Statistical Convergence and Summability Methods in Functional Analysis

Statistical convergence and summability methods extend classical notions of limit and series summation by relaxing uniformity requirements and incorporating density-based criteria. Within functional analysis, these approaches enable the study of convergence of sequences of functions and operators in Banach and Hilbert spaces under weaker conditions than pointwise or uniform convergence. By quantifying the proportion of terms that deviate from a limit, statistical convergence captures asymptotic behaviour in contexts where sporadic large deviations occur. Summability methods, often defined via matrix transformations or weighted means, offer tools to assign sums to divergent series and to accelerate convergence in approximation schemes. Applications range from Korovkin-type approximation theorems for positive linear operators to spectral analysis of infinite matrices and the stability of functional equations. The interplay between statistical convergence of order α, deferred Nörlund summability and weighted A-summability has led to refined approximation results, demonstrating global significance in numerical analysis, signal processing and probability theory.

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Recent advances have introduced pointwise and uniform statistical convergence of order α for sequences of real and vector‐valued functions. These studies establish equivalence between α‐statistical Cauchy criteria and convergence, and explore relations with strong wpβ‐summability, enriching the theory of function spaces with order‐sensitive density notions.

Deferred Nörlund statistical summability has been developed over Banach spaces to unite equi‐statistical convergence and summability. New Korovkin-type approximation theorems employ this framework, yielding sharper convergence rates for test functions and demonstrating superiority to classical and purely statistical schemes through concrete operator examples.

Weighted A-summability has been used to construct a statistical variant of the Korovkin theorem. By defining weighted regular matrices and establishing necessary and sufficient conditions for weighted A‐statistical convergence, these works deliver constructive approximation results and illustrate them via positive linear operators, reinforcing connections between summability matrices and functional approximation.

Statistical Convergence and Summability Methods in Functional Analysis publication trend

The graph below shows the total number of articles in statistical convergence and summability methods in functional analysis across all publications each year (not limited to Nature Index journals).

Technical terms

Statistical convergence: A sequence converges statistically to a limit if the set of indices where terms deviate beyond any given tolerance has natural density zero.

Summability method: A rule, often given by an infinite matrix or weighted mean, that assigns a value to a possibly divergent series by transforming partial sums into a new sequence with improved convergence properties.

Korovkin-type approximation theorem: A result characterising convergence of positive linear operators on a function space via their action on a small set of test functions, ensuring uniform approximation on compact sets.

Banach space: A complete normed vector space in which every Cauchy sequence converges, forming a standard setting for functional analysis and operator theory.

Order α convergence: A refinement of statistical convergence where the index density is measured with respect to n^α, allowing finer control over convergence rates.

References

1. On pointwise and uniform statistical convergence of order α for sequences of functions. Fixed Point Theory and Algorithms for Sciences and Engineering (2013).
2. Statistical Deferred Nörlund Summability and Korovkin-Type Approximation Theorem. Mathematics (2020).
3. Statistical weighted A-summability with application to Korovkin’s type approximation theorem. Journal of Inequalities and Applications (2016).