Order regularity of two-node Birkhoff interpolation with lacunary polynomials

Rank of Submatrices of the Pascal Matrix

“…The index matrix Ir ,ĉ defined in Corollary 3.2 is nonzero only at entries (α i , β i ). These are (0, 1), (1,2) and (2,4). Hence, …”

“…This kind of matrix has been considered in [4,5], but for square matrices. In [3] we showed that this generalized Vandermonde, for the case that x = 1, is related to the Pascal matrix as follows: Now we will apply our result to polynomial approximation.…”

“…has linearly independent rows (full row rank), and the matrix [2,7,11,14,17,20], [4,9,15] has linear independent columns (full column rank). The index matrix Ir ,ĉ defined in Corollary 3.2 is nonzero only at entries (α i , β i ).…”

“…[4,15]} is an ordered sub-pair of length 2. But this sub-pair is not maximal since the ordered sub-pair { [2,7,14], [4,9,15]} is of length 3, which, as it turns out, is maximal. Note that {[2, 7, 11], [9, 10, 15]} is also a maximal sub-pair, hence these need not be unique.…”

“…In the proof of that result, we showed that the invertibility of square submatrices of T is equivalent to the unique solvability of a two-point Birkhoff interpolation problem. This Birkhoff interpolation problem has been studied in [1,2,6,7], and generalized to lacunary interpolation (see [4,5]). …”