samples/texts/102047/page_10.md

On the other hand, by (4.4) and (4.6),

$$\dim ({\mathrm{\Pi }}_j\cap \mathcal{P}(X))=\dim ({\mathrm{\Pi }}_j\cap \mathcal{H}(X)),\mathrm{\forall }j,\backslash dim(\backslash Pi\_j\; \backslash cap\; \backslash mathcal\{P\}(X))\; =\; \backslash dim(\backslash Pi\_j\; \backslash cap\; \backslash mathcal\{H\}(X)),\; \backslash quad\; \backslash forall\; j,$$

and (5.3) now follows from the fact that $\Pi(M_X) = \mathcal{H}(X)$. $\square$

We note that the result is no longer valid if we drop the unimodularity assumption (cf. [R; Ex. 4.1]).

6. A remark on (2.11) Lemma. A careful examination of the proof of (2.11) Lemma shows that the details of the connection between the ideal $I^X$ and its kernel $I^X \perp$ enter into the argument in only a minor way. The only facts used are: (i) the map $\Pi \to L(\Pi): p \mapsto p(D)$ is a ring homomorphism; and (ii) for any basis $B$ in $X$, $\dim I^B \perp = 1$. This suggests the following result.

(6.1) PROPOSITION. Let $M: \Pi \to L(V)$ be a ring-homomorphism into the ring of linear maps on the linear space $V$. For a given multiset $X$ of directions, define

$$I_M^X\perp :=\bigcap _{p\in I^X}\ker M(p)\mathrm{.}I\_M^X\; \backslash perp\; :=\; \backslash bigcap\_\{p\; \backslash in\; I^X\}\; \backslash ker\; M(p).$$

Then

$$\dim I_M^X\perp \le \sum _{B\in \mathfrak{B}(X)}\dim I_B^X\perp \mathrm{.}\backslash dim\; I\_M^X\; \backslash perp\; \backslash leq\; \backslash sum\_\{B\; \backslash in\; \backslash mathfrak\{B\}(X)\}\; \backslash dim\; I\_B^X\; \backslash perp.$$

The proof of the proposition follows entirely that of (2.11) Lemma, with the obvious modifications whenever the induction hypothesis is applied.

It would be nice to identify other settings rather than the one utilized in this paper, where the above proposition is of use.

REFERENCES

[BH1] C. de Boor and K. Höllig, B-splines from parallelepipeds, J. d'Anal. Math., 42 (1982/3), 99-115.

[BH2] __________, Bivariate box splines and smooth pp functions on a three-direction mesh, J. Comput. Applied Math., 9 (1983), 13-28.

[BR1] C. de Boor and A. Ron, On multivariate polynomial interpolation, Constructive Approx., 6 (1990), 287-302.

[BR2] __________, On polynomial ideals of finite codimension with application to box spline theory, J. Math. Anal. Appl., to appear.

[DM1] W. Dahmen and C. A. Micchelli, Translates of multivariate splines, Linear Algebra and Appl., 52/3 (1983), 217-234.