We study a discrete convolution streaming problem. An input arrives as a stream of numbers $z = (z_0,z_1,z_2,\ldots)$, and at time $t$ our goal is to output $(Tz)_t$ where $T$ is a lower-triangular Toeplitz matrix. We focus on space complexity; we define a model for studying the memory-size of online continuous algorithms. In this model, algorithms store a buffer of $β(t)$ numbers in order to achieve their goal. We characterize space complexity using the language of generating functions. The matrix $T$ corresponds to a generating function $G(x)$. When $G(x)$ is rational of degree $d$, it is known that the space complexity is at most $O(d)$. We prove a corresponding lower bound; the space complexity is at least $Ω(d)$. In addition, for irrational $G(x)$, we prove that the space complexity is infinite. We also provide finite-time guarantees. For example, for the generating function $\frac{1}{\sqrt{1-x}}$ that was studied in various previous works in the context of differentially private continual counting, we prove a sharp lower bound on the space complexity; at time $t$, it is at least $Ω(t)$.

翻译：我们研究一个离散卷积流式处理问题。输入以数字流 $z = (z_0,z_1,z_2,\ldots)$ 的形式到达，在时刻 $t$ 我们的目标是输出 $(Tz)_t$，其中 $T$ 是一个下三角 Toeplitz 矩阵。我们重点关注空间复杂度；我们定义了一个模型来研究在线连续算法的内存大小。在该模型中，算法存储一个包含 $β(t)$ 个数字的缓冲区以实现其目标。我们使用生成函数的语言来刻画空间复杂度。矩阵 $T$ 对应一个生成函数 $G(x)$。当 $G(x)$ 是 $d$ 次有理函数时，已知空间复杂度至多为 $O(d)$。我们证明了一个对应的下界；空间复杂度至少为 $Ω(d)$。此外，对于无理生成函数 $G(x)$，我们证明其空间复杂度是无限的。我们还提供了有限时间保证。例如，对于先前在差分隐私持续计数背景下被广泛研究的生成函数 $\frac{1}{\sqrt{1-x}}$，我们证明了其空间复杂度的严格下界：在时刻 $t$，该复杂度至少为 $Ω(t)$。