INVERSE THEOREMS IN Lp FOR SOME MULTIDIMENSIONAL POSITIVE LINEAR OPERATORS

Let {L_n}_(n∈s) be positive linear operators in L_(I), I=0, 1] or 0, ∞). This paper considers their variants in L_p(I×I)L_(n,m)(F; x, y)=L_n(L_m(F(u, v); y); x)=L_m(L_n(F(u, v); x); y), n, m∈N.The characterization problem for these operators is solved which gives the inverse theorems in L_p for multidimensional Bernstein type operators.

INVERSE THEOREMS IN $L^{p}$ FOR SOME MULTIDIMENSIONAL POSITIVE LINEAR OPERATORS

Let $\{\{ {L_n}\} _{n \in N}}\]$ be positive linear operators in $\{L_p}(I),I = 0,1]or0,\infty )\]$, $I=0,1]$ or $\0,\infty )\]$ This paper considers their variants in $\{L_p}(I \times I)\]$ $$\{L_{n,m}}(F;x,y) = {L_n}({L_m}(F(u,v);y);x) = {L_m}({L_n}(F(u,v);x);y),n,m \in N.\]$$ The characterization problem for these operators is solved which gives the inverse theorems in $L_{p}$ for multidimensional Bernstein type operators.