Abstract
In this paper, we study the Strang‐Fix theory for approximation order in the weighted Lp ‐spaces and Herz spaces.
Highlights
In [10], Strang and Fix considered the relation between approximation order in L2(Rn) of a given function and properties of its Fourier transform
We say that the collection Φ = {φ1, · · ·, φN } of Cc(Rn) provides controlled Lp approximation of order k if for each f ∈ Lpk(Rn) there exist weights chj (h > 0, j = 1, · · ·, N ) such that
Strang and Fix proved that, when Φ consists of only one function, Φ provides controlled L2 -approximation of order k if and only if Φ satisfies the Strang-Fix condition of order k ([10])
Summary
In [10], Strang and Fix considered the relation between approximation order in L2(Rn) of a given function and properties of its Fourier transform. Strang and Fix proved that, when Φ consists of only one function, Φ provides controlled L2 -approximation of order k if and only if Φ satisfies the Strang-Fix condition of order k ([10]) They conjectured that this equivalence holds when Φ = {φ1, · · · , φN }. We say that Φ = {φ1, · · · , φN } provides local Lp -approximation of order k if for each f ∈ Lpk(Rn) there exist weights chj (h > 0, j = 1, · · · , N ) such that. De Boor and Jia proved that Φ provides local Lp -approximation of order k if and only if Φ satisfies the Strang-Fix condition of order k ([2]). We point out that the Strang-Fix theory for functions having noncompact support is given by, for example, Jia and Lei ([7])
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