I and I convergent function sequences

Transcription

1 Mathematical Communications 10(2005), I and I convergent function sequences F. Gezer and S. Karakuş Abstract. In this paper, we introduce the concepts of I pointwise convergence, I uniform convergence, I pointwise convergence and I uniform convergence of function sequences and then we examine the relation between them. Key words: pointwise and uniform convergence, I convergence, I convergence AMS subject classifications: 40A30 Received February 22, 2005 Accepted June 9, Introduction Steinhaus [20] introduced the idea of statistical convergence [see also Fast [10]]. If K is a subset of positive integers N, thenk n denotes the set {k K : k n} and K n denotes the cardinality of K n. The natural density of K [18] is given by δ(k) := 1 lim n n K n, if it exists. The number sequence x =(x k ) is statistically convergent to L provided that for every ε>0thesetk := K(ε) :={k N : x k L ε} has natural density zero; in that case we write st lim x = L [10, 12]. Hence x is statistically convergent to L if ( C 1 χ K(ε) 0(as n,foreveryε>0), where )n C 1 is the Cesáro mean of order one and χ K is the characteristic function of the set K. Properties of statistically convergent sequences have been studied in [2, 12, 16, 19]. Statistical convergence can be generalized by using a nonnegative regular summability matrix A in place of C 1. Following Freedman and Sember [11], we say that a set K N has A density if δ A (K) := lim n (Aχ K ) n = lim n k K a nk exists, where A =(a nk ) is a nonnegative regular matrix. The number sequence x =(x k )isa statistically convergent to L provided that for every ε>0thesetk (ε) hasa density zero[3, 11, 16]. Connor gave an extension of the notion of statistical convergence where the asymptotic density is replaced by a finitely additive set function. Let µ be a finitely additive set function taking values in [0, 1] defined on a field Γ of subsets of N such Department of Mathematics, Faculty of Sciences and Arts Sinop, Ondokuz Mayıs University, , Sinop, Turkey, fgezer@omu.edu.tr Department of Mathematics, Faculty of Sciences and Arts Sinop, Ondokuz Mayıs University, , Sinop, Turkey, skarakus@omu.edu.tr

2 72 F. Gezer and S. Karakuş that if A <, thenµ (A) =0;ifA B and µ (B) =0,thenµ (A) = 0; and µ (N) =1[4, 6]. The number of sequence x =(x k )isµ statistically convergent to L provided that µ {k N : x k L ε} =0foreveryε>0[4, 6]. Let X. A class S 2 X of subsets of X is said tobe an ideal in X provided that S is additive and hereditary, i.e. if S satisfies these conditions: (i) S, (ii) A, B S A B S, (iii) A S, B A B S [15]. An ideal is called non-trivial if X/ S. A non-trivial ideal S in X is called admissible if {x} S for each x X [14]. The non-empty family of sets Ϝ 2 X is a filter on X if and only if (i) / Ϝ, (ii) for each A, B Ϝ we have A B Ϝ,(iii) for each A Ϝ and each B A we have B Ϝ. I 2 X is a non-trivial ideal if and only if Ϝ := Ϝ (I) :={X A : A I} is a filter on X [17]. Let Ϝ be a filter. Ϝ has property (A) if for any given countable subset {A j } of Ϝ, thereexistsana Ϝ such that A \ A j < for each j [5]. Kostyrko, Mačaj and Šalát [14, 15] introduced two types of ideal convergence. Let I be a non-trivial ideal in N. A sequence x =(x k ) of real numbers is said tobe I convergent to L if for every ε>0theseta(ε) :={k N : x k L ε} belongs to I [14]. In this case we write I lim x = L. Let I be an admissible ideal in N. A sequence x =(x k )ofrealnumbersis said tobe I convergent to L if there is a set H I, such that for M = N\H = {m 1 <m 2 <...} we have lim x mk = L. InthiscasewewriteI lim x = L [14]. k For every admissible ideal I the following relation between them holds: Let I be an admissible ideal in N. IfI limit x = L, theni limit x = L [14]. Note that for some ideals the converse of this result holds (see [14, Example 3.1]). Kostyrko, Mačaj and Šalát have given the necessary and sufficent condition for equivalence of I and I convergences. This condition is similar to the additive property for null sets in [4, 11]. An admissible ideal I in N is said to satisfy the condition (AP) if for every countable system {A 1,A 2,...} of mutually disjoint sets belonging to I there exist sets B j N, (j =1, 2,...) such that the symmetric differences A j B j (j =1, 2,...) are finite and B = B j belong to I [14]. j=1 It is known that I limit x = L I limit x = L if and only if I has the additive property [14]. Some results on I convergence may be found in [7, 8, 14, 15]. Note that if we define I δa = {K N : δ A (K) =0}, I δc1 = {K N : δ (K) =0} and I µ = {K Γ: µ (K) =0}, then we get the definition of A statistical convergence, statistical convergence and µ statistical convergence, respectively. In this paper we give the I analogues of results given by Duman and Orhan [9]. Throughout the paper I will be an admissible ideal, D R and (f n ) a sequence of real functions on D. 2. I and I convergent function sequences Definition 1. (f n ) converges I pointwise to f ε>0 and x D, K x / I and n 0 = n 0(ε,x) K x n n 0 and n K x, f n (x) f (x) <ε.

3 I and I convergent function sequences 73 In this case we will write f n f (I convergent) ond. Definition 2. We say that (f n ) converges I uniform to f ε>0 and x D, K/ I and n 0 = n 0(ε) K n n 0 and n K, f n (x) f (x) <ε. In this case we will write f n f (I convergent) ond. Definition 3. (f n ) converges I pointwise to f ε>0 and x D, {n : f n (x) f (x) ε} I. In this case we will write f n f (I convergent) ond. Definition 4. The sequence (f n ) of bounded functions on D converges I- uniformly to f ε>0 and x D, {n : f n f ε} I, where the form. B(D) is the usual supremum norm on B (D), the space of bounded functions on D. In this case we will write f n f (I convergent) ond. As in the ordinary case the property of Definition 1 implies that of Definition 3; and, of course for bounded functions, the property of Definition 2 implies that of Definition 4. IfI satisfy the condition (AP), then Definitions 1 and 3 are equivalent, and Definition 2 and 4 are equivalent. The following result is a I analogue of the result that is well-known in analysis. Theorem 1. Let for all n, f n be continuous on D. Iff n f (I convergent) on D, thenf is continuous on D. Proof. Assume f n f (I convergent) ond. Then fo r every ε>0, there exists a set K / Iand n 0 = n 0(ε) K such that f n (x) f (x) < ε 3 for each x D and for all n n o and n K. Let x 0 D. Since f n0 is continuous at x 0 D, thereisaδ>0such that x x 0 <δimplies f n0 (x) f n0 (x 0 ) < ε 3 for each x D. Now for all x D for which x x 0 <δ,wehave f (x) f (x 0 ) f (x) f n0 (x) + f n0 (x) f n0 (x 0 ) + f n0 (x 0 ) f (x 0 ) <ε. Since x 0 D is arbitrary, f is continuous on D. Now from Theorem 5 we get the following. Corollary 1. Let all functions f n be continuous on a compact subset D of R, and let I satisfy the condition (AP). If f n f (I convergent) on D, thenf is continuous on D. The next example shows that neither of the converses of Theorem 5 and Corollary 6 are true. Example 1. Let K/ I and define f n :[0, 1) R by { 1 f n (x) = 2,n/ K x n 1+x,n K. n Then we have f n f =0(I convergent) on[0, 1). Hence we get f n f = 0(I convergent) on[0, 1). Though all f n and f are continuous on [0, 1), it follows from Definition 4 that the I convergent of(f n ) is not uniform for c n := sup f n (x) f (x) = 1 x [0,1) 2 and I lim c n = Now we will give the following result that is an analogue of Dini s theorem.

4 74 F. Gezer and S. Karakuş Theorem 2. Let I satisfy the condition (AP). Let D be a compact subset of R and (f n ) a sequence of continuous functions on D. Assume that f is continuous and f n f (I convergent) on D. Also, let (f n ) be monotonic decreasing on D; i.e. f n (x) f n+1 (x) (n =1, 2,...) for every x D. Thenf n f (I convergent) on D. Proof. Write g n (x) =f n (x) f (x). By hypothesis, each g n is continuous and g n 0(I convergent) ond, also(g n ) is a monotonic decreasing sequence on D. Now, since g n 0(I convergent) ond and I satisfy the condition (AP), g n 0(I convergent) ond. Hence for every ε>0andeachx D there exists K x / Iand a number n (x) =n (x, ε) K x such that 0 g n (x) < ε 2 for all n n (x) andn K x. Since g n (x) is continuous at x D, foreveryε>0 there is an open set J (x) which contains x such that gn(x) (t) g n(x) (x) < ε 2 for all t J (x). Hence given ε>0, by monotonicity we have 0 g n (t) g n(x) (t) =g n(x) (t) g n(x) (x)+g n(x) (x) gn(x) (t) g n(x) (x) + gn(x) (x) for every t J (x) andforalln n (x) andn K x. Since D J (x) and D is a compact set, by the Heine-Borel theorem D has a finite open covering such that D J (x 1 ) J (x 2 )... J (x m ). Now, let K := K x1 K x2... K xm and N := max {n (x 1 ),n(x 2 ),..., n (x m )}. Observe that K / I. Then 0 g n (t) <ε for every t D and for all n N and n K. Sog n 0(I convergent) ond. Consequently g n 0(I convergent) ond, which completes the proof. Now we will give the Cauchy criterion for I uniform convergence but we first need a definition and a lemma: Definition 5. (f n ) is I Cauchy if for every ε>0 and every x D there is an n (ε) N such that { n : fn (x) f n(ε) (x) } ε I x D Lemma 1. Let (f n ) be a sequence of a real function on D. (f n ) is I convergent if and only if (f n ) is I Cauchy. Proof. First we establish that a I convergent sequence is I Cauchy. Suppose that (f n )isi convergent to f. Since { n : f n (x) f (x) < 2} ε / I.We can select an n (ε) N such that fn(ε) (x) f (x) < ε 2. The triangle inequality now yields that { n : f n (x) f n(ε) (x) <ε } / I.Sinceεwasarbitrary, (f n )isi Cauchy. Now suppose that (f n )isi Cauchy. Select n (1) such that { n : fn (x) f n(1) (x) } < 1 / I and let A 1 = { n : fn (x) f n(1) (x) < 1 }. Suppose that n (1) <n(2) <n(3) <... < n (p) have been selected in such a fashion that if 1 r s p and A s = { n : fn (x) f n(s) (x) < 1/2 s 1 }

5 I and I convergent function sequences 75 then A r / I and n (s) A r. Select N such that { n : fn (x) f N (x) < 1/2 P +1} / I. Since N 1 A j { n : fn (x) f N (x) < 1/2 P +1} / I, there exists an n (p +1) such that n (p) <n(p +1)and N { A j n : fn (x) f N (x) < 1/2 P +1} 1 A p+1 = { n : fn (x) f n(p+1) (x) < 1/2 p } { n : f n (x) f N (x) < 1/2 P +1}. Observe that A p+1 / Iand n (p +1) A s for all s p +1. Note that since fn(p) (x) f n(p+1) (x) < 2 p, ( f n(p) (x) ) is Cauchy, and hence there exists an f (x) such that lim p f n(p) (x) = f (x). We claim that (f n ) is I convergent to f (x). Let ε>0 be given and select p N such that fn(p) (x) f (x) ε < and ε>2 p. 2 Note that if f n (x) f (x) ε, then fn(p) (x) f n (x) ε 2 > 21 p, and hence n is not an element of A p. It follows that {n : f n (x) f (x) ε} I and that (f n ) is I convergent to f (x). Theorem 3. Let I satisfy the condition (AP) and let (f n ) be a sequence of bounded functions on D. Then(f n ) is I uniformly convergent on D if and only if for every ε>0 there is an n (ε) N such that { n : } fn f B(D) n(ε) <ε / I (1) Note: The sequence (f n ) satisfying property (1) is said to be I uniformly Cauchy on D. Proof. Assume that (f n ) converges { I uniformly to } a function f defined on D. Let ε>0. Then we have n : f n f B(D) <ε / I. We can select an { n (ε) N such that n : f n(ε) f } B(D) <ε / I. The triangle inequality yields { that n : } fn f B(D) n(ε) <ε / I. Since ε is arbitrary, (f n )isi uniformly Cauchy on D. Conversely, assume that (f n )isi uniformly Cauchy on D. Letx D be fixed. By (1), for every ε>0thereisann (ε) N such that { n : fn (x) f n(ε) (x) } <ε / I. Hence {f n (x)} is I Cauchy, soby Lemma 10 we have that {f n (x)} converges I convergent to f (x). Then f n f (I convergent) ond. No w we shall sho w that this convergence must be uniform. Note that since I satisfy the condition (AP), by (1) there is a K/ Isuch that fn f B(D) n(ε) < ε 2 for all n n (ε) and n K. Soforeveryε>0thereisaK/ Iand n (ε) N such that f n (x) f m (x) <ε (2)

6 76 F. Gezer and S. Karakuş for all n, m n (ε) andn, m K and for each x D. Fixing n and applying the limit operator on m K in (2), we conclude that for every ε>0thereisak/ I and an n (ε) N such that f n (x) f (x) <εfor all n n 0 and for each x D. Hence f n f (I convergent) ond, consequently f n f (I convergent) on D. 3. Applications Using I uniform convergence, we can also get some applications. We merely state the following theorems and omit the proofs. Theorem 4. Let I satisfy the condition (AP). If a function sequence (f n ) converges I uniformly on [a, b] to a function f and f n is integrable on [a, b], then f is integrable on [a, b]. Moreover, I lim b a f n (x) dx = b a I lim f n (x) dx = b a f (x) dx Theorem 5. Let I satisfy the condition (AP). Suppose that (f n ) is a function sequence such that each (f n ) has a continuous derivative on [a, b]. If f n f (I convergent) on [a, b] and f n g (I convergent) on [a, b], then f n f (I convergent) on [a, b], wheref is differentiable, and f = g. 4. Function sequences that preserve I convergence This section is motivated by a paper of Kolk [13]. Recall that function sequence (f n ) is called convergence-preserving (or conservative) on D R if the transformed sequence {f n (x)} converges for each convergent sequence x =(x n ) from D [13]. In this section, analogously, we describe the function sequences which preserve the I convergence of sequences. Our arguments also give a sequential characterization of the continuty of I limit functions of I uniformly convergent function sequences. This result is complementary to Theorem 5. First we introduce the following definition. Definition 6. Let D R and (f n ) be a sequence of real functions on D. Then (f n ) is called a function sequence preserving I-convergence (or I-convergent conservative) on D if the transformed sequence {f n (x)} converges I for each I convergent sequence x =(x n ) from D. If(f n ) is I convergent conservative and preserves the limits of all I convergent sequences from D, then(f n ) is called I convergent regular on D. Hence, if (f n )isconservativeond, then(f n )isi convergent conservative on D. But the following example shows that the converse of this result is not true. Example 2. Let K/ I. Define f n :[0, 1] R by { 0,n K f n (x) = 1,n/ K

7 I and I convergent function sequences 77 Suppose that (x n )from[0, 1] is an arbitrary sequence such that I lim x = L. Then, for every ε>0, {n : f n (x n ) 0 ε} I. Hence I lim f n (x n )=0,so(f n ) is I convergent conservative on [0, 1]. But observe that (f n )isnotconservativeon [0, 1]. Now we give the first result of this section. But we need the following lemma: Lemma 2. Let I satisfy the condition (AP). If (fn r (x)) is a countable collection of sequences that are I convergent, then there exists λ : N N such that lim n fλ(k) r (x) exists for each r and {λ (k) :k N} / I. Proof. Let Ϝbe the filter generated by convergence in I convergence. Since each (fn r (x)) is I convergent, there is an A r Ϝ such that (fn r (x)) c Ar. Since Ϝ has property (A), there is an A Ϝ such that A \ A r < for each r. Suppose A = {n 1,n 2,...} where n 1 <n 2 <... and λ : N N satisfies λ (k) =n k for all k. Now lim n fλ(k) r (x) exists for each r and {λ (k) :k N} = A/ I. Theorem 6. Let I satisfy the condition (AP) and let (f k ) be a sequence of functions defined on closed interval [a, b] R. Then (f k ) is I convergent conservative on [a, b] if and only if (f k ) converges I uniformly convergent on [a, b] to a continuous function. Proof. Necessity. Assume that (f k )isi convergent conservative on [a, b]. Choose the sequence (v k ) = (t, t,...) for each t [a, b]. Since I lim v k = t, I lim f k (v k ) exists, hence I lim f k (v k )=f(t) forallt [a, b]. We claim that f is continuous on [a, b]. Toprove this we suppose that f is not continuous at a point t 0 [a, b]. Then there exists a sequence (u k )in[a, b] such that lim u k = t 0, but lim f (u k ) exists and lim f (u k ) f (t 0 ). Since f k f (I convergent) on[a, b] and I satisfy the condition (AP), we obtain f k f (I convergent) on[a, b]. Hence, for each j, {f k (u j ) f (u j )} 0(I convergent). Hence there exists λ : N N such that {λ (k) :k N} / Iand lim [ f λ(k) (u j ) f (u j ) ] =0 for each j. Now, by the diagonal process [1,p.192] we can choose an increasing index sequence (n k ) in such a way that {n k : k N} / Iand lim [f nk (u k ) f (u k )] = 0. Now define a sequence x =(t i )by t 0, i = n k and i is odd t i = u k,i= n k and i is even 0,otherwise. Hence t i t 0 (I convergent), which implies I lim t i = t 0. But if i = n k and i is odd, then lim f nk (t 0 ) = f (t 0 ), and if i = n k and i is even, then lim f nk (u k ) = lim [f nk (u k ) f (u k )] + lim f (u k ) f (t 0 ). Hence {f i (t i )} is not I convergent since the sequence {f i (t i )} converges to two different limit points and has two disjoint subsequences whose index set does not belong to I. So, the sequence {f i (t i )} is not I convergent, which contradicts the hypothesis. Thus f must be continuous on [a, b]. It remains toprove that (f k ) converges I convergent uniformly on [a, b] tof. Assume that (f k )isnoti uniformly convergent on [a, b] to f, then(f k )isnoti uniformly convergent on [a, b] tof. Hence, for an arbitrary index sequence (n k )with{n k : k N} / I, there exists a number ε 0 > 0

8 78 F. Gezer and S. Karakuş and numbers t k [a, b] such that f nk (t k ) f (t k ) 2ε 0 (k N). The bounded sequence x =(t k ) contains a convergent subsequence (t ki ), I lim t ki = α, say. By the continuity of f, lim f (t ki ) = f (α). Sothere is an index i 0 such that f(t ki ) f (α) <ε 0 (i i o ). For the same i s, we have f nki (t ki ) f (α) f nki (t ki ) f (t ki ) f(t ki ) f (α) ε 0. (3) Now, defining α, j = n ki and j is odd u j = t ki,j= n ki and j is even 0, otherwise, we get u j α (I convergent). Hence I lim u j = α. But if j = n ki and j is odd, then lim f (t ki )=f(α), and if j = n ki and j is even, then, by (3), lim f (t ki ) f (α). Hence {f i (t i )} is not I convergent since the sequence {f i (t i )} converges to two different limit points and has two disjoint subsequences whose index set does not belong to I So, the sequence {f i (t i )} is not I convergent, which contradicts the hypothesis. Thus (f k )mustbei uniformly convergent to f on[a, b]. Sufficiency. Assume that f n f (I convergent)on[a, b]andf is continuous. Let x =(x n )beai convergent sequence in [a, b] withi lim x n = x 0. Since I satisfy the condition (AP), x n x 0 (I convergent), sothere is an index sequence {n k } such that lim x nk = x 0 and {n k : k N} / I. By the continuity of f at x 0, lim f (x nk )=f(x 0 ). Hence f (x n ) f (x 0 )(I convergent). Let ε>0be given. Then there exists K 1 / Iand a number n 1 K 1 such that f (x n ) f (x 0 ) < ε 2 for all n n 1 and n K 1. By assumption I satisfy the condition (AP). Hence the I uniform convergence is equivalent to the I uniform convergence, so there exists a K 2 / Iand a number n 2 K 2 such that f n (t) f (t) < ε 2 for every t [a, b] foralln n 2 and n K 2. Let N := max {n 1,n 2 } and K := K 1 K 2. Observe that K/ I. Hence taking t = x n we have f n (x n ) f (x 0 ) f n (x n ) f (x n ) + f (x n ) f (x 0 ) <ε for all n N and n K. This shows that f n (x n ) f (x 0 )(I convergent) which necessarily implies that I lim f n (x n )=f(x 0 ), whence the proof follows. Theorem 6 contains the following necessary and sufficient condition for the continuity of I convergence limit functions of function sequences that converge I convergent uniformly on a closed interval. Theorem 7. Let I satisfy the condition (AP) and let (f k ) be a sequence of functions that converges I convergent uniformly on a closed interval [a, b] to a function f. The function f is continuous on [a, b] if and only if (f k ) is I convergent conservative on [a, b]. Now, we study the I convergence regularity of function sequences. If (f k )is I convergent regular on [a, b], then obviously I lim f k (t) =t for all t [a, b]. So, taking f (t) =t in Theorem 6, we immediately get the following. Theorem 8. Let I satisfy the condition (AP) and let (f k ) be a sequence of functions on [a, b]. Then (f k ) is I convergent regular on [a, b] if and only if I convergent uniformly on [a, b] to the function f defined by f (t) =t.

9 I and I convergent function sequences 79 References [1] R. G. Bartle, Elements of Real Analysis, John Wiley & Sons, Inc., New York, [2] J. Connor, The statistical and strong p Cesáro convergence of sequences, Analysis 8(1988), [3] J. Connor, On strong matrix summability with respect to a modulus and statistical convergence, Canad. Math. Bull. 32(1989), [4] J. Connor, Twovaluedmeasuresandsummability,Analysis10(1990), [5] J. Connor, R-Type summability methods, Cauchy criteria, P-sets and statistical convergence, Proc. Amer. Math. Soc. 115(1992), [6] J. Connor J. Kline, On statistical limit points and the consistency of statistical convergence, J. Math. Anal. Appl. 197(1996), [7] K. Demirci, I-limit superior and limit inferior, Mathematical Communications 6(2001), [8] K. Demirci S. Yardimci, σ core and I core of bounded sequences, J. Math.Anal. Appl. 290(2004), [9] O. Duman C. Orhan, µ statistically convergent function sequences, Czech. Math. Journal, 54(129) (2004), [10] H. Fast, Sur la convergence statistique, Colloq. Math. 2(1951), [11] A. R. Freedman J. J. Sember, Densities and summability, Pacific J. Math. 95(1981), [12] J. A. Fridy, On statistical convergence, Analysis5(1985), [13] E. Kolk, Convergence-preserving function sequences and uniform convergence, J. Math. Anal. Appl. 238(1999), [14] P. Kostyrko, M. Mačaj, T. Šalát, Statistical convergence and I- convergence, Real Anal. Exchange, toappear [15] P. Kostyrko, M. Mačaj, T. Šalát, I-convergence, Real Anal. Exchange, toappear [16] H. I. Miller, A measure theoretical subsequence characterization of statistical convergence, Trans. Amer. Math. Soc. 347(1995), [17] J. Nagata, Modern General Topology, North Holland, Amsterdam-London, [18] I. Niven, H. S. Zuckerman, An Introduction to the Theory of Numbers, fourth ed., Wiley, New York, 1980.

10 80 F. Gezer and S. Karakuş [19] T. Šalát, On statistically convergent sequences of real numbers, Math. Slovaca 30(1980), [20] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math. 2(1951),