Density by moduli and Wijsman statistical convergence

@article{Bhardwaj2016DensityBM,
  title={Density by moduli and Wijsman statistical convergence},
  author={Vinod Kumar Bhardwaj and Shweta Dhawan and Oleksiy Dovgoshey},
  journal={arXiv: Functional Analysis},
  year={2016}
}
In this paper, we generalized the Wijsman statistical convergence of closed sets in metric space by introducing the $f$-Wijsman statistical convergence these of sets, where $f$ is an unbounded modulus. It is shown that the Wijsman convergent sequences are precisely those sequences which are $f$-Wijsman statistically convergent for every unbounded modulus $f$. We also introduced a new concept of Wijsman strong Ces\`{a}ro summability with respect to a modulus, and investigate the relationships… 

Figures from this paper

KOROVKIN TYPE APPROXIMATION THEOREMS VIA f-STATISTICAL CONVERGENCE
The concept of f -statistical convergence which is, in fact, a generalization of statistical convergence, and is intermediate between the ordinary convergence and the statistical convergence, has
On Asymptotically f-statistical Equivalent Set Sequences in the Sense of Wijsman
The aim of this paper is to introduce a generalization of statistical convergence of asymptotically equivalent set sequences and examine some inclusion relations related to a new concept of Wijsman
Application of f-lacunary statistical convergence to approximation theorems
TLDR
The main object of this paper is to prove Korovkin type approximation theorems using the notion of f-lacunary statistical convergence, and as an application a corresponding Korvkin type theorem is established.
ON THE ASYMPTOTIC EQUIVALENCE OF UNBOUNDED METRIC SPACES
Let (X, d) be an unbounded metric space. To investigate the asymptotic behavior of (X, d) at infinity, one can consider a sequence of rescaling metric spaces (X, 1 rn d) generated by given sequence
On equivalence of unbounded metric spaces at infinity
Let (X, d) be an unbounded metric space. To investigate the asymptotic behavior of (X, d) at infinity, one can consider a sequence of rescaling metric spaces (X, 1 rn d) generated by given sequence

References

SHOWING 1-10 OF 33 REFERENCES
Density by Moduli and Lacunary Statistical Convergence
We have introduced and studied a new concept of -lacunary statistical convergence, where is an unbounded modulus. It is shown that, under certain conditions on a modulus , the concepts of lacunary
Wijsman convergence: A survey
A net 〈Aλ〉 of nonempty closed sets in a metric space 〈X, d〉 is declaredWijsman convergent to a nonempty closed setA provided for eachx εX, we haved(x, A)=limλd(x, A). Interest in this convergence
f-Statistical convergence of order α and strong Cesàro summability of order α with respect to a modulus
In this paper, following a very recent and new approach of Aizpuru et al. (Quaest. Math. 37:525-530, 2014), we further generalize a concept of α-density to that of fα$f_{\alpha}$-density, where f is
Density by moduli and statistical convergence
Abstract By using modulus functions we introduce a new concept of density for sets of natural numbers. Consequently, we obtain a generalization of the notion of statistical convergence which is
Density by Moduli and Statistical Boundedness
We have generalized the notion of statistical boundedness by introducing the concept of -statistical boundedness for scalar sequences where is an unbounded modulus. It is shown that bounded sequences
Functions transferring metrics to metrics
We study the properties of real functions f for which the compositions f ◦ d is a metric for every metric space (X, d). The explicit form is found for the invertible elements of the semigroup
Sequence spaces defined by a modulus
Ruckle[4] used the idea of a modulus function ƒ (see Definition 1 below) to construct the sequence space This space is an FK space, and Ruckle proved that the intersection of all such L ( f ) spaces
On convergence of closed sets in a metric space and distance functions
  • G. Beer
  • Mathematics
    Bulletin of the Australian Mathematical Society
  • 1985
Let CL(X) denote the nonempty closed subsets of a metric space X. We answer the following question: in which spaces X is the Kuratowski convergence of a sequence {Cn} in CL(X) to a nonempty closed
ON STATISTICAL CONVERGENCE
lim„ — {the number of k < n : | | > t} = 0. n These concepts are shown to be equivaleot. Also, Statistical convergence is studied as a regulär summability method, and it is shown that it cannot be
Inclusions between FK spaces and Kuttner's theorem
The result known as Kuttner's theorem [2] asserts that if 0 < p < 1 and A is a Toeplitz matrix then there is a sequence which is strongly Cesaro summable with index p but which is not A summable.
...
...