Abstract
In this work we introduce and investigate the ideas of statistical Riemann integrability, statistical Riemann summability, statistical Lebesgue integrability and statistical Lebesgue summability via deferred weighted mean. We first establish some fundamental limit theorems connecting these beautiful and potentially useful notions. Furthermore, based upon our proposed techniques, we establish the Korovkin-type approximation theorems with algebraic test functions. Finally, we present two illustrative examples under the consideration of positive linear operators in association with the Bernstein polynomials to exhibit the effectiveness of our findings.
Highlights
IntroductionLet [ a, b] ⊂ R, for every k ∈ N there defined a sequence (hk ) of functions such that hk : [ a, b] → R
We present the definitions of statistical Riemann integrability and statistical Riemann summability via deferred weighted summability mean
We present below the definitions of statistical Lebesgue integrability and statistical Lebesgue summability of a sequence of measurable functions via deferred weighted mean
Summary
Let [ a, b] ⊂ R, for every k ∈ N there defined a sequence (hk ) of functions such that hk : [ a, b] → R. Two eminent mathematicians Fast [1] and Steinhaus [2] independently introduced a new concept called statistical convergence in sequence space theory This nice concept is very useful for advanced study in pure and applied Mathematics. Srivastava et al [11] introduced and studied the concepts of deferred weighted summability mean and proved the Korovkin-type theorems and in the same year Srivastava et al [12]. Dutta et al [13] demonstrated some Korovkin-type approximation theorems via the usual deferred Cesàro summablity mean Such concepts have been generalised in many aspects. A sequence (hk )k∈N of functions is said to be statistically Riemann integrable to h (a function) on [ a, b] if, for each e > 0 and every x ∈ [ a, b], ∃ σe > 0, and for any tagged partition. We consider two illustrative examples involving suitable positive linear operators associated with the Bernstein polynomials to justify the effectiveness of our findings
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