Symbols and exact regularity of symmetric pseudo-splines of any arity

Abstract

Pseudo-splines form a family of subdivision schemes that provide a natural blend between interpolating schemes and approximating schemes, including the Dubuc–Deslauriers schemes and B-spline schemes. Using a generating function approach, we derive expressions for the symbols of the symmetric m-ary pseudo-spline subdivision schemes. We show that their masks have positive Fourier transform, making it possible to compute the exact Hölder regularity algebraically as a logarithm of the spectral radius of a matrix. We apply this method to compute the regularity explicitly in some special cases, including the symmetric binary, ternary, and quarternary pseudo-spline schemes.

Appendix: Regularity of m-ary subdivision

It is well known [20, 34] that for symmetric interpolatory schemes with positive Fourier transform, it is possible to determine the Hölder regularity exactly. In the report [18] it was shown that this is possible for non-interpolatory binary schemes as well. In this appendix we show that these results generalize to the general m-ary scheme (2.1) (cf. [22] for the ternary case). This is related to results described in [2, 29, 30], which show that the underlying mathematical reason for the correctness of the method is the validity of the finiteness conjecture for the joint spectral radius of subdivision submatrices derived from schemes with positive Fourier transform.

In this appendix we suppose that a(z) satisfies the conditions (2.5b) for polynomial generation up to some degree $r\ge 0$, and, after shifting the coefficients $a_k$ as necessary, that the mask ${\mathbf {b}}= (b_j)_j$ corresponding to b(z) is odd symmetric and centered at zero, i.e.,

$$\begin{aligned} {\mathbf {b}}= [b_p, \ldots , b_1, b_0, b_1,\ldots , b_p], \qquad b_p\ne 0, \end{aligned}$$

(6.1)

for some $p \ge 0$. Then the Fourier transform of ${\mathbf {b}}$,

$$\begin{aligned} B(\xi ) := b(\mathrm {e}^{-i\xi }) = b_0 + 2 \sum _{j = 1}^p b_j \cos (j\xi ), \qquad \xi \in \mathbb {R}, \end{aligned}$$

is real and periodic with period $2\pi$.

1.1 Regularity as a decay rate of differences of the data

The regularity of the limit function f is related to the decay rate of divided differences of the scheme. For each integer $s \ge 0$, let $f_{\ell ,j}^{[s]}$ denote the divided difference of the values $f_{\ell ,j-s},\ldots ,f_{\ell ,j}$ at the corresponding m-adic points $m^{-\ell }(j-s),\ldots ,m^{-\ell }j$. That is,

$$\begin{aligned} f_{\ell ,j}^{[0]} = f_{\ell ,j},\qquad f_{\ell ,j}^{[s]} = \frac{m^\ell }{s} \left( f_{\ell ,j}^{[s-1]} - f_{\ell ,j-1}^{[s-1]}\right) , \qquad s\ge 1. \end{aligned}$$

(6.2)

Under condition (2.5), there is a scheme for the $f_{\ell ,j}^{[s]}$ for $s=0,\ldots ,r+1$. Writing

$$\begin{aligned} a^{[s]}(z) = \sum _j a_j^{[s]} z^j := \frac{a(z)}{\sigma _m^s(z)},\qquad f_\ell ^{[s]}(z) := \sum _j f_{\ell ,j}^{[s]} z^j, \end{aligned}$$

this scheme takes the equivalent forms

$$\begin{aligned} f_{\ell + 1, j}^{[s]} = \sum _k a_{j-mk}^{[s]} f_{\ell ,k}^{[s]}, \qquad f_{\ell +1}^{[s]}(z) = a^{[s]}(z) f_\ell ^{[s]}(z^m). \end{aligned}$$

(6.3)

Consider the differences $g_{\ell ,j}^{[r]}$ (of the divided differences) of the data and the corresponding symbol, defined by

$$\begin{aligned} g_{\ell ,j}^{[r]} := f_{\ell ,j}^{[r]} - f_{\ell ,j-1}^{[r]},\qquad g_\ell ^{[r]}(z) := \sum _j g_{\ell ,j}^{[r]} z^j. \end{aligned}$$

The following lemma relates the decay rate of $g_{\ell ,j}^{[r]}$ to the regularity of the limit function f. It was shown to hold for binary schemes in [16, Theorem 4.9] and for ternary schemes in [22, Theorem 3.4.4], but also holds for schemes with general arity m.

Lemma 6.1

Suppose that, for large enough $\ell$,

$$\begin{aligned} |g^{[r]}_{\ell ,j}| \le K\lambda ^\ell , \end{aligned}$$

(6.4)

for some constants K and $\lambda < 1$. Then $f^{(r)} \in C^0$. Moreover, if $1/m< \lambda < 1$, then $f^{(r)} \in C^{-\log _m(\lambda )}$.

Proof

To simplify notation, let us drop the superscripts in $g^{[r]}_{\ell ,j}, g^{[r]}_\ell (z), f^{[r]}_{\ell ,j}$, $f^{[r]}_\ell (z)$, $f^{(r)}, a^{[r]}(z)$. Using the standard parametrization, let $L_\ell$ denote the piecewise linear function through the points $(m^{-\ell }j, f_{\ell ,j})$ at level $\ell$. We first bound the maximal difference between these piecewise linear functions at levels $\ell$ and $\ell +1$ in terms of the differences at level $\ell$. Since this maximum is attained at one of the breakpoints,

$$\begin{aligned} \Vert L_{\ell +1} - L_\ell \Vert _\infty = \max _j \left| f_{\ell +1,j} - h_{\ell +1,j} \right| , \end{aligned}$$

(6.5)

where, writing $j = mj' + \varepsilon$,

$$\begin{aligned} h_{\ell +1, mj' + \varepsilon } := \frac{m-\varepsilon }{m} f_{\ell ,j'} + \frac{\varepsilon }{m} f_{\ell ,j'+1},\qquad \varepsilon = 0, 1, \ldots , m - 1,\qquad j'\in \mathbb {Z}, \end{aligned}$$

with corresponding symbol

$$\begin{aligned} h_{\ell +1}(z) := \sum _j h_{\ell +1, j} z^j = \frac{(1 + z + \cdots + z^{m-1})^2}{mz^{m-1}} f_\ell (z^m). \end{aligned}$$

Therefore

$$\begin{aligned} f_{\ell +1}(z) - h_{\ell +1}(z)&= (1 + z + \cdots + z^{m-1}) d(z) f_\ell (z^m), \end{aligned}$$

with

$$\begin{aligned} d(z) := \frac{a(z)}{1 + z + \cdots + z^{m-1}} - \frac{1 + z + \cdots + z^{m-1}}{mz^{m-1}}. \end{aligned}$$

But $d(1) = a(1)/m - 1 = 0$ by (2.4b), so that $d(z) = (1 - z)e(z)$, with $e(z) = \sum _j e_j z^j$ a Laurent polynomial. Therefore

$$\begin{aligned} f_{\ell +1}(z) - h_{\ell +1}(z) = e(z)(1 - z^m) f_\ell (z^m) = e(z) g_\ell (z^m), \end{aligned}$$

or equivalently

$$\begin{aligned} f_{\ell +1,j} - h_{\ell +1,j} = \sum _k e_{j - mk} g_{\ell ,k}. \end{aligned}$$

Using (6.5), we obtain, for some constant $K_1$,

$$\begin{aligned} \Vert L_{\ell +1} - L_\ell \Vert _\infty = \max _j | f_{\ell +1,j} - h_{\ell +1,j} | \le \max _j \sum _k |e_{j - mk}| \cdot \max _k |g_{\ell ,k}| \le K_1 \lambda ^\ell , \end{aligned}$$

from which it follows that $[L_\ell ]_\ell$ is a Cauchy sequence. Equipped with the infinity norm $\Vert \cdot \Vert _\infty$, the space of bounded continuous functions on the real line is complete, and $[L_\ell ]_\ell$ converges uniformly to a continuous function f. Moreover,

$$\begin{aligned} \Vert f - L_\ell \Vert _\infty \le \sum _{\ell ' = \ell }^\infty \Vert L_{\ell '+1} - L_{\ell '}\Vert _\infty \le \sum _{\ell ' = \ell }^\infty K_1\lambda ^{\ell '} = K_2 \lambda ^\ell ,\quad K_2 := \frac{K_1}{1-\lambda }, \end{aligned}$$

(6.6)

so that $[L_\ell ]_\ell$ converges to f with rate $\lambda$. In addition, note that

$$\begin{aligned} |L_\ell (x) - L_\ell (y)| \le |x - y|\cdot \frac{\max _j g_{\ell ,j}}{m^{-\ell }} \le \frac{|x - y|}{m^{-\ell }} K \lambda ^\ell , \end{aligned}$$

(6.7)

implying

$$\begin{aligned} |f(x) - f(y)| \le |f(x) - L_\ell (x)| + |L_\ell (x) - L_\ell (y)| + |L_\ell (y) - f(y)|. \end{aligned}$$

(6.8)

It suffices to verify the Hölder condition locally. Let x, y be such that $m^{-\ell } \le |x - y| \le m^{-\ell + 1}$, so that

$$\begin{aligned} \lambda ^\ell = m^{\ell \log _m(\lambda )} \le |x - y|^{-\log _m(\lambda )},\qquad \frac{|x - y|}{m^{-\ell }} \le m. \end{aligned}$$

Combining (6.6)–(6.8) gives

$$\begin{aligned} |f(x) - f(y)| \le (2K_2 + m K)|x - y|^{-\log _m(\lambda )}, \end{aligned}$$

implying that $f\in C^{-\log _m(\lambda )}$ whenever $1/m< \lambda < 1$.

1.2 Growth rate of the differences of the data

How can we use (6.4) in the case that it holds with $\lambda \ge 1$? Then we do not know whether $f \in C^r$, but if $r \ge 1$ we can use the ‘reduction procedure’ of Daubechies, Guskov, and Sweldens [5] to obtain information about lower order derivatives. Although the procedure was shown to work for binary interpolatory schemes in [5], it also applies to the more general scheme (2.1).

Lemma 6.2

Suppose (2.5b) holds for some $r \ge 1$. If (6.4) holds for some $\lambda$, then there are constants $D_1, D_2, D_3$ such that

$$\begin{aligned} |g_{\ell ,j}^{[r-1]}| \le \left\{ \begin{array}{rl} \displaystyle D_1 \left( \frac{\lambda }{m}\right) ^\ell &{} \text { if }\lambda > 1,\\ \displaystyle (D_2 + D_3\ell ) \left( \frac{1}{m}\right) ^\ell &{} \text { if }\lambda = 1. \end{array} \right. \end{aligned}$$

Proof

Since $a^{[r]}(z) := a(z)/\sigma _m^r(z)$, one has $a^{[r]}(1) = a(1) = m$. Moreover, by the divisibility assumption (2.5b), $a^{[r]}$ is divisible by $(1 - z^m)/(1 - z)$ so that $a^{[r]}(\zeta _m^k) = 0$ for $k = 1,\ldots , m-1$. It follows that

$$\begin{aligned} \sum _k a_{mk}^{[r]} = \sum _k a_{mk+1}^{[r]} = \cdots = \sum _k a_{mk+m-1}^{[r]} = 1, \end{aligned}$$

which, together with (6.3), implies that there is a constant $K_1$ such that

$$\begin{aligned} |f_{\ell +1,mj+s}^{[r]} - f_{\ell ,j}^{[r]}| \le K_1 \max _j |g_{\ell ,j}^{[r]}|, \qquad s=0,1,\ldots ,m-1. \end{aligned}$$

So, for any level $\ell \ge 1$, if we represent any $j \in \mathbb {Z}$ in m-ary form as $j = j_\ell$, where

$$\begin{aligned} j_{\ell '} = m j_{\ell '-1} + s_\ell , \qquad {\ell '}=\ell ,\ell -1,\ldots ,1, \end{aligned}$$

for some $j_0 \in \mathbb {Z}$ and $s_1,\ldots ,s_\ell \in \{0,1,\ldots ,m-1\}$, then

$$\begin{aligned} |f_{\ell ,j}^{[r]} - f_{0,j_0}^{[r]}| \le \sum _{\ell '=1}^\ell | f_{\ell ',j_{\ell '}}^{[r]} - f_{\ell '-1,j_{\ell '-1}}^{[r]} | \le K_1 K (1 + \lambda + \cdots + \lambda ^{\ell -1}). \end{aligned}$$

Hence,

$$\begin{aligned} |f_{\ell ,j}^{[r]}| \le K_2 + K_1 K (1 + \lambda + \cdots + \lambda ^{\ell -1}) \end{aligned}$$

for some constant $K_2$, and since

$$\begin{aligned} g_{\ell ,j}^{[r-1]} = m^{-\ell } r f_{\ell ,j}^{[r]}, \end{aligned}$$

this gives the result in the two cases $\lambda > 1$ and $\lambda = 1$.

By applying this procedure recursively, it follows that if (6.4) holds for any $\lambda$ with $1/m< \lambda < m^r$, then $f \in C^{r-\log _m(\lambda )}$ if $\log _m(\lambda )$ is not an integer, and $f \in C^{r-\log _m(\lambda ) - \varepsilon }$ for any small $\varepsilon > 0$ if $\log _m(\lambda )$ is an integer.

1.3 Growth rate of the iterated scheme for the differences

If $p=0$ in (6.1), then $b_0 = 1$ by (2.4b) and (2.5b). In this case the scheme (2.1) is the m-ary B-spline scheme of degree r. Since (6.4) holds with $\lambda = 1$, we conclude using Lemma 6.2 that the limit function belongs to $C^\beta$ for any $\beta < r$, which is well known.

Therefore we assume from now on that $p \ge 1$. With r in (2.5b) fixed, write $g_{\ell ,j} = g_{\ell ,j}^{[r]}$ and $g_\ell (z) = \sum _j g_{\ell ,j} z^j$. Then

$$\begin{aligned} g_{\ell +1}(z) = b(z) g_\ell (z^m), \end{aligned}$$

(6.9)

with b(z) as in (2.5b), or equivalently,

$$\begin{aligned} g_{\ell +1,j} = \sum _k b_{j-mk} g_{\ell ,k}. \end{aligned}$$

(6.10)

In the following lemma we rephrase the bound (6.4) for the data $g_{\ell ,j}$ as a bound for their scheme.

Lemma 6.3

The bound (6.4) holds, for some constant K, if there is some constant $K'$ such that

$$\begin{aligned} \max _j |b_{\ell ,j}| \le K' \lambda ^\ell . \end{aligned}$$

(6.11)

Proof

Iterating (6.9) gives

$$\begin{aligned} g_\ell (z) = b_\ell (z) g_0(z^{m^\ell }), \end{aligned}$$

(6.12)

where

$$\begin{aligned} b_\ell (z) := b(z) b(z^m) \cdots b(z^{m^{\ell -1}}). \end{aligned}$$

(6.13)

But then

$$\begin{aligned} b_{\ell +1}(z) = b(z) b_\ell (z^m), \end{aligned}$$

(6.14)

and so $b_\ell (z)$ is the Laurent polynomial of the data $b_{\ell ,j}$, where $b_{0,j} = \delta _{j,0}$ and

$$\begin{aligned} b_{\ell +1,j} = \sum _k b_{j-mk} b_{\ell ,k}. \end{aligned}$$

(6.15)

In particular, $b_{1,j} = b_j$. Since (6.12) can be written as

$$\begin{aligned} g_{\ell ,j} = \sum _k b_{\ell ,j-m^\ell k} g_{0,k}, \end{aligned}$$

(6.16)

it follows that

$$\begin{aligned} |g_{\ell ,j}| \le \max _j |b_{\ell ,j}| \sum _k |g_{0,k}|, \end{aligned}$$

and so (6.4) holds if (6.11) holds for some constant $K'$.

The following lemma provides the reason why the bound (6.11) is easier to verify than (6.4), in the case of a nonnegative Fourier transform. For a direct proof see the report [18] or [34]. It is also a direct consequence of Herglotz’ theorem, which states that the condition of the lemma is equivalent to ${\mathbf {b}}$ being a positive definite sequence; see [2].

Lemma 6.4

If ${\mathbf {b}}$ as in (6.1) has Fourier transform $B(\xi ) \ge 0$ for all $\xi$, then

$$\begin{aligned} \max _j |b_{\ell ,j}| = b_{\ell ,0} \qquad \text {for all } \ell \ge 0. \end{aligned}$$

1.4 Growth rate as a spectral radius

For an odd symmetric mask ${\mathbf {b}}$ with nonnegative Fourier transform, it follows that (6.4) holds if $b_{\ell ,0} \le K \lambda ^\ell$ for large enough $\ell$. One way to determine such $\lambda$ is using a subvector of $[b_{\ell ,j}]_j$ that includes the central coefficients $b_{\ell ,0}$ and is ‘self-generating’ in the following sense.

Lemma 6.5

For $\ell \ge 0$, the finite submatrix and subvectors

$$\begin{aligned} {\mathbf {M}}:= [b_{j-mk}]_{j,k=-\lfloor \frac{p-1}{m-1}\rfloor ,\ldots ,\lfloor \frac{p-1}{m-1}\rfloor },\quad {\mathbf {b}}_\ell := \left[ b_{\ell ,-\lfloor \frac{p-1}{m-1}\rfloor },\ldots ,b_{\ell ,\lfloor \frac{p-1}{m-1}\rfloor }\right] ^T, \end{aligned}$$

(6.17)

satisfy ${\mathbf {b}}_{\ell +1} = {\mathbf {M}}{\mathbf {b}}_\ell$.

Proof

If $k\ge \lfloor \frac{p-1}{m-1}\rfloor + 1$ in (6.15) and the corresponding coefficient $b_{j-mk}\ne 0$, then $j - mk \ge -p$ implying that

$$\begin{aligned} j\ge -p + mk\ge - p + m\left\lfloor \frac{p-1}{m-1}\right\rfloor + m \ge \left\lfloor \frac{p-1}{m-1}\right\rfloor + 1. \end{aligned}$$

So any such $b_{\ell ,k}$ will not contribute to the linear combination for $b_{\ell +1,j}$ with $j\le \lfloor \frac{p-1}{m-1} \rfloor$. Similarly, if $k\le -\lfloor \frac{p-1}{m-1}\rfloor - 1$ in (6.15) and the corresponding coefficient $b_{j-mk}\ne 0$, then $j - mk \le p$ implying that

$$\begin{aligned} j \le p + mk\le p - m\left\lfloor \frac{p-1}{m-1} \right\rfloor - m \le - \left\lfloor \frac{p-1}{m-1} \right\rfloor - 1. \end{aligned}$$

So any such $b_{\ell ,k}$ will not contribute to the linear combination for $b_{\ell +1,j}$ with $j\ge - \left\lfloor \frac{p-1}{m-1} \right\rfloor$. By (6.15), it follows that ${\mathbf {b}}_{\ell +1} = {\mathbf {M}}{\mathbf {b}}_\ell$ for $\ell \ge 0$.

Theorem 6.1

Let $\rho$ be the spectral radius of ${\mathbf {M}}$. If $B(\xi ) \ge 0$ for all $\xi$, then

$$\begin{aligned} \lim _{\ell \rightarrow \infty } b_{\ell ,0}^{1/\ell } = \rho , \end{aligned}$$

(6.18)

If $\rho > 1/m$, a lower bound for the regularity of the scheme (2.1) is $r - \log _m(\rho )$.

Proof

Using Lemma 6.4 and ${\mathbf {b}}_\ell = {\mathbf {M}}^\ell {\mathbf {b}}_0$ by Lemma 6.5,

$$\begin{aligned} b_{\ell ,0} = \Vert {\mathbf {b}}_\ell \Vert _\infty \le \Vert {\mathbf {M}}^\ell \Vert _\infty \Vert {\mathbf {b}}_0 \Vert _\infty = \Vert {\mathbf {M}}^\ell \Vert _\infty . \end{aligned}$$

On the other hand, by (6.16) the matrix ${\mathbf {M}}^\ell$ takes its entries from $[b_{\ell ,j}]_j$, so that its maximum absolute row sum $\Vert {\mathbf {M}}^\ell \Vert _\infty$ satisfies

$$\begin{aligned} \Vert {\mathbf {M}}^\ell \Vert _\infty \le \left( 2\left\lfloor \frac{p-1}{m-1} \right\rfloor + 1\right) \max _j |b_{\ell ,j}| = \left( 2\left\lfloor \frac{p-1}{m-1} \right\rfloor + 1\right) b_{\ell ,0}. \end{aligned}$$

Taking $\ell$-th roots and the limit $\ell \rightarrow \infty$ one obtains (6.18). It follows from (6.18) and Lemma 6.3 that (6.4) holds with $K = 1$ for any $\lambda > \rho$, and this proves the lower bound on the regularity of the scheme.

1.5 A smaller matrix

Due to the assumption that ${\mathbf {b}}$ is odd symmetric, the limit (6.18) can also be computed as the spectral radius of a matrix roughly half the size of ${\mathbf {M}}$, using a ‘folding procedure’ [18, 34]. Since $b_{\ell ,-j} = b_{\ell ,j}$ for all j, the vector of coefficients

$$\begin{aligned} {\mathbf {b}}_\ell := \left[ b_{\ell ,0},b_{\ell ,1},\ldots ,b_{\ell ,\lfloor \frac{p-1}{m-1}\rfloor }\right] ^T, \end{aligned}$$

also includes $b_{\ell ,0}$ and is self-generating as well. Indeed, from (6.15),

$$\begin{aligned} b_{\ell +1,j} = b_j b_{\ell ,0} + \sum _{k \ge 1} (b_{j-mk} + b_{j+mk}) b_{\ell ,k}, \end{aligned}$$

and, using that $b_{-j} = b_j$, one obtains

$$\begin{aligned} b_{\ell +1,j} = b_j b_{\ell ,0} + \sum _{k \ge 1} (b_{|j-mk|} + b_{j+mk}) b_{\ell ,k}. \end{aligned}$$

It follows that ${\mathbf {b}}_{\ell +1} = {\mathbf {M}}{\mathbf {b}}_\ell$, where ${\mathbf {M}}$ is the matrix of dimension $\lfloor \frac{p-1}{m-1}\rfloor + 1$,

$$\begin{aligned} {\mathbf {M}}= [m_{j,k}]_{j,k=0,\ldots ,\lfloor \frac{p-1}{m-1}\rfloor }, \qquad m_{j,k} = \left\{ \begin{array}{ll} b_j &{}\quad k = 0,\\ b_{|j-mk|} + b_{j+mk} &{}\quad k \ge 1. \end{array} \right. \end{aligned}$$

(6.19)

1.6 Optimality

In this section we show that under a slightly stricter condition, the lower bound on the regularity of Theorem 6.1 is optimal. For related results in the binary and ternary case, see [34] and [22, §3.4].

Theorem 6.2

If $B(\xi ) > 0$ for all $\xi$, the lower bound $r - \log _m(\rho )$ of Theorem 6.1 is optimal.

To prove this we first establish a lemma that shows that the bound is optimal whenever the cardinal function $\phi$ of the scheme (2.1) has $\ell ^\infty$-stable integer translates. The main point in proving this lemma is that the stability allows us to bound divided differences of the scheme by corresponding divided differences of the limit function.

Following Jia and Micchelli [26], we say that $\phi$ has $\ell ^\infty$-stable integer translates if there is some constant $K_\infty > 0$ such that for any sequence ${\mathbf {c}}= [c_j]_j$ in $\ell ^\infty (\mathbb {Z})$,

$$\begin{aligned} \left\| \sum _j c_j \phi (\cdot - j) \right\| _{L^\infty (\mathbb {R})} \ge K_\infty \Vert {\mathbf {c}}\Vert _{\ell ^\infty (\mathbb {Z})}. \end{aligned}$$

(6.20)

Lemma 6.6

Suppose $\phi$ has $\ell ^\infty$-stable integer translates and $f\in C^{q + \alpha }$ for some integer $q\ge 0$ and $0< \alpha < 1$. Then for any integer $r \ge q$, there is a constant K such that

$$\begin{aligned} |g_{\ell ,j}^{[r]}| \le K m^{\ell (r-q-\alpha )}. \end{aligned}$$

(6.21)

Proof

The limit function for general initial data $f_{0,k}$ can be expressed as the linear combination

$$\begin{aligned} f(x) = \sum _k f_{0,k} \phi (x-k). \end{aligned}$$

As is well known [21], $\phi$ satisfies the refinement equation

$$\begin{aligned} \phi (x) = \sum _j a_j \phi (mx - j), \end{aligned}$$

(6.22)

and therefore, for any $\ell \ge 0$,

$$\begin{aligned} f(x) = \sum _k f_{\ell ,k} \phi (m^\ell x - k). \end{aligned}$$

(6.23)

We can use this equation to relate any divided difference of f of the form

$$\begin{aligned} \tilde{f}_{\ell , y}^{[q]} := [m^{-\ell }(y-q),m^{-\ell }(y-q+1),\ldots ,m^{-\ell }y]f,\qquad y\in \mathbb {R}, \end{aligned}$$

to the divided differences of the scheme. Putting $x = m^{-\ell }(y-j)$ in (6.23),

$$\begin{aligned} f\big (m^{-\ell }(y-j)\big ) = \sum _k f_{\ell ,k-j} \phi (y-k), \end{aligned}$$

and, using the cases $j=0,1,\ldots ,q$, and the linearity of divided differences,

$$\begin{aligned} \tilde{f}_{\ell ,y}^{[q]} = \sum _k f_{\ell ,k}^{[q]} \phi (y-k). \end{aligned}$$

Similarly, if

$$\begin{aligned} \tilde{g}_{\ell ,y}^{[q]} := \tilde{f}_{\ell ,y}^{[q]} - \tilde{f}_{\ell ,y-1}^{[q]}, \end{aligned}$$

then

$$\begin{aligned} \tilde{g}_{\ell ,y}^{[q]} = \sum _k g_{\ell ,k}^{[q]} \phi (y-k). \end{aligned}$$

Using that f has compact support, if f has regularity $q + \alpha$, there is a constant $K'>0$ such that for any $\xi _0,\xi _1 \in \mathbb {R}$,

$$\begin{aligned} |f^{(q)}(\xi _1) - f^{(q)}(\xi _0)| \le K' |\xi _1 - \xi _0|^\alpha , \end{aligned}$$

and, by the mean value theorem for divided differences, for each $\ell$ and y,

$$\begin{aligned} |\tilde{g}_{\ell ,y}^{[q]}| = |f^{(q)}(\xi _1) - f^{(q)}(\xi _0)| / q!, \end{aligned}$$

for $\xi _0,\xi _1 \in \big (m^{-\ell }(y-q-1),m^{-\ell }y\big )$. Therefore, for any y,

$$\begin{aligned} |\tilde{g}_{\ell ,y}^{[q]}| \le K'' m^{-\ell \alpha }, \end{aligned}$$

where $K'' = K' (q+1)^\alpha / q!$. Therefore,

$$\begin{aligned} \left\| \sum _\ell g_{\ell ,k}^{[q]} \phi (\cdot - k) \right\| _{L^\infty (\mathbb {R})} \le K'' m^{-\ell \alpha }, \end{aligned}$$

and by (6.20) it follows that for any $k \in \mathbb {Z}$,

$$\begin{aligned} |g_{\ell ,k}^{[q]}| \le K^{-1}_\infty K'' m^{-\ell \alpha }. \end{aligned}$$

Finally, by applying the divided difference definitions (6.2) recursively, $r-q$ times, we obtain (6.21).

Lemma 6.7

If $\phi$ has $\ell ^\infty$-stable integer translates, then the lower bound $r - \log _m(\rho )$ of Theorem 6.1 is optimal.

Proof

Let f be the limit of the scheme with any initial data for which $g_{0,j}^{[r]} = \delta _{j,0}$, $-p+1 \le j \le p-1$, and with only a finite number of initial data $f_{0,j}$ non-zero. Then f has compact support. Suppose that $f \in C^{r-\log _m(\rho ) + \varepsilon }$ for some small $\varepsilon > 0$ and write the exponent as

$$\begin{aligned} r-\log _m(\rho ) + \varepsilon = q + \alpha , \qquad q \in \mathbb {N}_0, \qquad 0< \alpha < 1. \end{aligned}$$

If $\rho > 1/m$, we have $r \ge q$, and so Lemma 6.6 can be applied, implying

$$\begin{aligned} |g_{\ell ,j}^{[r]}| \le K m^{\ell (\log _m(\rho ) - \varepsilon )} = K \rho ^\ell m^{-\ell \varepsilon }. \end{aligned}$$

Hence,

$$\begin{aligned} \limsup _{\ell \rightarrow \infty } \left| g_{\ell ,0}^{[r]}\right| ^{1/\ell } \le \rho m^{-\varepsilon }. \end{aligned}$$

By choice of the $g_{0,j}^{[r]}$, however, $g_{\ell ,0}^{[r]} = b_{\ell ,0}$, which contradicts (6.18).

Using this lemma we can now prove Theorem 6.2 by comparing the cardinal function $\phi$ with B-splines, which are known to be stable. A similar idea was used by Dong and Shen [11, Lemma 2.2] to show that binary pseudo-splines are stable.

Proof of Theorem 6.2

By Lemma 6.7, it is sufficient to show that $\phi$ has $\ell ^\infty$-stable integer translates if $B(\xi ) > 0$ for all $\xi$. We apply some results by Jia and Micchelli [26]. Consider the (continuous) Fourier transform of $\phi$, defined as

$$\begin{aligned} \widehat{\phi }(\xi ) := \int _{\mathbb {R}} \phi (x) \mathrm {e}^{-i\xi x} \, \mathrm {d}x, \qquad \xi \in \mathbb {R}. \end{aligned}$$

Since the scheme (2.1) reproduces constants,

$$\begin{aligned} \sum _k \phi (x-k) = 1, \qquad x \in \mathbb {R}. \end{aligned}$$

As a 1-periodic function, it has a Fourier series expansion

$$\begin{aligned} \sum _k \phi (x-k) = \sum _{n\in \mathbb {Z}} c_n \mathrm {e}^{2\pi i n x}, \end{aligned}$$

with Fourier coefficients

$$\begin{aligned} \delta _{n,0} = c_n = \int _0^1 \sum _k \phi (x-k) \mathrm {e}^{-2\pi i n x} \mathrm {d}x = \int _\mathbb {R}\phi (x) \mathrm {e}^{-2\pi i n x} \mathrm {d}x = \widehat{\phi }(2\pi n). \end{aligned}$$

In particular $\widehat{\phi }(0) = 1$. Together with the Fourier transform of (6.22),

$$\begin{aligned} \widehat{\phi }(\xi ) = m^{-1} A(\xi /m) \widehat{\phi }(\xi /m), \end{aligned}$$

if follows that

$$\begin{aligned} \widehat{\phi }(\xi ) = \prod _{\ell =1}^\infty \big (m^{-1} A(\xi /m^\ell )\big ). \end{aligned}$$

By [26, Theorem 3.5], $\phi$ has $\ell ^\infty$-stable integer translates precisely when

$$\begin{aligned} \sup _{k \in \mathbb {Z}} \left| \widehat{\phi }(\xi + 2 \pi k)\right| > 0, \qquad \hbox {for all } \xi \in \mathbb {R}. \end{aligned}$$

(6.24)

Consider again the case that the scheme admits a factorization (2.5b). Then

$$\begin{aligned} A(\xi ) = m \mathrm {e}^{-(m-1)(r+1)i\xi /2} \left( \frac{\sin (m\xi /2)}{m\sin (\xi /2)}\right) ^{r+1}B(\xi ), \end{aligned}$$

where, since $A(0) = m$ under the assumption of convergence, $B(0) = 1$. For the B-spline scheme of degree r we have $b(z) = 1$, in which case we can write its symbol as $a_r(z) = (1+z+\cdots + z^{m-1})^{r+1} / m^r$. The cardinal function $\phi _r$ is the B-spline of degree r centered at 0, and we have, after shifting,

$$\begin{aligned} \mathrm {e}^{(r+1)i\xi /2} \cdot \widehat{\phi }_r(\xi )&= \prod _{\ell =1}^\infty \left( \frac{\sin (m^{-\ell +1}\xi /2 )}{m\sin (m^{-\ell } \xi /2 )} \right) ^{r+1} \\&= \left( \frac{\sin (\xi /2)}{\xi /2}\right) ^{r+1} \lim _{\ell \rightarrow \infty } \left( \frac{m^{-\ell } \xi /2}{\sin (m^{-\ell } \xi /2 )}\right) ^{r+1}\\&= \left( \frac{\sin (\xi /2)}{\xi /2}\right) ^{r+1} =: \mathrm {sinc}^{r+1}(\xi /2). \end{aligned}$$

It then follows that

$$\begin{aligned} \widehat{\phi }(\xi ) = \widehat{\phi }_r(\xi ) \prod _{\ell =1}^\infty B\big (\xi /m^\ell \big ). \end{aligned}$$

Since the condition (6.24) holds for the B-spline $\phi _r$, we deduce that $\phi$ has $\ell ^\infty$-stable integer translates if $B(\xi ) > 0$ for all $\xi$.