Asymptotic Estimates for the Sequences of the Operators on the Space of Multivariate Periodic Functions

In the paper we prove asymptotic estimates for sequences of linear positive operators on the space C2π( Rᵏ) C 2 π R k . A sample result: Let Kₙ:R→ [ 0,∞ ) K n : R → 0 , ∞ be a sequence of even continuous 2π 2 π -periodic functions such that 1/ 2π ∫ -ππKₙ( t) dt=1 1 2 π ∫ - π π K n t d t = 1 for all n∈ N n ∈ N and let Vₙ:C2π( Rᵏ) → C2π( Rᵏ) V n : C 2 π R k → C 2 π R k be the operator defined by $${aligned} & Vₙ( f) ( x₁,...,xₖ) ={1}{( 2π ) ᵏ}∫ _{[ -π ,π ] ᵏ}f( x₁-t₁,...,xₖ-tₖ) \\ & Kₙ( t₁) · · · Kₙ( tₖ) dt₁· · · dtₖ. {aligned}$$ V n f x 1 , . . . , x k = 1 2 π k