Let \(\{\xi_ j| j\in {\mathbb{Z}}\}\) and \(\{x_ j| j\in {\mathbb{Z}}\}\) be two doubly infinite sequences. Denote the class of polynomial spline functions of degree n with knot sequence \(\{\xi_ j| j\in {\mathbb{Z}}\}\) by \({\mathcal S}_ n\). Then every \(S\in {\mathcal S}_ n\) can be written uniquely in the form \(S(x)=\sum^{\infty}_{-\infty}c_ jM_ j(x)\) where \(M_ j(x)=M_ j(x| \xi_ j,\xi_{j+1},...,\xi_{j+n+1})\) is the B-spline with knots at \(\xi_ j,\xi_{j+1},...,\xi_{j+n+1}\). This allows us to write the interpolation conditions \(S(x_ j)=y_ j\), \(j\in {\mathbb{Z}}\), \(S\in {\mathcal S}_ n\), in the form \(Mc=y\) where \(M=(M_ j(x_ i)),\quad c=(c_ j)\) and \(y=(y_ j).\)
The set \(\{M_{i,i+k}| i\in {\mathbb{Z}}\},\) is called the k-th band of M. Define \(m\equiv m(M)=\max \{q-p | p-th\quad band\) and q-th band of M have nonzero element\(s\}\), and \(m=\infty\) if each band of M has a nonzero element.
The null space of M is isomorphic to the space \({\mathcal N}=\{S\in {\mathcal S}_ n | S(x_ j)=0\quad \forall j\in {\mathbb{Z}}\}\) of null splines and for the case where the knots and nodes are N-periodic with \(x_{j+N}=x_ j+1\) and \(\xi_{j+N}=\xi_ j+1,\) \(j\in {\mathbb{Z}}\), we prove Theorem 1: (a) The dimension of \({\mathcal N}\) is m, and \(m=\infty\) if and only if \(\exists\) a nontrivial null spline with compact support. (b) If \(m<\infty\), then \({\mathcal N}\) is spanned by m eigensplines \(S_ j(x)\) with \(S_ j(x+1)=\lambda_ jS_ j(x), j=1,2,...,m\) where \(\lambda_ 1<...<\lambda_ m\) and \((-1)^ N\lambda_ j>0, j=1,2,...,m\). Corollary: Given \(y=(y_ j)\in \ell^{\infty}\), the interpolation problem \(S(x_ j)=y_ j,\quad j\in {\mathbb{Z}},\quad S\in {\mathcal S}_ n\cap L^{\infty}(R)\) is uniquely solvable if and only if \(m<\infty\) and \((- 1)^ N\lambda_ j\neq 1 \forall j=1,2,...,m\). For the case \(N=2\), precise values of m and conditions for which \(\lambda_ j\neq 1\) are given.
The matrix M possesses the block Toeplitz structure and it is shown that the eigenvalues \(\lambda_ j\), \(j=1,2,...,m\) of the eigensplines \(S_ j\in N\) are the nonzero zeros of the determinant of its symbol. The results of this paper are supplements to the earlier works of A. S. Cavaretta jun., W. Dahmen, C. A. Micchelli and P. W. Smith (see references therein) and have been substantially improved and generalized by T. N. T. Goodman [ibid. 95, 39-57 (1983; Zbl 0526.41004)].