Directional extensions of univariate operators and Boolean sum approximation

Many approximation processes for multivariate real-valued functions are constructed on the basis of parametric extensions of univariate approximation operators. A parametric extension of a univariate operator with respect to the ν-th component is defined in such a way that it acts on a multivariate function as if all variables except of the ν- th one were fixed. We generalize this approach by introducing directional extensions of univariate operators, which are defined similarly as parametric extension, but with respect to some arbitrary direction in ℝ d . In particular, we investigate approximation by Boolean sums of directional extension operators.