New Definitons about Ai- Statistical Convergence with Respect to a Sequence of Modulus Functions and Lacunary Sequences

In this paper, we generalize the concept of I statistical convergence, which is a recently introduced summability method, with an in…nite matrix of complex numbers and a modulus function. Lacunary sequences will also be included in our de…nitions. The name of our new method will be AI -lacunary statistical convergence with respect to a sequence of modulus functions. We also de…ne strongly AI -lacunary convergence and we give some inclusion relations between these methods. 1. Introduction As is known, convergence is one of the most important notions in Mathematics. Statistical convergence extends the notion. We can easily show that any convergent sequence is statistically convergent, but not conversely. Let E be a subset of N; the set of all natural numbers. d(E) := limn 1 n n P j=1 E(j) is said to be natural density of E whenever the limit exists, where E is the characteristic function of E: Statistical convergence was given by Zygmund in the …rst edition of his monograph published in Warsaw in 1935 [23]. It was formally introduced by Fast [6] and Steinhaus [21] and later was reintroduced by Schoenberg [20]. It has become an active area of research in recent years. This concept has applications in di¤erent …elds of mathematics such as number theory by Erdös and Tenenbaum [5] ; measure theory by Miller [14] ; trigonometric series by Zygmund [23] ; summability theory by Freedman and Sember [7] ; etc. Following this very important de…nition, the concept of lacunary statistical convergence was de…ned by Fridy and Orhan [9]. Also, Fridy and Orhan gave the relationships between the lacunary statistical convergence and the Cesàro summability. Freedman and Sember. established the connection between the strongly Cesàro summable sequences space j 1j and the strongly lacunary summable sequences space N in their work [7] published in 1978. I convergence has emerged as a kind of generalization form of many types of convergence. This means that, if we choose di¤erent ideals we will have di¤erent convergences. Koystro et. al. introduced this concept in a metric space [11] : We will explain this situation with two examples later. Before de…ning I convergence, the de…nitions of ideal and …lter will be needed. An ideal is a family of sets I 2N such that (i) ; 2 I, (ii) A;B 2 I implies A [ B 2 I, iii) For each A 2 I and each B A implies B 2 I. An ideal is called 2000 Mathematics Subject Classi…cation. 40G15, 40A35. Key words and phrases. Lacunary sequence, ideal convergence, modulus function. 1 Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 20 February 2018 doi:10.20944/preprints201802.0126.v1 © 2018 by the author(s). Distributed under a Creative Commons CC BY license. 2 ÖMER K· IŞ· I, HAF· IZE GÜMÜŞ, AND EKREM SAVAS non-trivial if N = 2 I and a non-trivial ideal is called admissible if fng 2 I for each n 2 N: A …lter is a family of sets F 2N such that (i) ; = 2 F , (ii) A;B 2 F implies A \B 2 F , iii) For each A 2 F and each A B implies B 2 F . If I is an ideal in N then the collection, F (I) = fA N : NnA 2 Ig forms is a …lter in N which is called the …lter associated with I. Now let’s remember the de…nition of modulus function. The notion of a modulus function was introduced by Nakano [15]. We recall that a modulus f is a function from [0;1) to [0;1) such that (i) f (x) = 0 if and only if x = 0, (ii) f (x+ y) = f (x)+f (y) for x, y 0, (iii) f is increasing and (iv) f is continuous from the right at 0. It follows that f must be continuous on [0;1) : Connor [2], Bilgin [1], Maddox [13], Kolk [10], Pehlivan and Fisher [16] and Ruckle [17] used a modulus function to construct sequence spaces. Now let S be the space of sequence of modulus function F = (fk) such that limx!0+ supk fk (x) = 0. Throughout this paper the set of all modulus functions determined by F and it will be denoted by F = (fk) 2 S for every k 2 N. In this paper, we intend to unify these approaches and use ideals to introduce the notion of AI-lacunary statistically convergence with respect to a sequence of modulus functions. 2. Definitions and Notations First we recall some of the basic concepts which will be used in this paper. Let A = (aki) be an in…nite matrix of complex numbers. We write Ax = (Ak (x)), if Ak (x) = 1 X i=1 akixk converges for each k. De…nition 2.1. [6]A number sequence x = (xk) is said to be statistically convergent to the number L if for every " > 0; lim n!1 1 n jfk n : jxk Lj "gj = 0: In this case we write st limxk = L: As we said before, statistical convergence is a natural generalization of ordinary convergence i.e. if limxk = L; then st limxk = L: By a lacunary sequence we mean an increasing integer sequence = fkrg such that k0 = 0 and hr = kr kr 1 ! 1 as r ! 1: Throughout this paper the intervals determined by will be denoted by Ir = (kr 1; kr]. De…nition 2.2. [9] A sequence x = (xk) is said to be lacunary statistically convergent to the number L if for every " > 0; lim r!1 1 hr jfk 2 Ir : jxk Lj "gj = 0: In this case we write S limxk = L or xk ! L(S ): De…nition 2.3. [9] The sequence space N , which is de…ned by N = ( (xk) : lim r!1 1 hr X k2Ir jxk Lj = 0 ) : Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 20 February 2018 doi:10.20944/preprints201802.0126.v1