论文标题
Wiener-HOPF差异方程和半色调插值与可集成的卷积内核
Wiener-Hopf difference equations and semi-cardinal interpolation with integrable convolution kernels
论文作者
论文摘要
令$ h \ subset \ mathbb {z}^d $为半空间晶格，与固定坐标相对于固定坐标定义（例如\ $ h = \ $ h = \ mathbb {z}^{d-1} \！\！\！\！\！\！\！\！\！\！\！ $ \ mathbb {z}^d $，即\ $ h = \ {j \ in \ mathbb {z}^d：0 \ preceq j \} $。我们考虑了$ h $的插值问题，从系列扩展的空间中，就腐烂的内核$ ϕ $的$ h $ chhifts而言。使用$ \ mathbb {z}^d $上的$ ϕ $在$ \ mathbb {z}^d $上的符号的wiener-hopf分解，我们在$ h $上得出了半粒子插值的一些基本属性，例如存在和唯一性，lagrange系列表示，差异性表征，变性表征，comvernical to contriatiation to Crevental to Cardinal Interpolation。我们的主要结果证明，内核$ ϕ $的特定代数或指数衰减被转移到$ H $上的Lagrange函数上，例如在$ H $上进行插值。这些结果表明适用于各种示例，包括高斯，Matérn，广义逆多QuaDric，盒子单频和多结式B-Spline内核。
Let $H\subset\mathbb{Z}^d$ be a half-space lattice, defined either relative to a fixed coordinate (e.g.\ $H = \mathbb{Z}^{d-1}\!\times\!\mathbb{Z}_+$), or relative to a linear order $\preceq$ on $\mathbb{Z}^d$, i.e.\ $H = \{j\in\mathbb{Z}^d : 0\preceq j\}$. We consider the problem of interpolation at the points of $H$ from the space of series expansions in terms of the $H$-shifts of a decaying kernel $ϕ$. Using the Wiener-Hopf factorization of the symbol for cardinal interpolation with $ϕ$ on $\mathbb{Z}^d$, we derive some essential properties of semi-cardinal interpolation on $H$, such as existence and uniqueness, Lagrange series representation, variational characterization, and convergence to cardinal interpolation. Our main results prove that specific algebraic or exponential decay of the kernel $ϕ$ is transferred to the Lagrange functions for interpolation on $H$, as in the case of cardinal interpolation. These results are shown to apply to a variety of examples, including the Gaussian, Matérn, generalized inverse multiquadric, box-spline, and polyharmonic B-spline kernels.
