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Some new lacunary statistical convergence with ideals.

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Summary

This study explores lacunary statistical convergence in sequence spaces using Musielak-Orlicz functions. Researchers examined the relationships between this convergence and summability, extending the analysis to probabilistic normed spaces.

Keywords:
Musielak-Orlicz functionideal convergencelacunary sequencesprobabilistic normed space

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Area of Science:

  • Real Analysis
  • Functional Analysis
  • Sequence Spaces

Background:

  • The study of convergence in sequence spaces is a fundamental area of mathematical analysis.
  • Musielak-Orlicz functions provide a generalized framework for defining sequence spaces.
  • Statistical convergence and its lacunary variants offer alternative modes of convergence.

Purpose of the Study:

  • To introduce and define lacunary statistical convergent sequence spaces using Musielak-Orlicz functions.
  • To investigate the relationship between lacunary statistical convergence and lacunary statistical summability.
  • To extend the study of lacunary statistical convergence to probabilistic normed spaces and explore its topological properties.

Main Methods:

  • Definition of lacunary statistical convergent sequence spaces based on Musielak-Orlicz functions.
  • Analysis of the relationship between lacunary statistical convergence and lacunary statistical summability using standard convergence techniques.
  • Extension of these concepts to probabilistic normed spaces, employing tools from functional analysis and topology.

Main Results:

  • Established the concept of lacunary statistical convergent sequence spaces defined by Musielak-Orlicz functions.
  • Demonstrated connections between lacunary statistical convergence and lacunary statistical summability.
  • Investigated the behavior of lacunary statistical convergence in probabilistic normed spaces, including some topological aspects.

Conclusions:

  • The paper successfully introduces and analyzes a new class of sequence spaces.
  • The findings contribute to the understanding of convergence modes in generalized sequence spaces.
  • The extension to probabilistic normed spaces opens avenues for further research in topological analysis of these spaces.