Annales Mathematiae et Informatiae 38 (2011) pp. 27 36 http://ami.ektf.hu On I (q) -onvergene J. Gogola a, M. Mačaj b, T. Visnyai b a University of Eonomis, Bratislava, Slovakia e-mail: gogola@euba.sk b Faulty of Mathematis, Physis and Informatis Comenius University, Bratislava, Slovakia e-mail: visnyai@fmph.uniba.sk, maaj@fmph.uniba.sk Submitted Otober 28, 2010 Aepted Marh 10, 2011 Abstrat In this paper we will study the properties of ideals I (q) notion of I-onvergene of sequenes of real numbers. related to the We show that I (q) -onvergene are equivalent. We prove some results about modified and I (q) Olivier s theorem for these ideals. For bounded sequenes we show a onnetion between I (q) 1. Introdution -onvergene and regular matrix method of summability. In papers [9],and [10] the notion of I-onvergene of sequenes of real numbers is introdued and its basi properties are investigated. The I-onvergene generalizes the notion of the statistial onvergene (see[5]) and it is based on the ideal I of subsets of the set N of positive integers. Let I 2 N. I is alled an admissible ideal of subsets of positive integers, if I is additive (i.e. A, B I A B I), hereditary (i.e. A I, B A B I), ontaining all singletons and it doesn t ontain N. Here we present some examples of admissible ideals. More examples an be found in the papers [7, 9, 10, 12]. (a) The lass of all finite subsets of N form an admissible ideal usually denote by I f. (b) Let ϱ be a density funtion on N, then the set I ϱ = {A N : ϱ(a) = 0} is an admissible ideal. We will use the ideals I d, I δ, I u related to asymptoti,logarithmi,uniform density,respetively. For those densities for definitions see [9, 10, 12, 13]. 27
28 J. Gogola, M. Mačaj, T. Visnyai () For any q (0, 1 the set I (q) = {A N : a A a q < } is an admissible ideal. The ideal I (1) = {A N : a A a 1 < } is usually denoted by I. It is easy to see, that for any q 1 < q 2 ; q 1, q 2 (0, 1) I f I (q1) I (q2) I I d (1.1) In this paper will we study the ideals I (q). In partiular the equivalene between I (q), I (q), Olivier s like theorems for this ideals and haraterization of I (q) - onvergent sequenes by regular matries. 2. The equivalene between I (q) and I (q) -onvergene Let us reall the notion of I-onvergene of sequenes of real numbers, (f.[9, 10]). Definition 2.1. We say that a sequene x = (x n ) n=1 I-onverges to a number L and we write I lim x n = L, if for eah ε > 0 the set A(ε) = {n : x n L ε} belongs to the ideal I. I-onvergene satisfies usual axioms of onvergene i.e. the uniqueness of limit, arithmetial properties et. The lass of all I-onvergent sequenes is a linear spae. We will also use the following elementary fat. Lemma 2.2. Let I 1, I 2 be admissible ideals suh that I 1 I 2. If I 1 lim x n = L then I 2 lim x n = L. In the papers [9, 10] there was defined yet another type of onvergene related to the ideal I. Definition 2.3. Let I be an admissible ideal in N. A sequene x = (x n ) n=1 of real numbers is said to be I -onvergent to L R (shortly I lim x n = L) if there is a set H I, suh that for M = N \ H = {m 1 < m 2 <...} we have, lim x m k = L. k It is easy to prove, that for every admissible ideal I the following relation between I and I -onvergene holds: I lim x n = L I lim x n = L. Kostyrko, Šalát and Wilzynski in [9] give an algebrai haraterization of ideals I, for whih the I and I -onvergene are equal; it turns out that these ideals are with the property (AP). Definition 2.4. An admissible ideal I 2 N is said to satisfy the property (AP) if for every ountable family of mutually disjoint sets {A 1, A 2,...} belonging to I there exists a ountable family of sets {B 1, B 2,...} suh that A j B j is a finite set for j N and j=1 B j I.(A B = (A \ B) (B \ A)).
On I (q) -onvergene 29 For some ideals it is already known whether they have property (AP)(see [9, 10, 12, 13]). Now, will show the equivalene between I (q) and I (q) -onvergene. Theorem 2.5. For any 0 < q 1 the ideal I (q) has a property (AP). Proof. It suffies to prove that any sequenes (x n ) n=1 of real numbers suh that I (q) lim x n = ξ there exist a set M = {m 1 < m 2 <... < m k <...} N suh that N \ M I (q) I (q) and lim k x m k = ξ. For any positive integer k let ε k = 1 2 k and A k = {n N : x n ξ 1 lim x n = ξ, we have A k I (q), i.e. a A k a q <. 2 k }. As Therefore there exist an infinite sequene n 1 < n 2 <... < n k... of integers suh that for every k = 1, 2,... a>n k a A k Let H = [(n k, n k+1 A k ]. Then a q a H a>n 1 a A 1 a q + a>n 2 a A 2 a q < 1 2 k a q +... + a>n k a A k a q +... < < 1 2 + 1 2 2 +... + 1 2 k +... < + Thus H I (q). Put M = N \ H = {m 1 < m 2 <... < m k <...}. Now it suffies to prove that lim x 1 m k = ξ. Let ε > 0. Choose k 0 N suh that < ε. Let k 2 k 0 m k > n k0. Then m k belongs to some interval (n j, n j+1 where j k 0 and doesn t belong to A j (j k 0 ). Hene m k belongs to N \ A j, and then x mk ξ < ε for every m k > n k0, thus lim x m k = ξ. k 3. Olivier s like theorem for the ideals I (q) In 1827 L. Olivier proved the results about the speed of onvergene to zero of the terms of a onvergent series with positive and dereasing terms.(f.[8, 11]) Theorem A. If (a n ) n=1 is a non-inreasing sequenes and n=1 a n < +, then lim n a n = 0.
30 J. Gogola, M. Mačaj, T. Visnyai Simple example a n = 1 n if n is a square i.e. n = k2, (k = 1, 2,...) and a n = 1 2 n otherwise shows that monotoniity ondition on the sequene (a n ) n=1 an not be in general omitted. In [14] T.Šalát and V.Toma haraterized the lass S(T ) of ideals suh that n=1 for any onvergent series with positive terms. a n < + I lim n a n = 0 (3.1) Theorem B. The lass S(T ) onsists of all admissible ideals I P(N) suh that I I. From inlusions (1.1) is obvious that ideals I (q) do not belong to the lass S(T ). In what follows we show that it is possible to modify the Olivier s ondition n=1 a n < + in suh a way that the ideal I (q) will play the role of ideal I in Theorem B. Lemma 3.1. Let 0 < q 1. Then for every sequene (a n ) n=1 suh that a n > 0, n = 1, 2,... and n=1 a n q < + we have I (q) lim n a n = 0. Proof. Let the onlusion of the Lemma 3.1 doesn t hold. Then there exists ε 0 > 0 suh that the set A(ε 0 ) = {n : n a n ε 0 } doesn t belong to I (q). Therefore m q k = +, (3.2) where A(ε 0 ) = {m 1 < m 2 <... < m k <...}. By the definition of the set A(ε 0 ) we have m k a mk ε 0 > 0, for eah k N. From this m q k aq m k ε q 0 > 0 and so for eah k N a q m k ε q 0 m q k (3.3) From (3.2) and (3.3) we get aq m k = +, and hene n=1 aq n = +. But it ontradits the assumption of the theorem. Let s denote by S q (T ) the lass of all admissible ideals I for whih an analog Lemma 3.1 holds. From Lemma 2.2 we have: Corollary 3.2. If I is an admissible ideal suh that I I (q) then I S q (T ). Main result of this setion is the reverse of Corollary 3.2. Theorem 3.3. For any q (0, 1 the lass S q (T ) onsists of all admissible ideals suh that I I (q). Proof. It this suffiient to prove that for any infinite set M = {m 1 < m 2 <... < m k <...} I (q) we have M I, too. Sine M I (q) we have < +. m q k
On I (q) -onvergene 31 Now we define the sequene (a n ) n=1 as follows a mk = 1 m k (k = 1, 2,...), a n = 1 for n N \ M. 10n Obviously a n > 0 and n=1 a n q < + by the definition of numbers a n. Sine I S q (T ) we have I lim n a n = 0. This implies that for eah ε > 0 we have in partiular M = A(1) I. A(ε) = {n : n a n ε} I, 4. I (q) -onvergene and regular matrix transformations I (q) -onvergene is an example of a linear funtional defined on a subspae of the spae of all bounded sequenes of real numbers. Another important family of suh funtionals are so alled matrix summability methods inspired by [1, 6]. We will study onnetions between I (q) -onvergene and one lass of matrix summability methods. Let us start by introduing a notion of regular matrix transformation (see [4]). Let A = (a nk ) (n, k = 1, 2,...) be an infinite matrix of real numbers. The sequene (t n ) n=1 of real numbers is said to be A-limitable to the number s if lim s n = s, where s n = a nk t k (n = 1, 2,...). If (t n ) n=1 is A-limitable to the number s, we write A lim t n = s. We denote by F (A) the set of all A-limitable sequenes. The set F (A) is alled the onvergene field. The method defined by the matrix A is said to be regular provided that F (A) ontains all onvergent sequenes and lim t n = t implies A lim t n = t. Then A is alled a regular matrix. It is well-known that the matrix A is regular if and only if satisfies the following three onditions (see [4]): (A) K > 0, n = 1, 2,... a nk K; (B) k = 1, 2,... lim a nk = 0 (C) lim a nk = 1
32 J. Gogola, M. Mačaj, T. Visnyai Let s ask the question: Is there any onnetion between I-onvergene of sequene of real numbers and A-limit of this sequene? It is well know that a sequene (x k ) of real numbers I d-onverges to real number ξ if and only if the sequene is strongly summable to ξ in Caesaro sense. The omplete haraterization of statistial onvergene (I d -onvergene) is desribed by Fridy-Miller in the paper [6]. They defined a lass of lower triangular nonnegative matries T with properties: n a nk = 1 n N if C N suh that d(c) = 0, then lim They proved the following assertion: k C a nk = 0. Theorem C. The bounded sequene x = (x n ) n=1 is statistially onvergent to L if and only if x = (x n ) n=1 is A-summable to L for every A in T. Similar result for I u -onvergene was shown by V. Baláž and T. Šalát in [1]. Here we prove analogous result for I (q) -onvergene. Following this aim let s define the lass T q lower triangular nonnegative matries in this way: Definition 4.1. Matrix A = (a nk ) belongs to the lass T q if and only if it satisfies the following onditions: n (I) a nk = 1 lim (q) If C N and C I (q), then lim k C a nk = 0, 0 < q 1. It is easy to see that every matrix of lass T q is regular. As the following example shows the onverse does not hold. Example 4.2. Let C = {1 2, 2 2, 3 2, 4 2,..., n 2,...} and q = 1. Obviously C = I. Now define the matrix A by: I (1) a nk = a 11 = 1, a 1k = 0, k > 1 1 2k ln n, k l2, k n a nk = 1 l ln n, k = l2, k n a nk = 0, k > n It is easy to show that A is lower triangular nonnegative regular matrix but does not satisfy the ondition (q) from Definition 4.1. k<n 2 k C for n. Therefore A / T 1. a n 2 k = 1 ln n 2 (1 + 1 2 +... + 1 n ) ln n 2 ln n = 1 2 0
On I (q) -onvergene 33 Lemma 4.3. If the bounded sequene x = (x n ) n=1 is not I-onvergent then there exist real numbers λ < µ suh that neither the set {n N : x n < λ} nor the set {n N : x n > µ} belongs to ideal I. As the proof is the same as the proof on Lemma in [6] we will omit it. Next theorem shows onnetion between I (q) -onvergene of bounded sequene of real numbers and A-summability of this sequene for matries from the lass T q. Theorem 4.4. Let q (0, 1. Then the bounded sequene x = (x n ) n=1 of real numbers I (q) -onverges to L R if and only if it is A-summable to L R for eah matrix A T q. Proof. Let I (q) lim x n = L and A T q. As A is regular there exists a K R suh that n = 1, 2,... a nk K. It is suffiient to show that lim b n = 0, where b n = a nk.(x k L). For ε > 0 put B(ε) = {k N : x k L ε}. By the assumption we have B(ε) I (q). By ondition (q) from Definition 4.1 we have lim a nk = 0 (4.1) k B(ε) As the sequene x = (x n ) n=1 is bounded, there exists M > 0 suh that k = 1, 2,... : x k L M (4.2) Let ε > 0. Then b n k B( ε 2K ) a nk x k L + M k B( ε 2K ) a nk + M ε 2K k / B( ε 2K ) a nk x k L k B( ε 2K ) a nk + k / B( ε 2K ) a nk ε 2 (4.3) By part (q) of Definition 4.1 there exists an integer n 0 suh that for all n > n 0 ε a nk < 2M k B( ε 2K ) Together by (4.3) we obtain lim b n = 0. Conversely, suppose that I (q) lim x n = L doesn t hold. We show that there exists a matrix A T q suh that A lim x n = L does not hold, too. If I (q) lim x n = L L then from the firs part of proof it follows that A lim x n = L
34 J. Gogola, M. Mačaj, T. Visnyai L for any A T q. Thus, we may assume that (x n ) n=1 is not I (q) -onvergent, and by the above Lemma 4.3 there exist λ and µ (λ < µ), suh that neither the set U = {k N : x k < λ} nor V = {k N : x k > µ} belongs to the ideal I (q). It is lear that U V =. If U / I (q) then i U i q = + and if V / I (q) then i V i q = +. Let U n = U {1, 2,..., n} and V n = V {1, 2,..., n}. Now we define the matrix A = (a nk ) by the following way: Let s (1)n = i U n i q for n U, s (2)n = i V n i q for n V and s (3)n = n n / U V. As U, V / I (q) a nk = we have lim s (j)n = +, j = 1, 2, 3. a nk = k q s (1)n n U and k U n, a nk = 0 n U and k / U n, a nk = k q s (2)n n V and k V n, a nk = 0 n V and k / V n, a nk = k q s (3)n n / U V, a nk = 0 k > n, i=1 i q for Let s hek that A T q. Obviously A is a lower triangular nonnegative matrix. Condition (I) is lear from the definition of matrix A. Condition (q): Let B I (q) and b = k B k q < +. Then k B a nk 1 s (3)n for n. Thus A T q. For n U a nk x k = 1 s (1)n on other hand for n V a nk x k = 1 s (2)n k B {1,...,n} n k q χ U (k)x k < n Therefore A lim x n does not exist. k q χ B (k) λ s (1)n k q χ V (k)x k > µ s (2)n Corollary 4.5. If 0 < q 1 < q 2 1, then T q2 T q1. b s (3)n 0 n k q χ U (k) = λ n k q χ V (k) = µ. Proof. Let B I (q2) \ I (q1) and let (x n ) = χ B (n), n = 1, 2,... Clearly I (q2) lim x n = 0 and I (q1) lim x n does not exist. Let A be the matrix onstruted from the sequene (x n ) n=1 as in the proof of Theorem 4.4. In partiular A T q1 and A lim x n does not exist. Therefore A / T q2.
On I (q) -onvergene 35 Further we show some type well-known matrix whih fulfills ondition (I). Let (p j ) j=1 be the sequene of positive real numbers. Put P n = p 1 + p 2 +... + p n. Now we define matrix A = (a nk ) in this way: a nk = p k P n k n a nk = 0 k > n. This type of matrix is alled Riesz matrix. Espeially we put p n = n α, where 0 < α < 1. Then a nk = k α 1 α + 2 α +... + n α k n a nk = 0 k > n. This speial lass of matrix we denote by (R, n α ). It is lear that this matrix fulfills onditions (I) and (q). For this lass of matrix is true following impliation: I (q) lim x k = L (R, n α ) lim x k = L where (x k ) is a bounded sequene, 0 < q 1, 0 < α < 1. Converse does not hold. It is suffiient to hoose the harateristi funtion of the set of all primes P. Then (R, n α ) lim x k = 0, but I (q) lim x k does not exist, beause n P n q = +, where P is a se of all primes. Hene the lass (R, n α ) of matries belongs to T \ T q. Problem 4.6. If we take any admissible ideal I and define the lass T I of matries by replaing the ondition (I) in Definition 4.1 by ondition:if C N and C I, I admissible ideal on N then lim k C a nk = 0 then it is easy to see that the if part of Theorem 4.4 holds for I too. The question is what about only if part. Referenes [1] Baláž, V. - Šalát, T.: Uniform density u and orresponding I u onvergene, Math. Communiations 11 (2006), 1 7. [2] Brown,T.C. - Freedman,A.R.: The uniform density of sets of integers and Fermat s Last Theorem, C.R. Math. Rep. Aad. Si. Canada XII (1990), 1 6. [3] Buk,R.C.: The measure theoreti approah to density, Amer. J. Math. 68 (1946), 560 580. [4] Cooke, C.: Infinite matries and sequenes spaes, Moskva (1960). [5] Fast, H.: Sur la onvergene statistique, Colloq. Math. 2 (1951), 241 244. [6] Fridy,J.A.-Miller,H.I.: A matrix haraterization of statistikal onvergene Analysis 11 (1986), 59 66. [7] Gogola, J. - Červeňanský, J.: I (q) -onvergene of real numbers, Zborník vedekýh prá MtF STU v Trnave 18 (2005), 15 18
36 J. Gogola, M. Mačaj, T. Visnyai [8] Knopp, K: Theorie ubd Anwendung der unendlihen Reisen, Berlin (1931). [9] Kostyrko,P. - Šalát, T. - Wilzyński W.: I-onvergene, Real Anal. Exhange 26 (2000-2001), 669 686. [10] Kostyrko, P. - Mačaj, M. - Šalát, T. - Sleziak, M.: I-onvergene and extremal I-limit poits, Mathematia Slovaa 55 (2005), No. 4, 443 464. [11] Olivier, L: Remarques sur les series infinies et lem onvergene, J. reine angew. Math 2 (1827), 31 44. [12] Pašteka, M. - Šalát, T. - Visnyai, T.: Remark on Buk s measure density and generalization of asymptoti density, Tatra Mt. Math. Publ. 31 (2005), 87 101. [13] Sember,J.J. - Freedman,A.R.: On summing sequenes of 0 s and 1 s, Roky Mount J. Math 11 (1981), 419 425. [14] Šalát, T. - Toma, V.: A lassial Olivier s theorem and statistial onvergene, Annales Math. Blaise Pasal 1 (2001), 10 18. [15] Šalát, T. - Visnyai,T.: Subadditive measures on N and the onvergene of series with positive terms, Ata Math.(Nitra) 6 (2003), 43 52.