Shifted Splines: Concepts and Applications

• Shifted splines are piecewise polynomial functions built with basis functions shifted to prescribed nodes, enabling local approximation and exact reproduction of powers.
• They are constructed by aligning local polynomial pieces via Faulhaber’s formula and discrete convolution to achieve flexible approximation schemes with controlled smoothness.
• In advanced applications, shifted splines underpin SIAC convolution kernels and Box spline formulations, yielding higher-order convergence in numerical PDE solvers and interpolation.

A shifted spline formula refers generically to an explicit or constructive method of forming piecewise-polynomial functions whose building blocks or spline basis functions are centered (“shifted”) at prescribed nodes, rather than uniformly or symmetrically distributed about the origin. Such constructions are central to spline theory, quasi-interpolation, box spline analysis, and the design of convolution kernels for numerical methods. Shifted splines generalize classical splines by incorporating translation parameters or nonuniform centers, facilitating more flexible approximation schemes, particularly when reproducing powers, constructing shift-invariant spaces, or enforcing desired boundary or smoothness conditions. In modern approximation theory and computational mathematics, shifted splines find applications ranging from efficient power function approximation to the enhancement of numerical PDE solvers.

1. Shifted Polynomial Approximation via Centered Spline Building Blocks

The construction and deployment of shifted splines most directly arises in the context of approximating power functions, particularly $X^j$, for real $X$ and non-negative integer or real exponent $j$. The key development is the formation of a local polynomial $P(m,X,N)$ of degree $m$ in $X$, constructed by rearrangement of Faulhaber's formula, which can approximate $X^{2m+1}$ to high accuracy (percentage error under $1\%$) in a neighborhood of a fixed node $N \in \mathbb{N}$. The length of this accurate interval increases with $N$, enabling flexible local approximation properties. By repeating these centered approximants on consecutive intervals $[N,N+1]$, $[N+1,N+2]$, etc., one obtains a naive "spline": a piecewise-polynomial function with $0$-th order continuity at integer knots but without higher-derivative continuity enforced at the join points. Generalization to arbitrary non-integer exponents $j$ is obtained via construction of splines from such shifted polynomial pieces (Kolosov, 24 Feb 2025).

2. Formal Structure of Shifted Spline Formulas

A general shifted spline formula can be written as

$$S_j(X;N) = \sum_{i=0}^{n-1} \sum_{r=0}^{m} c_{i,r} (X - x_i)^r \chi_{[x_i, x_{i+1})}(X)$$

where $\{x_i\}$ denotes a knot sequence (possibly centered around $N$), $c_{i,r}$ are the coefficients of the polynomial on the $i$-th interval, and $\chi_{[x_i, x_{i+1})}$ is the characteristic function of the interval. Classical "full" spline constructions further require matching of derivatives up to order $d$ at each knot: $$S_j^{(\ell)}(x_i^-) = S_j^{(\ell)}(x_i^+), \quad \ell = 0,1,\ldots,d,$$ as well as end-point interpolation constraints. These conditions lead to a banded linear system for the $c_{i,r}$, which can often be solved in closed form via discrete convolution or by using Bernoulli-sum identities when the building block polynomials are derived from power sum or Faulhaber-type formulas. However, in some studies, including (Kolosov, 24 Feb 2025), only the piecewise $C^0$ case is treated, and the explicit derivation of higher-order smooth, shifted spline formulas (along with error bounds and coefficient representations) is left for standard spline theory references.

3. Shifted Box Spline and B-Spline Formulations

In higher dimensions and for more general geometric settings, shifted splines are represented by shifted Box splines or B-splines. For a vector space $V$ with lattice $\Lambda$ and a list of direction vectors $X = [\alpha_1,\ldots,\alpha_N] \subset \Lambda$, a shifted Box spline is defined via a translation vector $\xi = \sum_{k=1}^N r_k \alpha_k$, where $r = (r_1,\ldots,r_N) \in \mathbb{R}^N$. The Box spline shifted by $\xi$ becomes $B_r(X)(v) = B(X)(v + \xi)$. These splines admit explicit closed-form, piecewise-polynomial representations involving truncated powers and combinatorial coefficients in each alcove of the domain, as seen in the de Boor–Laurent formula: $$B_r(X)(v) = \frac1{(N−d)!} \sum_{|I|=d} \frac1{|\det(\alpha_I)|} \sum_{S \subset \{1..N\}\setminus I}(-1)^{|S|}\, \bigl\langle \alpha_I^*,\, v−\xi + \sum_{k\in S}\alpha_k\bigr\rangle_+^{N−d}$$ where the sum is over $d$-element subsets $I$ and their complements, and $\langle\cdot\rangle_+^{N-d}$ denotes the truncated power (Vergne, 2013).

4. Shifted Splines in Quasi-Interpolation and Shift-Invariant Spaces

Shifted splines play a central role in quasi-interpolation theory and the analysis of principal shift-invariant spaces. For instance, shifted thin-plate splines of the form

$$\varphi(x) = (c^{2d} + \|x\|^{2d}) \log(c^{2d} + \|x\|^{2d})$$

with $x \in \mathbb{R}^n$, $n$ even, $d \in \mathbb{N}$, are used to construct quasi-Lagrange operators: $$Q_h[f](x) = \sum_{j \in \mathbb{Z}^n} f(jh) \Psi\left(\frac{x}{h} - j\right), \quad \Psi(x) = \sum_{\alpha \in \mathbb{Z}^n} \mu_\alpha \varphi(x - \alpha),$$ where the coefficients $\mu_\alpha$ are determined to ensure reproduction of all polynomials of total degree $n + 2d - 1$. For suitable $d$, such shifted splines improve the uniform approximation error rate by a factor of $h^{2(d-1)}$ compared to the classical ($d=1$) thin-plate spline. These constructions are essentially optimal for the associated shift-invariant spaces, achieving best-possible error of $O(h^{n+2d})$ (Ortmann et al., 23 Jun 2024).

5. Moment Reproduction, Convolution Kernels, and the SIAC Paradigm

In the context of finite element and discontinuous Galerkin methods, shifted splines emerge as the core components of smoothness-increasing accuracy-conserving (SIAC) convolution kernels. Here, uniform integer shifts of B-splines of degree $d$ are linearly combined to produce a kernel

$$K_d(x) = \sum_{\gamma=-d}^{d} c_\gamma B_d(x - \gamma),$$

with coefficients determined so that convolution with any monomial up to degree $2d$ reproduces that monomial exactly. The system for the coefficients is banded, and for uniform knots, entries admit a divided-difference representation: $$M_{0;\delta,\gamma} = \frac1{(d+1)!} \sum_{\ell=0}^{d+1} (-1)^\ell \binom{d+1}{\ell} (\gamma + \ell)^{d+1+\delta}, \quad \gamma = -d, \ldots,d, \quad \delta = 0, \ldots, 2d.$$ The vector $c = M_0^{-1}e_1$ yields rational coefficients for the optimal SIAC kernel. Convolution with such shifted splines post-processes discontinuous Galerkin outputs and yields superconvergence rates $O(h^{2d+1})$—two orders higher than the native $O(h^{d+1})$ rate (Peters, 2014).

6. Explicit Formulae and Special Cases

Shifted spline formulas admit closed-form expressions in specific cases. In the one-dimensional, parameter-free scenario with $d=1$ and $X=[1,1,\ldots,1]$ ($N$ times), the shifted B-spline of order $N$ is

$$B_r^{(N)}(t) = \frac1{(N-1)!} \sum_{k=0}^{N} (-1)^k \binom{N}{k} (t - r - k)_+^{N-1},$$

with explicit partitioning and support. Concrete multivariate and mixed-directional cases are given using shifted Box splines, which generalize univariate splines to arbitrary vector spaces and lattices (Vergne, 2013). For illustrative purposes, numerical examples in (Kolosov, 24 Feb 2025) apply this machinery to approximate real powers near integer nodes using $P(m,X,N)$ pieces.

7. Theoretical Foundations and Limitations

The systematic theory of shifted spline formulas encompasses: construction of shifted local polynomial or spline pieces, determination of the knot sequence, enforcement of smoothness and polynomial reproduction conditions, solution of associated linear systems (often utilizing Faulhaber, Bernoulli, or divided-difference identities), and derivation of error bounds—typically via Peano kernels or modulus-of-continuity techniques. However, some works, such as (Kolosov, 24 Feb 2025), present only the basic building blocks and naive piecewise constructions, without the full spline apparatus (continuity of derivatives, closed-form coefficients, or rigorous error analysis). For such details, comprehensive treatments refer to foundational spline texts or advanced studies on B-splines, quasi-interpolation, and shift-invariant spaces.

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References:

(Kolosov, 24 Feb 2025) An efficient method of spline approximation for power function (Vergne, 2013) Poisson summation formula and Box splines (Peters, 2014) Computing SIAC spline coefficients (Ortmann et al., 23 Jun 2024) On Quasi-Interpolation and their associated shift-invariant space using a new class of generalized Thin Plate Splines and Inverse Multiquadrics