On the norms of the Dubuc–Deslauriers subdivision schemes

A note on magnitude bounds for the mask coefficients of the interpolatory Dubuc–Deslauriers subdivision scheme

“…This provides a re-proof and extension of the impressive recent result of Deng et al in [2], who considered the important case p = 1 and established (1.5) below, in order to disprove the conjecture of Conti et al in [1] that sup n∈N S a [n] ∞→∞ is finite.…”

“…Corollary 1.3 says that, for arbitrarily small τ > 0, one only needs to use the first n τ terms of the mask in order to obtain the essential size of the full sum (that is logarithmic growth in n) and that the sum of the terms |a Before giving the proofs of the above results in the subsequent section, we remark that except for the bound (1.5) it is not possible to obtain any of the above results from the approach in [2] since their analysis only yields estimates on the full sum σ n = n i=1 |a It is convenient for us to write [r] := {1, 2, . .…”

“…Our approach via Theorem 1.1 is substantially more revealing and robust than the approach in [2], since the latter relies on some delicate combinatorial identities and a telescoping sum approach which avoids estimating the magnitude of each coefficient of the mask. Our direct approach using Theorem 1.1 allows us to simultaneously prove (1.4) and (1.5), with (1.4) highlighting that sup n∈N S a [n] p →∞ is finite for each p > 1 with the exact rate of 'blow-up' as p → 1+; this gives a somewhat different perspective on the falsity of the aforementioned conjecture that sup n∈N S a [n] ∞→∞ is finite.…”

“…Applying the duplication formula Γ(x)Γ(x + where r 0 and s 0 are the largest integers less than or equal to r and s respectively derived in [3] and used in [2]. However, we will make use of Stirling's approximation…”