Positive-definite splines of special form

The Wiener algebra of absolutely convergent Fourier integrals: an overview

“…Trigub can be found in, e.g., [154,174] and [160]. ] (integral part), and for no r greater than that indicated.…”

“…It is of considerable interest to give a full description of the extremal points of a convex set of such "splines" of a given degree; see also [174].…”

On some properties of Buhmann functions

“…In special cases, we get the following known functions: µδϕ δ,µ,1,δ (x) = 1−|x| δ µ + and ϕ 1,µ,ν,2ν−1 (x) ≡ h µ,ν (x) ≡ 2 ν−1 Γ(ν) µ ψ µ,ν−1 (x). Note that the functions h µ,ν were introduced by Zastavnyi in [7,8] and the functions ψ µ,ν−1 with µ, ν ∈ N, were introduced by Wendland in [9]. For r ∈ Z + and k ∈ N, we have h r+k,r+1 (x) ≡ B(r + k, 2r + 1)A r,2k−1 (x).…”

“…In the theory of approximation and interpolation by functions of the form ξ∈Ξ λ ξ ϕ( x−ξ 2 ), where Ξ is a finite system of points from R m and ϕ( · 2 ) is a function with compact support continuous in R m , an important role is played by the following two properties of the Fourier transform (see, e.g., [5][6][7]): (i) F m (ϕ( · 2 ))(u) > 0 for all u ∈ R m and (ii) there exist positive constants c i , γ > 0 such that c 1 ≤ F m ϕ( · 2 ) (u) 1+ u 2 γ ≤ c 2…”

“…For r ∈ Z + and k ∈ N, we have h r+k,r+1 (x) ≡ B(r + k, 2r + 1)A r,2k−1 (x). The functions A r,2k−1 were introduced by Trigub in [3,[10][11][12][13][14][15][16] (for details, see [8]). Here, B(α, β) :…”