Korovkin‐type results for multivariate functions which are periodic with respect to one variable

Let K be a compact metric space and letC 2 π( K × R )denote the real Banach space of all continuous functions f : K × R → R which are 2π‐periodic with respect to the second variable. We prove the following Korovkin‐type result: Letψ : K × K → [ 0 , ∞ )be a continuous algebraic separating function such thatψ ( x , x ) = 0for allx ∈ K , and letV n : C 2 π( K × R ) → C 2 π( K × R )be a sequence of positive linear operators. Iflim n → ∞V n ( ψ ( · , x ) ⊗ 1 ) ( x , t ) = 0uniformly with respect to( x , t ) ∈ K × Randlim n → ∞V n ( 1 ⊗ f ) = 1 ⊗ funiformly onK × Rfor allf ∈ { 1 , sin , cos } , thenlim n → ∞V n ( f ) = funiformly onK × Rfor everyf ∈ C 2 π( K × R ) .As a corollary we deduce: IfK = [ a , b ] , thenlim n → ∞V n ( f ) = funiformly on[ a , b ] × Rfor everyf ∈ C 2 π( [ a , b ] × R )if and only iflim n → ∞V n ( f ) = funiformly on[ a , b ] × Rfor everyf ∈ { 1 ⊗ 1 , e 1 ⊗ 1 , e 2 ⊗ 1 , 1 ⊗ sin , 1 ⊗ cos } , wheree 1 ( x ) = xande 2 ( x ) = x 2 .