Polynomial Scaling with Dimension

• Polynomial scaling with dimension is a framework for understanding how the complexity of polynomial spaces changes as the number of variables increases.
• It employs Strang–Fix conditions and matrix analysis to determine which multivariate polynomials can be exactly reproduced by function shifts.
• These insights guide the design of refinable wavelets and high-dimensional numerical algorithms by distinguishing between affine invariant and shift-invariant subspaces.

Polynomial scaling with dimension describes how the complexity, structure, and functional properties of polynomial spaces, approximation methods, or solution sets change as the underlying ambient dimension (number of variables or spatial/geometric parameters) increases. This interplay arises in approximation theory, multivariate analysis, computational mathematics (including numerical PDEs and high-dimensional integration), and signal processing. Core aspects include the characterization of polynomial spaces via refinability and invariance conditions, the algebraic and analytic machinery that ties the ability to reproduce polynomials to function shifts and scalings, and the efficient extraction of polynomial subspaces through spectral or matrix-based approaches.

1. Strang–Fix Conditions and Polynomial Reproduction

The classical Strang–Fix conditions articulate necessary and sufficient criteria for the reproduction of algebraic polynomials of degree ≤ $L$ by integer shifts of a compactly supported function $f$. Formally, for a polynomial $P$ in the space $\Pi$ of multivariate polynomials in $d$ variables,

$$(P(-iD)\,f)(2T k) = 0\quad \text{for all}\ k \in \mathbb{Z}^d \setminus \{0\},$$

as presented in equations (1.2) and (2.9) of the source. In the Fourier domain, this translates to the vanishing of derivatives of $\widehat{f}$ at the lattice $2\pi k$ up to the prescribed order. The critical feature is that these conditions are equivalently expressed as the null-space of matrices constructed from the derivatives of $f$. The minimal order $L$ for which the corresponding matrix's null-space (Definitions 2.3–2.5) achieves maximal dimension determines the maximal degree of polynomials that can be generated by the integer translates.

This foundational theory generalizes to affinely invariant spaces through the work of Dahmen and Micchelli, which extend these conditions to allow for invariance under affine (not just translation and scaling) transformations.

2. Elliptic Scaling Functions and Differential Operator Kernels

Elliptic scaling functions are compactly supported refinable functions $\varphi$ designed so that the first non-trivial term in the Taylor series of their refinement mask $m_0(\omega)$ (in the Fourier variable) is a positive definite quadratic form $W(\omega) = \omega^T Q^2 \omega$ (see (3.11)). The central refinement equation is,

$$\varphi(x) = \sum_{k} h_k\,|\det A|^{-1/2}\,\varphi(Ax - k),$$

and in the frequency domain,

$$\widehat{\varphi}(\omega) = \prod_{j=1}^\infty m_0((A^T)^{-j}\omega),$$

with isotropic dilation matrix $A$. This structure directly links each reproduced polynomial $P$ to the null-space of the homogeneous elliptic differential operator $W(-iD)$: $W(-iD) P = 0.$ For example, for the quincunx dilation (see matrix (4.2)), $Q=I$, and $W(\omega)=|\omega|^2$, so the Laplacian $\Delta$ arises, constraining the reproduction to polyharmonic polynomials. This construction fundamentally differentiates elliptic scaling functions from other families (e.g., classical B-splines), as here the reproducing property is enforced by analytic conditions on the kernel of a differential operator determined by the underlying scaling geometry.

3. Matrix Approach to Reproduction Space Analysis

The identification and extraction of polynomial spaces reproducible by a function $f$ are accomplished via a matrix-theoretic formalism. Define the $[P_L(x)]$ row-matrix of monomials up to degree $L$, and $[D_L]$ aggregating derivatives of $f$ up to the same degree. The matrix $A_L^f$ is then defined so that its kernel consists of coefficient vectors $v$ yielding a polynomial $P(x) = [P_L(x)]\,v$ such that

$(P(-iD)f)(2Tk) = 0~ \forall~k.$

Thus, $\ker A_L^f$ parameterizes all polynomials reproducible under integer shifts of $f$. This approach unifies both the classical Strang–Fix setting and the generalized scenarios, supporting characterization not only of the full shift-invariant span but also of the largest affinely invariant subspaces. This machinery provides constructive means to determine, for any compactly supported $f$, precisely which algebraic polynomials are contained in the span of its translates, crucial for high-dimensional approximation and wavelet constructions.

4. Generalized Strang–Fix and Affine Invariance

The generalized Strang–Fix conditions, developed to consider settings beyond strictly scale-invariant spaces, formalize the reproduction property as

$$(P(-iD)f)(2Tk) = 0,\quad k\in \mathbb{Z}^d\setminus\{0\},$$

with analysis shifted from vanishing derivatives per se to the structure of the null-space of $A_L^f$. The central result (Theorem 2.7) shows that, for functions associated with isotropic dilation (i.e., scalar-multiplied orthogonal matrices), the polynomial subspace generated by the integer translates is typically affinely invariant—meaning stability under the full group of affine maps (translational and dilation symmetries). Thus, in most practically relevant settings of wavelet and multiscale analysis, the polynomial scaling with dimension is governed by these affine invariance properties.

5. Scale-Invariance Versus Shift-Invariance

The distinction between scale-invariant (affinely invariant) and merely shift-invariant polynomial spaces is sharp and essential. Scale-invariant spaces are stable under integer lattice dilations (e.g., those arising from self-similar function spaces or multiresolution analyses). In contrast, shift-invariance alone admits polynomial spaces lacking such scaling symmetry. The paper demonstrates that while stationary elliptic scaling functions (with fixed dilation and mask) produce affinely invariant reproduction spaces, certain nonstationary constructions (see Section 4.3), in which the mask varies with scale, can be engineered to yield only shift-invariant (not scale-invariant) spaces. This is achieved by designing the mask so that its vanishing moment structure does not persist under dilation by $A$, thereby breaking the affinity symmetry.

6. Homogeneous Elliptic Differential Operator as Filter

For elliptic scaling functions, the reproduction property is mediated by the null-space of a homogeneous elliptic differential operator associated with the underlying scaling geometry. Given dilation matrix $A$, the corresponding positive definite matrix $Q$ yields the quadratic form $W(\omega)$, which in turn defines the operator $W(-iD)$ satisfying

$W(-iD) P = 0,$

for any polynomial $P$ reproduced. For instance, in the quincunx or diagonal dilation cases, the null-space consists of polyharmonic polynomials annihilated by the Laplacian or its higher-rank analogs. This characterization links the analytic properties of the scaling function (via its Fourier expansion) directly to the algebraic and geometric structure of the polynomial space scalable with dimension.

7. Nonstationary Elliptic Scaling Functions and Shift-Invariant Polynomial Spaces

Extending the theory, nonstationary elliptic scaling functions—with masks and functions depending explicitly on the scale parameter $j$—permit richer reproduction behavior. Their construction (see formulas (4.14), (4.15), and condition (4.17)) modifies the Maclaurin expansion of the mask to, for instance, nullify affine invariance. Under certain conditions (notably, by selecting polynomials $X$ not invariant under scaling and designing the mask accordingly), the resulting polynomial space is only shift-invariant: translation symmetric but not stable under dilation. The existence of such cases, particularly in specific matrix dilation scenarios (e.g., quincunx, diagonal matrices), is established by explicit construction, demonstrating the possibility of polynomial reproduction confined to shift-invariant but not scale-invariant subspaces.

Connections and Influence on High-Dimensional Approximation

The interplay between polynomial scaling and dimension—mediated by functional refinability, affine invariance, and the structure of associated elliptic operators—directly informs the construction and analysis of compactly supported wavelets, multiresolution analyses, and function approximation in high-dimensional settings. The matrix and analytic criteria for polynomial reproduction enable precise identification of scalable subspaces, fundamental in the development of efficient numerical algorithms (such as sparse grid approximations, high-dimensional PDE solvers, and generalized multilevel decompositions). The distinction between affine and shift-invariant generated polynomial spaces also impacts the design of tight frames and the choice of basis in computational applications, especially in the context of data with complex scaling symmetries.

In summary, polynomial scaling with dimension in the context of reproduction by function shifts and scalings is controlled by a combination of analytic (Strang–Fix-type) conditions, the algebraic structure of associated matrices encoding differential properties, and the choice of scaling function (stationary or nonstationary, elliptic or otherwise). The dimension of the generated polynomial space depends on the interplay between these parameters and the underlying geometry of the dilation, with direct implications for applications in approximation theory, computational mathematics, and the analysis of wavelet bases.