A scheme for interpolation by Hankel translates of a basis function

This work discusses interpolation of complex-valued functions defined on the positive real axis II by certain special subspaces, in a variational setting that follows the approach of Light and Wayne [W. Light, H. Wayne, Spaces of distributions, interpolation by translates of a basis function and error estimates, Numer. Math. 81 (1999) 415–450]. The set of interpolation points will be a subset {a1,…,an}{a1,…,an} of II and the interpolants will take the form u(x)=∑i=1nαi(τaiϕ)(x)+∑j=0m−1βjpμ,j(x)(x∈I), where μ≥−1/2,ϕμ≥−1/2,ϕ is a complex function defined on II (the so-called basis function), pμ,j(x)=x2j+μ+1/2(j∈Z+,0≤j≤m−1) is a Müntz monomial, τz(z∈I) denotes the Hankel translation operator of order μμ, and αi,βj(i,j∈Z+,1≤i≤n,0≤j≤m−1) are complex coefficients. An estimate for the pointwise error of these interpolants is given. Some numerical examples are included.