An algebraic characterization of B-splines

B-splines of order k can be viewed as a mapping N taking (k+1)-tuple increasing real numbers a0<⋯<ak and giving result certain piecewise polynomial function. Looking at this whole, basic properties B-spline functions imply that it has the following algebraic properties: (1) N(a0,…,ak) local support contained in interval [a0,ak]; (2) allows refinement, i.e. for every a∈∪j=0k−1(aj,aj+1) we have if (α0,…,αk+1) is rearrangement points {a0,…,ak,a}, ‘old’ function linear combination ‘new’ N(α0,…,αk) N(α1,…,αk+1); (3) translation dilation invariant. It easy to see derivatives satisfy (1)–(3) well. Let F (k+1)-tuples some generalized In paper show under additional mild condition on size supports F(a0,…,ak) relative [a0,ak], are already sufficient deduce non-zero multiple (some derivative of) However, somewhat surprisingly, explicitly give examples choices satisfying but not form.