[Paper Review] Kernels of discrete convolutions and subdivision operators

Tomas Sauer|arXiv (Cornell University)|Mar 30, 2014

Advanced Numerical Analysis Techniques4 references3 citations

TL;DR

This paper establishes that kernels of discrete convolution and stationary subdivision operators in multiple dimensions consist precisely of exponential polynomials, with the structure of these kernels determined by the multiplicities of common zeros of multivariate Laurent polynomials—extending classical one-dimensional results to higher dimensions using Gröbner's multiplicity theory for zero-dimensional ideals.

ABSTRACT

We consider kernels of discrete convolution operators or, equivalently, homogeneous solutions of partial difference operators and show that these solutions always have to be exponential polynomials. The respective polynomial space in connected directly though somewhat intricately to the multiplicity of the common zeros of certain multivariate polynomials, a concept introduced by Gröbner in the description of kernels of partial differential operators with constant coefficients. These results can are then used to determine the kernels of stationary subdivision operators as well.

Motivation & Objective

• To characterize the kernels of discrete convolution operators on ℤ^s as exponential polynomial spaces.
• To extend the classical one-dimensional result on convolution kernel structure to multiple dimensions.
• To apply Gröbner's multiplicity theory for zero-dimensional ideals to describe the structure of kernels in higher dimensions.
• To characterize the kernels of stationary subdivision operators using symmetric zeros of their symbols.
• To establish a correspondence between symmetric zeros of the subdivision symbol and the resulting kernel structure.

Proposed method

• Uses the symbol of a convolution operator, a Laurent polynomial h*(z), to represent the operator as h*(τ⁻¹), where τ is the shift operator.
• Applies the duality between solutions of h*(τ⁻¹)c = 0 and the zeros of h*(z), showing that exponential sequences e_θ are in the kernel iff h*(θ⁻¹) = 0.
• Introduces Gröbner's multiplicity theory for zero-dimensional ideals to describe the structure of solutions when zeros have multiplicity.
• Defines the inner product (f,g) = (f(D)g)(0) to analyze D-invariant polynomial subspaces and their annihilators.
• Uses the transformation z ↦ z^Ξ to relate subdivision operators to convolution operators via periodicity and symmetry.
• Establishes that symmetric zeros of the subdivision symbol a* correspond to common zeros of the associated a_ξ* polynomials, enabling kernel characterization.

Experimental results

Research questions

• RQ1 What is the precise structure of the kernel of a discrete convolution operator on ℤ^s with a finitely supported mask?
• RQ2 How do multiplicities of common zeros of multivariate Laurent polynomials determine the form of solutions to homogeneous partial difference equations?
• RQ3 How can the kernel of a stationary subdivision operator in multiple dimensions be characterized in terms of its symbol?
• RQ4 What is the role of symmetric zeros in the symbol of a subdivision operator in determining the kernel?
• RQ5 How does the multiplicity of a symmetric zero relate to the degree of polynomial growth in the kernel sequences?

Key findings

• The kernel of any discrete convolution operator on ℤ^s consists exactly of exponential polynomials, with the polynomial degree determined by the multiplicity of the corresponding zero in the symbol.
• For a convolution operator with symbol h*(z), the kernel is isomorphic to the direct sum over θ ∈ Θ of e_θ · Π_{k_θ−1}, where k_θ is the multiplicity of the zero at θ⁻¹.
• The kernel of a stationary subdivision operator S_a is given by ⨁_{θ∈Θ} P_θ e_θ, where P_θ is a polynomial space of degree k_θ, provided that θ^−Ξ⁻¹ is a symmetric zero of order k_θ.
• Symmetric zeros of the subdivision symbol a* are in one-to-one correspondence with common zeros of the associated a_ξ* polynomials for ξ ∈ Ξ.
• The multiplicity of a symmetric zero θ^−Ξ⁻¹ in a* is equivalent to the vanishing of all derivatives of order up to k_θ of a* at that point.
• The transformation z ↦ z^Ξ allows the reduction of the subdivision kernel problem to a convolution kernel problem via periodicity, enabling the use of Gröbner’s multiplicity theory.