. Rhoades, S. M. Mohammad Monotonicity Peseving Matices b n b, p p 0. Thee ae also some non-summability matices that peseve deceasing sequences. A factoable matix is a lowe tiangula matix with enties a n = a n b, whee a n depends only upon n and b depends only upon. Theoem 3. Let A be a positive factoable matix. If a n is non-inceasing the A peseves deceasing sequences. Poof: a n a, = a n b a b = (a n a ) b 0. Let (p ) be a nonnegative sequence with p 0 > 0, and n define = p. A weighted mean matix (N, p) is a tiangula matix with enties p /. A weighted mean matix is the special case of the factoable matix. Coollay 4. Evey nonnegative weighted mean matix peseves deceasing sequences. Poof: The poof is easy to veify since is noninceasing. It is clea that evey uppe tiangula matices ae monotonicity peseving matices if they have deceasing in columns; i.e., a n a,. The following theoem gives anothe appoach fo evey uppe tiangula matices. Theoem 4. Let A be positive uppe tiangula matix with enties c n c,+1, fo = n, n + 1, then A is a monotonicity peseving matix. Poof: y n y = c n x =n = c, x = c n x c,+1 x +1. =n =n Since {x n } is a positive deceasing sequence and and A is a positive matix with c n c,+1, then y n y 0 whee x n x 0. An infinite matix will be called a band matix if it has only a finite numbe of nonzeo diagonals. The width of a band matix efes to the numbe of nonzeo diagonals. An uppe band matix A has finite numbe of the diagonals on o above the main diagonal. Coollay 5. Let A be an uppe banded matix with enties a j a j+1,+1 0. Then A is a monotonicity peseving matix. In [1] Bennett emaed that matices of the fom a 1 a 2... a n a ( 1 a 2... a n a 1 a 2... a n 0.), (8) with a 1, a 2,..., a n 0, ae monotonicity-peseving. Coollay 6. Let A be an uppe banded matix with enties a ij = a j i+1, whee j = i, i + 1,..., n + i 1, and a 1, a 2,..., a n 0, is monotonicity-peseving. Coollay 7. Bidiagonal matices ae monotonicity deceasing if each a n a,+1. Poof: Let y = Ax whee A is a bidiagonal matix with two nonzeo diagonals a nj and a n. Then, y n y = a ni x i +1 a,i x i +1 = a ni x i a,i+1 x i+1 a ni x i a,i+1 x i = (a ni a,i+1 )x i 0. We shall call a matix D an fold weighted shift if, fo some positive intege, it has enties d n = { a = n, n + 1,, n +, 0 othewise whee a is a positive sequence. Saaya Ünivesitesi Fen Bilimlei Enstitüsü Degisi, vol. 21, no. 3: pp ,

5 F. Aydın Agun, B. E. Rhoades, S. M. Mohammad Monotonicity Peseving Matices Coollay 8. A fold weighted shift D matix peseves deceasing sequences if {a } is a deceasing sequences. REFERENCES [1] G. Bennett, Monotonicity peseving Matices, Analysis, Vol. 24, No. 4, pp , Dec [2] E. Endl, Untesuchungen ube Momentpobleme bei Vefahen vom Hausdoffschen typus, Math. Anal. Vol. 139, pp , Oct [3] G. H. Hady, Divegent seies, Vol. 334, Ameican Mathematical Soc., [4] F. Hausdoff, Summationsmethoden und Momentfolgen, I, Math. Z. Vol. 9, pp , Feb [5] F. Hausdoff, Summationsmethoden und Momentfolgen, II, Math. Z. Vol. 9, pp , Sep [6] A. Jaimovsi, The poduct of summability methods; new classes of tansfomations and thei popeties, Tech. Note, Contact No. AF61, pp , Saaya Ünivesitesi Fen Bilimlei Enstitüsü Degisi, vol. 21, no. 3: pp ,