Abstract
A method for constructing a new kind of spline basis functions ('2n(x)) with compact support on R was described by Ueno et al. (2007). These basis functions for sampling approximations consist of a linear combination of the cardinal B splines, but the construction is complicated and therefore were constructed basis functions '2n(x) only for n ≤ 5. We discuss a new construction of the basis functions and its approximation properties are considered.
Highlights
In [5] the spline basis functions φ2n(x) which are symmetric with narrower compact support are constructed satisfying the following conditions (C1) φ2n(−x) = φ2n(x), (C2) suppφ2n(x) = [−n − 1, n + 1], Received: December 14, 2015
In [5] it is shown that if φ2n(x) is a specific linear combination of {N2n+2, · · ·, Nn+2} and satisfies (C3), it enjoys the condition (C4) which is equivalent to the moment condition
The general rule for choosing the specific linear combination mentioned above is complicated. (C1)-(C4) properties can not guarantee a unique basis depending on how a specific linear combination of the cardinal B-spline is chosen
Summary
In [5] the spline basis functions φ2n(x) which are symmetric with narrower compact support are constructed satisfying the following conditions (C1) φ2n(−x) = φ2n(x),. As mentioned in [5], the construction of such basis functions is interesting from both the sampling and interpolating approximation point of view. The general rule for choosing the specific linear combination mentioned above is complicated. (C1)-(C4) properties can not guarantee a unique basis depending on how a specific linear combination of the cardinal B-spline is chosen. The aim of this paper is to give a new construction of functions φ2n that can overcome the above mentioned complication.
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