Maximal convergence theorems for functions of squared modulus holomorphic type in R2R2 and some applications

In this paper we extend the theory of maximal convergence introduced by Walsh to functions of squared modulus holomorphic type, i.e.F(x,y)=|g(x+iy)|2,(x,y)∈L,L⊂R2compact,where g   is holomorphic in an open connected neighborhood of {x+iy∈C:(x,y)∈L}{x+iy∈C:(x,y)∈L}. We introduce in accordance to the well-known complex maximal convergence number for holomorphic functions a real maximal convergence number for functions of squared modulus holomorphic type and prove several maximal convergence theorems. We achieve that the real maximal convergence number for F is always greater or equal than the complex maximal convergence number for g and equality occurs if L   is a closed disk in R2R2. Among other various applications of the resulting approximation estimates we show that for functions F   of squared holomorphic type which have no zeros in B¯2,r≔{(x,y)∈R2:x2+y2≤r} the relationlimsupn→∞En(B¯2,r,F)n=limsupn→∞En(∂B2,r,F)nis valid, where En(B¯2,r,F)≔inf{max(x,y)∈B¯2,r|F(x,y)-Pn(x,y)|:Pn:R2→R a polynomial of degree ⩽n}⩽n}.