The convergence of a family of integral operators with a positive kernel

The aim of this study is to investigate the convergence of a family of integral operators with a positive kernel in L p ( − ∞ , ∞ ) ( 1 < p < ∞ ) space generalized as L λ ( f , x ) = ∫ − ∞ ∞ ∑ k = 1 ∞ P k , λ f ( x + α k , λ t ) K λ ( t ) d t within these integral operators, Λ ⊂ R is an index set, for ∀ λ ∈ Λ , λ ≥ 0 , P k , λ and α k , λ are real numbers and ∑ k = 1 ∞ | P k , λ | ≤ M < ∞ , where M is independent of λ and for ∀ λ ∈ Λ , ∑ k = 1 ∞ P k , λ = 1 and sup k , λ { α k , λ } = α ∗ < ∞ .