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| Cardinal SplineInterpolation Problem(CIP) is to consider finding a cardinal splinefunction s(x) of degree n such that s(v)= yv ,∀v∈Z, with given information (yv)v∈Z. Schoenberg solved this problem and the corresponding Cardinal Hermite Spline Interpolation Problem(CHIP) and obtained manygraceful results in about 1970. In this paper, the authors will considerCardinal (ρ0,ρ1,ρr-1) Birkhoff SplineInterpolation Problem(CBIP) using the similar method fromSchoenberg. For this purpose, the cardinal splines spaces S2m-1,∧1 with Birkhoff knots is introduced,where ∧1:={θ0,θ1,...,θr-1}∈ {0,...,2m-1} is an ordered set. The lacunary interpolation problemconsidered is to find the interpolation functions on Z with derivativesinformation of order ∧2:={ρ0,ρ1,...,ρr-1} in the spline space S2m-1,∧1. Here ∧2:={ρ0,ρ1,...,ρr-1}∈ {0,...,2m-1} is an ordered set.The necessary condition and several sufficient conditions on ∧1, ∧2 of the regularity(i.e. existing a uniquesolution) of CBIP are gained, and some results in CHIP areproved to be also true in CBIP. |
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| Keywords:approximation of function; cardinal spline; splinefunction; Birkhoff interpolation; lacunary interpolation |
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