Research Article Hahn Sequence Space of Modals

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1 International Scholarly Research Notices, Article ID , 6 pages Research Article Hahn Sequence Space of Modals T. Balasubramanian 1 and S. Zion Chella Ruth 2 1 Department of Mathematics, Kamaraj College, Tuticorin, Tamilnadu , India 2 Department of Mathematics, Dr. G. U. Pope College of Engineering, Sawyerpuram, Tuticorin, Tamilnadu , India Correspondence should be addressed to S. Zion Chella Ruth; ruthalwin@gmail.com Received 6 June 2014; Revised 14 October 2014; Accepted 15 October 2014; Published 9 November 2014 Academic Editor: Mahdi Sanati Copyright 2014 T. Balasubramanian and S. Zion Chella Ruth. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original wor is properly cited. The history of modal intervals goes bac to the very first publications on the topic of interval calculus. The modal interval analysis is used in Computer graphics and Computer Aided Design (CAD), namely, the computation of narrow bounds on Bezier and B-Spline curves. Since modal intervals are used in many fields, we introduce a new sequence space h(gi) called the Hahn sequence space of modal intervals. We have given some new definitions and theorems. Some inclusion relation and some topological properties of this space are investigated. Also dual spaces of this space are computed. 1. Introduction Interval arithmetic was first suggested by Dwyer [1 in Furthermore, Moore and Yang [2, 3 have developed applications to differential equations. Chiao in 2002 [4 introduced sequence of interval numbers and defined usual convergence of sequences of interval number. Recently, Zararsız and Şengönül [5 introduced null, bounded, and convergent sequence space of modals. Hahn in 1922 [6 defined h space and G. Goes and S. Goes [7 in1970studiedthefunctional analytic properties of this space. The Hahn sequence space was initiatedby Rao in 1990 [8. The present paper is devoted to the study of Hahn sequence space of modal intervals. LetusdenotethesetofallrealvaluedclosedintervalbyI, the set of positive integers by N,andthesetofallrealnumbers by R. Any element of I is called interval number and it is denoted by x. That is x ={x R :x x x}. An interval number x is a closed subset of real numbers. Let x and x be, respectively, first and last points of the interval number x. Therefore, when x > x, x is not an interval number. But in modal analysis [x, x is a valid interval. A modal x ={[x, x : x, x R} is defined by a pair of real numbers x, x. Letus denote the set of all modals by gi.letussupposethat x, y gi. Then the algebraic operations between x and y are defined in the Kaucher arithmetic [9. For a modal x = [x, xdual operator is defined as dual x =[x, x. Thus,if x gi,then x dual x = [0,0 = 0, dual x gi.let us suppose that x gi;then x is called symmetric modal if x = x or vice versa. ThesetofallmodalsgI is a metric space with the metric d defined by d( x 1, x 2 )=max { x 1 x 2, x 1 x 2 }. (1) If x, y giand x x, y y then the set gi is reduced ordinary set of interval numbers which is complete metric space with the metric d defined in (1) [4. If we tae x 1 = [a, a and x 2 = [b, b, weobtaintheusualmetricofr with d( x 1, x 2 )= a b,wherea, b R. 2. Definitions and Preinaries Let f be a function from N to gi which is defined by f() = x, x =( x ).Then( x ) is called sequence of modals. Wewilldenotethesetofallsequencesofmodalsbyw(gI). Fortwosequencesofmodals( x ) and ( y ), the addition, scalar product, and multiplication are defined as follows: ( x + y )=[x +y, x + y, (α x )=[αx,αx, α R, ( x y )= [min(x y,x y, x y, x y ), max(x y,x y, x y, x y ), respectively. The set w(gi) is a vector space since the vector space rules are clearly provided. The zero element of w(gi) is

2 2 International Scholarly Research Notices the sequence θ =( θ ) = ([0, 0), alltermsofwhicharezero interval. If ( x ) w(gi)then inverse of ( x ), according to addition, is dual( x ). Let λ(gi) w(gi). Ifasequencespacecontainsa sequence ( e n ) of modals with the property that for every u λ(gi) there is a unique sequence of scalars ( t n ) such that n d( u, t 1 e t n e n ) 0 then ( e n ) is called a Schauder modal basis for λ(gi).theseries t e which has the sum u is then called the expansion of u with respect to ( e n ),andwe write u = t e. Let λ(gi) and μ(gi) be linear space of modals. Then a function A : λ(gi) μ(gi) is called a linear transformation if and only if, for all u 1, u 2 λ(gi) and all t 1, t 2 gi, A( t 1 u 1 + t 2 u 2 )= t 1 A u 1 + t 2 A u 2. Proposition 1. If ( x ), ( y ),and( r ) are sequences of symmetric modal, then the following equality holds: ( x ){( y ) ( r )} = ( x )( y ) ( x )( r ). (2) Definition 2. Asequence x =( x ) w(gi)of modals is said to be convergent to the modal x 0 if for each ε>0there exists a positive integer n 0 such that d( x, x 0 )<εfor all n 0 and we denote it by writing x = x 0.Thus, x = x 0 x =x 0 and x = x 0. Definition 3. Asequenceofmodals, x =( x ) w(gi),issaid to be modal fundamental sequence if for every ε>0there exists 0 Nsuch that d( x, x n )<εwhenever n,> 0. Definition 4. Asequenceofmodalsw(gI) is said to be solid if y =( y ) w(gi)whenever y x for all Nand x =( x ) w(gi). Definition 5. A sequence of generalized intervals w(gi) is said to be monotone if w(gi) contains the canonical preimage of all its step spaces. Definition 6 (Weierstrass M-test). Let f : gi gi be given. If there exists an M 0such that d[ f ( x), 0 M and the series M converges, then the series f ( x) is uniformly and absolutely convergent in gi. The following spaces are needed for our wor: c 0 (gi) = { u =( u ) w(gi): d( u, 0) = 0} c(gi)={ u =( u ) w(gi): d( u, u 0 )=0} bs (gi) = { u =( u ) w(gi):sup n l 1 (gi) = { u =( u ) w(gi): bv (gi) = { u =( u ) w(gi): bv 0 (gi) = { u =( u ) w(gi): d( u, 0) = 0} {d ( n n 1 σ (gi) = { u =( u ) w(gi):sup n n d( u, 0)} < } d( u, 0) < } d( u u +1, 0) < } d( u u +1, 0) <, u, 0) < } E(gI)={ u =( u ) w(gi):( u ) E(gI)} istheintegratedspaceofe(gi) de (gi) = { u =( u ) w(gi):( 1 u ) E(gI)} 3. Main Results is the differentiated space of E(gI). Define a sequence y =( y ) which will be frequently used as the T-transform of a sequence x = ( x ) w(gi).thatis, y =(T x) =( x x +1 ), 1. We introduce the sequence spaces h(gi) and h (gi) as the set of all sequences such that T-transforms of them are in l(gi). That is, h(gi)={ u =( u ) w(gi): d( u, 0) = 0}, d((t u), 0) <, h (gi) = { u =( u ) w(gi):supd((t u), 0) < }. (4) h(gi) is a normed space with the norm u = d((t u), 0). (3) l (gi) = { u =( u ) w(gi):supd( u, 0) < } cs (gi) = { u =( u ) w(gi): n d( n u, u 0 )=0} =0 Example 7. Consider the sequence u =( u ) defined by [1, 1, u =( u )={ 1 n [0, 0, >n. (5)

3 International Scholarly Research Notices 3 Note that d((t u), 0) = d(( u u +1 ), 0) = 0 which is convergent. Also d( u, 0) = 0. Hence u =( u ) h(gi). Theorem 8. h(gi) and h (gi) are complete metric spaces with the metrics d h and d h defined by d h ( u, Ṽ) = d[(t u), (TṼ), (6) Hence d((t u), 0) + + d[(t u),(t u (i) ) d[(t u (i) ),(T u (j) ) d[(t u (j) ), 0. (14) d h ( u, Ṽ) = supd[(t u), (TṼ), (7) respectively, where u =( u ) and Ṽ =(Ṽ ) are the elements of the space h(gi) or h (gi). Also from (10) and (11), d[(t u), 0) = 0. Hence u h(gi). Since{ u (i) } is an arbitrary fundamental sequence, the space h(gi) is complete. Similarly, we can prove h (gi) is complete space. Proof. Let { u (i) } be any fundamental sequence in the space h(gi), where{ u (i) }={ u (i) 0, u(i) 1,...}.Thenforagivenε>0, there exists a positive integer n 0 (ε) such that d h ( u (i), u (j) )= d[(t u (i) ),(T u (j) ) <ε, We obtain for each fixed Nfrom (8) that i, j n 0 (ε). (8) d[(t u (i) ),(T u (j) ) <ε for every i, j n 0 (ε) (9) Theorem 9. The space h(gi) is monotone. Proof. Let n<m;itfollowsfrom d( u n, 0) d ( u n u n+1, 0) + d ( u n+1 u n+2, 0) + +d( u m 1 u m, 0) + d ( u m, 0) that n 1 u(n) d ( u u +1, 0) + nd ( u n u n+1, 0) (15) which leads to the fact that {(T u (i) ) } is a fundamental sequence in gi for every fixed N. Since gi is complete, {(T u (i) ) } converges to (T u) as i. Consider the sequence {(T u) 1,(T u) 2,...};wehavefrom (9) for each m Nand i, j n 0 (ε) that m d[(t u (i) ),(T u (j) ) d h ( u (i), u (j) )<ε, for i, j n 0 (ε). (10) Tae any i n 0 (ε) and tae it as j firstandthenlet m in (8);weobtain d h ( u (i), u) <ε. (11) + (m 1) d( u m 1 u m, 0) + md ( u m, 0) = u(m). (16) The sequence ( u (n) ) is monotone increasing; it thus follows from u = n u (n) that u = n u (n) = sup n u (n).thiscompletestheproof. Theorem 10. The space h(gi) is not rotund. Proof. Consider the sequences u =( u ) and Ṽ =(Ṽ ) defined by u =( u )={[1, 1, 0, 0,...}, Ṽ =(Ṽ )={[1/2, 1/2,, [1/2, 1/2,, 0, 0,...}. (17) Therefore u (i) u. We have to prove u h(gi). Since { u (i) } is a fundamental sequence in h(gi),wehave Now, d((t u (i) ), 0), d( u, 0) = 0. (12) Then Also, d((t u), 0) <, d[ u, 0 = 0, d((tṽ), 0) <. (18) d[ṽ, 0 = 0. (19) d[(t u), 0 d [(T u),(t u (i) ) +d[(t u (i) ),(T u (j) ) + d [(T u (j) ), 0. (13) Thus u = ( u ) and Ṽ = (Ṽ ) are in h(gi) and d h ( u, 0) = d h (Ṽ, 0) = 1. Note that u = Ṽ,but d h (( u+ṽ)/2, 0) = 1. Therefore h(gi) is not rotund.

4 4 International Scholarly Research Notices Theorem 11. The space h(gi) is not solid. Proof. Consider the sequence Since [1, 1, u =( u )={ 1 n [0, 0, >n, Ṽ =(Ṽ )=([( 1), ( 1) ). (20) d[ṽ, 0 = d [ u, 0 = 1, d ((T u), 0) = 0, (21) it immediately follows that u =( u ) h(gi). However, it is trivial that d((tṽ), 0) = 2. d((tṽ), 0) d((t u), 0), which implies Ṽ =(Ṽ ) h(gi).thiscompletes the proof. 4. Dual Space of h(gi) Definition 12. The α-dual, β-dual, and γ-dual of s(gi) w(gi) are, respectively, defined by {s (gi)} α ={( u ) w(gi):( u Ṽ ) l 1 (gi) (Ṽ ) s(gi)} {s (gi)} β ={( u ) w(gi):( u Ṽ ) cs(gi) (Ṽ ) s(gi)} {s (gi)} γ ={( u ) w(gi):( u Ṽ ) bs(gi) (Ṽ ) s(gi)}. (22) Itistrivialthatthefollowinginclusionshold:{s(gI)} α {s(gi)} β {s(gi)} γ. Theorem 13. Let E(gI) and E 1 (gi) bethesetsofsequencesof modals. Then the following statements hold. (i) E(gI) [E(gI) ββ. (ii) [E(gI) βββ = [E(gI) β. (iii) If E 1 (gi) E(gI) then [E 1 (gi) β [E(gI) β. The same results hold for dual also. Proof. Let u =( u ), Ṽ =(Ṽ ),and w =( w ) be sequences of modal intervals. (i) Suppose u =( u ) [E(gI) ββ ;then( u Ṽ ) cs(gi) for at least one Ṽ =(Ṽ ) [E(gI) β.butṽ =(Ṽ ) [E(gI) β implies that ( u Ṽ ) cs(gi)for all w =( w ) E(gI). This means that u is not a member of E(gI). Hence E(gI) [E(gI) ββ. The proof for (ii) follows similarly. (iii) Suppose u =( u ) [E 1 (gi) β ;then( u Ṽ ) cs(gi) for all Ṽ =(Ṽ ) E 1 (gi). SinceE 1 (gi) E(gI), Ṽ =(Ṽ ) E(gI).Thus[E 1 (gi) β [E(gI) β. Define the sequence y = ( y ) which will be frequently used as the B-transform of a sequence x =( x ) w(gi). That is, y = (B x) = 1 x i. (23) i=1 The Cesaro space of l (gi) is the set of all sequences such that the B-transforms of them are in l (gi).thatis,σ(l (gi)) = { x =( x ), sup d[(b x), 0 < }. Theorem 14. σ(l (gi)) is a complete metric space with the metric d σ ( u, Ṽ) = sup d[(b u), (BṼ) where u = ( u ) and Ṽ =(Ṽ ) are the elements of space σ(l (gi)). Proof. Let { u (i) } be any fundamental sequence in the space σ(l (gi)) where { u (i) }={ u (i) 0, u(i) 1,...}.Thenforagivenε>0, there exists a positive integer n 0 (ε) such that d σ ( u (i), u (j) )=supd[(b u (i) ),(B u (j) ) <ε, (24) i, j n 0 (ε). We obtain for each fixed Nfrom (24) that d[(b u (i) ),(B u (j) ) <ε (25) which leads to the fact that (B u (i) ) is a fundamental sequence for every fixed N.SincegI is complete, (B u (i) ) (B u) as i. Consider the sequence {(B u) 1,(B u) 2,...}.Wehavefrom (25) for each m Nand i, j n 0 (ε) that supd[(b u (i) ),(B u (j) ) d σ ( u (i), u (j) )<ε, for,2...m, i,j n 0 (ε). (26) For any i n 0 (ε), taing it j first and letting m in (24),weobtain d σ ( u (i), u) <ε. (27) Finally, we proceed to prove u σ(l (gi)). Since { u (i) } is a fundamental sequence in σ(l (gi)), we have sup d[(b u (i) ), 0 <. Now, sup d[(b u), 0 d [(B u),(b u (i) ) + d [(B u (i) ),(B u (j) ) +d[(b u (j) ), 0, d[(b u), 0 supd[(b u),(b u (i) ) Hence u σ(l (gi)). + supd[(b u (i) ),(B u (j) ) + supd[(b u (j) ), 0 <. (28)

5 International Scholarly Research Notices 5 Since { u (i) } is an arbitrary fundamental sequence, the space σ(l (gi)) is complete. Theorem 15. The β-dual and γ-dual of h(gi) are σ(l (gi)). Proof. Let u = ( u ) h(gi) and (Ṽ ) σ(l (gi)). Since ( u ) h(gi),wehave d[ u, 0 = 0. Therefore for given ε>0, there exists n 0 such that d[ u, 0 < ε. Since (Ṽ ) σ(l (gi)), sup d[(bṽ), 0 <. Thus d[ṽ, 0 < for all and n. Hence there exists an M>0such that d[ṽ, 0 < M for all and n. Now, d[ u Ṽ, 0 < d [ u, 0 d [Ṽ, 0 < εm. (29) u Ṽ converges uniformly by Weierstrass M-test. Thus σ(l (gi)) [h (gi) β. (30) u =( u ) h(gi)implies > d((t u), 0) = d[ u (+1) u +1, 0 d( u +1, 0). d(( u u +1 ), 0) (33) The last series is convergent since h(gi) l 1 (gi). Hence also d[ u (+1) u +1, 0 < and therefore h(gi) bv(gi). Hence h(gi) l 1 (gi) bv (gi). (34) Conversely, (32) implies for ( u ) l 1 (gi) bv(gi) d((t u), 0) = d(( u u +1 ), 0) Conversely, suppose (Ṽ ) [h(gi) β. Then the series u Ṽ converges for all ( u ) h(gi).thisalsoholdsfor thesequenceofmodals( u ) defined by ( u )=([ 1,1)for all N. u Ṽ = [ 1, 1[V, V = max{ V, V converges uniformly. Thus sup d[(bṽ), 0 <. Hence (Ṽ ) σ(l (gi)), [h (gi) β σ(l (gi)). (31) Thus ( u ) h(gi). Therefore, + d[ u (+1) u +1, 0 d[ u, 0 = 0. d( u +1, 0) <, (35) From (30) and (31), [h(gi) β =σ(l (gi)).thiscompletesthe proof. Theorem 16. Consider (i) h(gi) l 1 (gi) c 0 (gi). (ii) h(gi) = l 1 (gi) bv(gi) = l 1 (gi) bv 0 (gi). Proof. (i) Let u =( u ) h(gi).then d((t u), 0) < and d[ u, 0 = 0. Consider d( u, 0) d((t u), 0) <. Therefore, u =( u ) l 1 (gi). Andalsosince d[ u, 0 = d[ u, 0 = 0, u =( u ) c 0 (gi). Therefore, u =( u ) l 1 (gi) c 0 (gi). Henceh(gI) l 1 (gi) c 0 (gi). (ii) For,2,..., ( u u +1 )= u +1 + u u +1 u +1 = u +1 +[ u (+1) u +1. (32) l 1 (gi) bv (gi) h (gi). (36) Hence from (34) and (36), we have shown that h(gi) = l 1 (gi) bv(gi). Similarly, we can prove other equalities. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. References [1 P. S. Dwyer, Linear Computation, Wiley, New Yor, NY, USA, [2 R.E.Moore,Automatic Error Analysis in Digital Computation, LSMD-48421, Locheed Missiles and Space Company, [3R.E.MooreandC.T.Yang, IntervalanalysisI, Locheed Missiles and Space Company LMSD , Locheed Missiles and Space Company, [4 K. P. Chiao, Fundamental properties of interval vector maxnorm, Tamsui Oxford Mathematical Sciences,vol.18, no. 2, pp , 2002.

6 6 International Scholarly Research Notices [5 Z. Zararsız and M. Şengönül, Some contributions to modals analysis, Thai Mathematics,vol.12,no.1,pp , [6 H. Hahn, Über folgen linearer operationen, Monatshefte für Mathemati und Physi,vol.32,no.1,pp.3 88,1922. [7 G. Goes and S. Goes, Sequences of bounded variation and sequences of Fourier coefficients. I, Mathematische Zeitschrift, vol. 118, pp , [8 K. C. Rao, The Hahn sequence space, Bulletin of the Calcutta Mathematical Society,vol.82,no.1,pp.72 78,1990. [9 E. Kaucher, Interval analysis in the extended interval space R, Computing, supplement 2, pp , 1980.

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