| Abstract: | Cardinal interpolation by integer-translates of shifted box splines $M_{n, alpha } : = M_{nnn} ( cdot + alpha )$ on the three-direction mesh is studied. It was recently shown by Sivakumar that for even integers n the imaginary part of a certain rotation of the symbol of $M_{n, alpha } $ does not vanish on the torus $T^2 $ for all a in the shift region $ Lambda = ( - frac{1}{2}, frac{1}{2})^2 cap { {(s,t):| {s - t} | < frac{1}{2}} }$, and consequently, cardinal interpolation at $ mathbb{Z}^2 $ by using $M_{n, alpha } ( cdot - j)$, $j in mathbb{Z}^2 $, is poised for all even n and all $ alpha in A$. For odd n, however, since both the real and imaginary parts of the rotated symbol have nonempty zero sets on $T^2 $ for certain $ alpha in Lambda $ close to the corners of $ partial Lambda $, the analysis in Sivakumar's work does not directly apply to the study of this situation. In this paper we prove that the above mentioned zero sets are disjoint for all odd integers n and all a $ alpha in Lambda $, and hence, the symbol of $M_{n, alpha } $ never vanishes on $T^2$. In other words, the cardinal interpolation operators corresponding to $M_{n, alpha } $, $ alpha in Lambda $ and $n = 1,2, cdots $, are invertible. |
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