On the sampling and recovery of bandlimited functions via scattered translates of the Gaussian Original Research Article

Let λλ be a positive number, and let (xj:j∈Z)⊂R(xj:j∈Z)⊂R be a fixed Riesz-basis sequence, namely, (xj)(xj) is strictly increasing, and the set of functions View the MathML source{R∋t↦eixjt:j∈Z} is a Riesz basis (i.e., unconditional basis) for L2[−π,π]L2[−π,π]. Given a function f∈L2(R)f∈L2(R) whose Fourier transform is zero almost everywhere outside the interval [−π,π][−π,π], there is a unique sequence (aj:j∈Z)(aj:j∈Z) in ℓ2(Z)ℓ2(Z), depending on λλ and ff, such that the function View the MathML sourceIλ(f)(x)≔∑j∈Zaje−λ(x−xj)2,x∈R, Turn MathJax on is continuous and square integrable on (−∞,∞)(−∞,∞), and satisfies the interpolatory conditions Iλ(f)(xj)=f(xj)Iλ(f)(xj)=f(xj), j∈Zj∈Z. It is shown that Iλ(f)Iλ(f) converges to ff in L2(R)L2(R), and also uniformly on RR, as λ→0+λ→0+. In addition, the fundamental functions for the univariate interpolation process are defined, and some of their basic properties, including their exponential decay for large argument, are established. It is further shown that the associated interpolation operators are bounded on ℓp(Z)ℓp(Z) for every