Abstract
In this study, we introduce and examine the concepts of q-double Cesaro matrices and q-statistical convergence and q-statistical limit point of double sequences. Also, we give some relations connected to these concepts.
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The idea of statistical convergence was given by Zygmund [1] in the first edition of his monograph published in Warsaw in 1935. The concept of statistical convergence was introduced by Steinhaus [2] and Fast [3] and later reintroduced by Schoenberg [4] independently. Statistical convergence has been discussed with different names in many branches of mathematics in the theory of Fourier analysis, Ergodic theory, Number theory, Measure theory, Trigonometric series, Turnpike theory and Banach spaces. Many authors have studied statistical convergence in the name of summability theory by Çınar et al. [5], Connor [6], Connor and Kline [7], Et et al. [8], Fridy ([9, 10]), Colak and Altın [11], Moricz [12], Mursaleen et al. [13] and many others.
Now we recall some basic definitions about q-integer. For any \(n\in \mathbb {N},\)q integer of n denotes \(\left[ n\right] _{q}\) ,which is defined as follows
In 1910, Jackson [14] was defined and studied q-integral. Later Lupas [15] introduced q-Brenstein polynomials. Bustoz et al. [16] studied q-Fourier theory and defined q-Hausdorff summability. Recently, q-integers and their applications have studied by many mathematicians such as Aktuğlu and Bekar [17].
Let \(A=( a_{jk}^{mn})\) be a four-dimensional infinite matrix of real numbers for all \(m,n=0,1,2,\ldots\) The sums
are called the A-transforms of the double sequence x. We say that a sequence x is A-summable to the limit s if the A-transform of x exists for all \(m,n=0,1,2,\ldots\) and convergent in the Pringsheim [18] sense, that is
The matrix corresponding to the first-order double Cesaro mean defined by
A matrix \(A=\left[ a_{jk}^{mn}\right]\) is RH-regular if and only if
It is obvious that the double Cesaro matrix defined by \(\left( 1\right)\) satisfies the above five conditions.
In the following theorem, we give without proof q-analog of the double Cesaro matrix.
\(C_{\left( 1,1\right) }\left( q\right) =\left( c_{jkmn}\left( q^{j+k}\right) \right)\) with
and \(C_{\left( 1,1\right) }^{2}\left( q\right) =\left( c_{jkmn}^{2}\left( q^{j+k}\right) \right)\) with
We give the following results without proof.
\(\left( i\right)\) \(C_{\left( 1,1\right) }\left( q\right)\) is conservative for each \(q\in \mathbb {R},\)
\(\left( ii\right)\) \(C_{\left( 1,1\right) }\left( q\right)\) is regular for each \(q\ge 1.\)
FormalPara Corollary 1If \(q_{1}\ne q_{2}\) then \(C_{\left( 1,1\right) }\left( q_{1}\right) \ne C_{\left( 1,1\right) }\left( q_{2}\right) .\) If \(q_{1}>1\) then \(C_{\left( 1,1\right) }\left( q^{j+k}\right)\) is regular, but \(C_{\left( 1,1\right) }\left( q^{-\left( j+k\right) }\right)\) is not regular.
FormalPara Theorem 2\(C_{\left( 1,1\right) }\left( q_{1}^{j+k}\right)\) is equivalent to \(C_{\left( 1,1\right) }\left( q_{2}^{j+k}\right)\) for \(1<q_{1}<q_{2}.\)
LetAbe a nonnegative regular matrix. Freedman and Sember [19] defined a density function by
where\(\chi _{k}\)denotes characteristic function of\(K\subseteq N.\)
Using \(C_{1}\) instead of A and ordinary limit of \(\lim \inf\) in \(\left( 2\right) ,\) we obtain natural density function such as
provided that limit exists.
Using\(C_{1}\left( q\right)\)instead of\(C_{1},\)Aktuğlu and Bekar [17] definedq-density as follows:
On generalizing the notion of the density of subsets of\(\mathbb {N}\), Tripathy [20] introduced the notion of the asymptotic density for subsets of\(\mathbb {N}\times \mathbb {N}\)as follows:
A subset E of \(\mathbb {N}\times \mathbb {N}\) is said to have density \(\delta \left( E\right)\) if
provided that limit exists.
We can define theq-density function\(\delta _{q}^{2}\left( K\right)\)as follows:
IfKis finite subset of\(N\times N,\)then obviously\(\delta _{q}^{2}\left( K\right) =0.\)
q-density function \(\delta _{q}^{2}\left( K\right)\) is well-defined as seen follows example:
i) Let \(K=\left\{ \left( 2j,2k\right) :j,k\in N\right\} ,\) then
Since \(q\ge 1\) we have \(\frac{1}{\left( 1+q\right) ^{2}}\le \frac{q}{ \left( 1+q\right) ^{2}}\le \frac{q^{2}}{\left( 1+q\right) ^{2}}\)
ii) Let \(K=\left\{ \left( j^{2},k^{2}\right) :j,k\in N\right\} .\) Then we have
and
Hence
So we have
A real double sequence \(x=\left( x_{jk}\right)\) is said to be q-statistically convergent to L if for every \(\varepsilon >0,\)\(\delta _{q}^{2}\left( K_{\varepsilon }\right) =0,\) where \(K_{\varepsilon }=\left\{ \left( j,k\right) :j\le n\,{\text { and }}\,k\le m:\left| x_{jk}-L\right| \ge \varepsilon \right\} .\) In this case we write \(st_{2}^{q}-\lim x=L.\)
q-Statistical convergence of double sequences is different from statistical convergence of double sequences. For this consider a double sequence defined by
Then \(x=\left( x_{jk}\right)\) is not statistically convergent, but q-statistically convergent to 0, since \(st_{2}^{q}-\lim x=0.\)
FormalPara Theorem 3If a double sequence\(x=\left( x_{jk}\right)\)for which there is a double sequence\(y=\left( y_{jk}\right)\)that is convergent\(x_{jk}=y_{jk}\)for almost all\(\left( q\right)\)then\(x=\left( x_{jk}\right)\)isq-statistical convergence.
FormalPara ProofOmitted. \(\square\)
FormalPara Definition 2Let \(C_{\left( 1,1\right) }^{q}\) be a double q-Cesaro matrix. A double sequence \(x=\left( x_{jk}\right)\) is said to be strongly \(C_{\left( 1,1\right) }^{q}\)-summable if there is a complex number L such that
Let \(x=\left( x_{jk}\right)\) be a double sequence. A double sequence \(x=\left( x_{jk}\right)\) is said to be \(C_{\left( 1,1\right) }^{q}\)-uniformly integrable if for each \(\varepsilon >0\) there exist \(N=N\left( \varepsilon \right)\) and \(h=h\left( \varepsilon \right)\) such that \(t>h\)
Let\(x=\left( x_{jk}\right)\)be a double sequence. Then\(x=\left( x_{jk}\right)\)is strongly\(C_{\left( 1,1\right) }^{q}\)-summable toLif and only ifxis\(C_{\left( 1,1\right) }^{q}\)- uniformly integrable toLand\(C_{\left( 1,1\right) }^{q}\)-statistically convergent toL.
FormalPara Definition 4If \(\left( x_{j_{n},k_{n}}\right)\) a subsequence of \(x=\left( x_{jk}\right)\) and let \(K=\left\{ \left( j_{n},k_{n}\right) :j_{1}<j_{2}<\cdots ;k_{1}<k_{2},\ldots \right\} .\) If \(\delta _{q}^{2}\left( K\right) =0,\) then \(\left( x_{j_{n},k_{n}}\right)\) is called subsequence of q-density zero or q-thin subsequence. On the other hand \(\left( x_{j_{n},k_{n}}\right)\) is a q-nonthin subsequence of x if K does not have density zero.
FormalPara Definition 5Let \(x=\left( x_{jk}\right)\) be a double sequence. The number L is q-statistical limit point of the double sequence \(x=\left( x_{jk}\right)\) provided that there exist a q-nonthin subsequence of \(x=\left( x_{jk}\right)\) that converges to L. By \(\wedge _{x}^{q}\) we denote the set of q-statistical limit points of double sequence \(\left( x_{jk}\right) .\)
FormalPara Definition 6Let \(x=\left( x_{jk}\right)\) be a double sequence and L be any real number. We say that a real number L is said to be q-statistical cluster point of the double sequence \(x=\left( x_{jk}\right)\) provided that for \(\varepsilon >0\)
By \(\Gamma _{x}^{q},\) we denote the set of all q-statistical cluster points of the double sequence \(x=\left( x_{jk}\right) .\)
FormalPara Theorem 5If\(x=\left( x_{jk}\right)\)is a double sequence if\(\left( x_{jk}\right)\)q-statistical convergenceLthen\(\wedge _{x}^{q}=\Gamma _{x}^{q}=L.\)
FormalPara ProofSuppose that \(\left( x_{jk}\right)\)q-statistical convergence L and \(x\in \Gamma _{x}^{q}.\) Suppose that there exist at least \(\ell \in \Gamma _{x}^{q}\) such that \(\ell \ne L.\) Thus there exists \(\varepsilon >0\) such that
holds. Hence
Since \(\left( x_{jk}\right)\)q-statistical convergence L we can write
which is a contradiction \(\ell \in \Gamma _{x}^{q}.\) Therefore \(\Gamma _{x}^{q}=L.\) On the other hand, since \(\left( x_{jk}\right)\)q-statistical convergence L, we get \(\wedge _{x}^{q}=\Gamma _{x}^{q}=L.\)\(\square\)
References
Zygmund A (1979) Trigonometric series. Cambridge University Press, Cambridge, UK
Steinhaus H (1951) Sur la convergence ordinaire et la convergence asymptotique. Colloq Math 2:73–74
Fast H (1951) Sur la convergence statistique. Colloq Math 2:241–244
Schoenberg IJ (1959) The integrability of certain functions and related summability methods. Amer Math Monthly 66:361–375
Et M, Çınar M, Karakaş M (2013) On \(\lambda\) -statistical convergence of order \(\alpha\) of sequences of function. J Inequal Appl 204:8
Connor JS (1988) The statistical and strong \(p-\)Cesàro convergence of sequences. Analysis 8:47–63
Connor JS, Kline J (1996) On statistical limit points and the consistency of statistical convergence. J Math Anal Appl 197:392–399
Et M, Şengül H (2014) Some Cesaro-type summability spaces of order \(\alpha\) and lacunary statistical convergence of order \(\alpha\). Filomat 28(8):1593–1602
Fridy JA (1985) On statistical convergence. Analysis 5:301–313
Fridy JA (1993) Statistical limit points. Proc Am Math Soc 118(4):1187–1192
Çolak R, Altin Y (2013) Statistical convergence of double sequences of order \(\alpha\). J Funct Spaces Appl Art. ID 682823, 5 pp
Móricz F (2003) Statistical convergence of multiple sequences. Arch Math (Basel) 81:82–89
Mursaleen M, Edely OHH (2003) Statistical convergence of double sequences. J Math Anal Appl 288:223–231
Jackson FH (1910) On a \(q\)-definite integrals. Q J Pure Appl Math 41:193–203
Lupaş AA (1987) \(q-\)analogue of the Bernstein operator. Seminar on numerical and statistical calculus Cluj-Napoca No 9
Bustoz J, Gordillo LF (2005) \(q-\)Hausdorff summability. J Comput Anal Appl 7:35–48
Aktuğlu H, Bekar Ş (2011) \(q-\)Cesáro matrix and \(q-\)statistical convergence. J Comput Appl Math 235:4717–4723
Pringsheim A (1900) Zur Theorie der zweifach unendlichen Zahlenfolgen. Math Ann 53:289–321
Freedman AR, Sember JJ (1981) Densities and summability. Pacific J Math 95:293–305
Tripathy BC (2003) Statistically convergent double sequences. Tamkang J Math 34:321–327
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Çinar, M., Et, M. q-Double Cesaro Matrices and q-Statistical Convergence of Double Sequences. Natl. Acad. Sci. Lett. 43, 73–76 (2020). https://doi.org/10.1007/s40009-019-00808-y
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DOI: https://doi.org/10.1007/s40009-019-00808-y