On an Interpolation Problem for Functions of Several Variables and Spline Functions

Abstract

Let ω denote the space of all doubly-infinite complex-valued sequences (u_K) (k∈Z) and let ω¹, denote the space of all double, doubly infinite, complex-values sequences (u_kr) (k,r∈Z). Also, ω↑ will denote the set of all doubly-infinite real-valued strictely increasing sequences. Suppose that x=(x_K)_k∈Z ∈ ω↑, y=(y_r)_r∈Z ∈ ω↑ are fixed sequences and denote a:=inf x_k ≥-∞, b:=sup x_k ≤ +∞, c:=inf y_r ≥-∞ and d:=sup y_r ≤ +∞. For D⊂Z² denote D^~: = (x_k,y_r):(k,r)∈ D}. With certain sets D⊂Z² a plane region D^- will be associated. For a set D⊂Z² and a prescribed complex sequence z ≡ (z_kr) ((k,r)∈ D), the problem of finding a function F belonging to a preassigned linear space L of functions from D^- into C and such that z_kr, =F(x_k, y_r) (∀(k,r)∈ D) is called the interpolation problem IP(z;L,x,y). The symbol IP(z;L,x,y) (for given fixed x,y and D) will also denote the set of all its solutions (which may be empty). It is the object of this paper to consider the existence and nature of the solutions of IP(z;L,x,y) for certain choices of the space L. When x_k = k and y_r = r for k, r∈Z, we obtain the cardinal interpolation problem CIP(z,L). The IP(z;L,x,y) and in particular when D is a finite set was considered extensively by several author’s. A survey of these results is given in the excellent survey paper by L.L. Schumaker [9]. One case of the CIP(z;L) when D is an infinite set was considered by Ju. N. Subbotin [10]. The authors acknowledge support from the Israel Commission for Basic Research, and the Natural Sciences and Engineering Research Council of Canada, during the preparation of this paper.