Some new lacunary statistical convergence with ideals.

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.