Abstract In this paper, we study an asymptotic expansion for a general family of Bernstein–Durrmeyer type operators $$L_n^{j,i}$$ L n j , i , $$i,j\in {\mathbb {N}}_0$$ i , j ∈ N 0 . For $$i=j=1$$ i = j = 1 , these operators reduce to the genuine Bernstein–Durrmeyer operators, while for $$i=j=0$$ i = j = 0 they coincide with the classical Bernstein–Durrmeyer operators. We derive an explicit asymptotic expansion for $$L_n^{j,i}$$ L n j , i under suitable smoothness conditions on the approximated function.

Abstract Part I of the paper considered infinite orthogonal sums of regular subspaces in a Kreĭn space (that is, of subspaces which are themselves Kreĭn spaces). How precisely these sums should be defined and conditions for when such sums are themselves regular were determined. These included, for example, a boundedness condition for the sum of the corresponding orthogonal projections. The same problem is addressed here for (quasi-)pseudo-regular subspaces. Such subspaces are orthogonal direct sums of a regular space and isotropic, or neutral, subspaces. Alternate characterizations of such subspaces are given, and infinite orthogonal sums are examined via unconditional, or Moore-Smith, sums of operator ranges.

Abstract We characterize the topologizability and power boundedness of convolution and dual convolution operators on power series spaces. We determine necessary conditions for a Toeplitz operator to be m-topologizable, and power bounded on $$\Lambda _{1}(n)$$ Λ 1 ( n ) and $$\Lambda _{\infty }(n)$$ Λ ∞ ( n ) , and consequently on $$H(\mathbb {C})$$ H ( C ) and $$H(\mathbb {D})$$ H ( D ) .

Abstract The second eigenfunction of the Neumann Laplacian on convex, planar domains is considered. Inspired by the famous hot spots conjecture and a related result of Steinerberger, we show that potential critical points of this eigenfunction (and, in particular, interior “hot spots”) cannot be located “near the center” of the domain. The region in which critical points are excluded is described explicitly.