Linear independence of pseudo-splines

Abstract

Abstract. In this paper, we show that the shifts of a pseudo-spline are linearly independent. This is stronger than the (more obvious) statement that the shifts of a pseudo-spline form a Riesz system. In fact, the linear independence of a compactly supported (refinable) function and its shifts has been studied in several areas of approximation and wavelet theory. Furthermore, the linear independence of the shifts of a pseudo-spline is a necessary and sufficient condition for the existence of a compactly supported function whose shifts form a biorthogonal dual system of the shifts of the pseudo-spline.

Multigrid methods: Grid transfer operators and subdivision schemes

“…When L = 0 the symbol in (4.1) is the symbol of the B-spline subdivision scheme of degree 2J − 1 and, when L = J − 1, one gets the symbol of the (2J)-point Dubuc-Deslauries interpolatory subdivision scheme. For more details on binary pseudo-splines see [14,19,20,23].…”

“…Therefore, it is left to show that the corresponding basic limit functions are ℓ ∞ -stable. In [19], the authors addressed this issue. We present an alternative proof of ℓ ∞ -stability of primal pseudo splines for completeness.…”

Biorthogonal wavelets and tight framelets from smoothed pseudo splines

“…[ 6 ] Let be a compactly supported refinable function with finitely supported refinement mask a . The shifts of ϕ are linearly independent if and only if : ϕ is stable ; the symbol ã does not have any symmetric zeros on .…”

“…Consider with , . Since coefficients of the polynomial form a strictly positive and increasing sequences, using Proposition 2.3 of [ 6 ], we can have that all zeros of any polynomial are contained in the open unit disk . Suppose has symmetric zeros and .…”

“…For the box splines, Li extended the results of [ 1 ] and [ 2 ], investigated pseudo box splines, and analyzed their properties, including stability and regularity in [ 5 ]. Dong showed that the shifts of arbitrarily given pseudo splines are linearly independent in [ 6 ]. Bin and Shen in [ 7 ] further studied the construction of biorthogonal wavelets from the pseudo splines and derived a dual refinable function with prescribed regularity.…”