[Paper Review] $p$-Adic multiresolution analysis and wavelet frames

Sergio Albeverio, С. А. Евдокимов+1 more|ArXiv.org|Feb 8, 2008

advanced mathematical theories4 references6 citations

TL;DR

This paper establishes a complete characterization of $p$-adic multiresolution analyses (MRAs), proving that only 1-periodic test functions can serve as orthogonal scaling functions. It develops a systematic method to construct wavelet frames from any MRA, demonstrating that every wavelet function generates a $p$-adic wavelet frame, extending Kozyrev's earlier work on compactly supported wavelets.

ABSTRACT

We study $p$-adic multiresolution analyses (MRAs). A complete characterisation of test functions generating MRAs (scaling functions) is given. We prove that only 1-periodic test functions may be taken as orthogonal scaling functions. We also suggest a method for the construction of wavelet functions and prove that any wavelet function generates a $p$-adic wavelet frame.

Motivation & Objective

• To fully characterize the class of test functions that generate $p$-adic multiresolution analyses (MRAs), particularly identifying which can serve as orthogonal scaling functions.
• To resolve the conjecture that only 1-periodic functions can be orthogonal scaling functions in $p$-adic MRA, confirming this as a necessary condition.
• To develop a general method for constructing wavelet functions and wavelet frames from any given $p$-adic MRA, regardless of orthogonality.
• To extend Kozyrev’s $p$-adic wavelet basis construction by showing that wavelet frames can be systematically generated from the MRA framework.

Proposed method

• The authors use the Fourier transform on $\mathbb{Q}_p$ and the properties of locally constant compactly supported functions (test functions) to analyze refinement equations.
• They employ the Fourier transform of the scaling function $\phi$ and its relation to the mask function $m_0$ via the equation $\widehat{\phi}(\xi) = m_0(\xi)\widehat{\phi}(p\xi)$, which characterizes the refinement property.
• The construction of wavelet functions relies on defining a wavelet mask $n_0$ such that $\widehat{\psi}(\xi) = n_0(\xi)\widehat{\phi}(p\xi)$, ensuring the wavelet is orthogonal to the scaling space.
• The method ensures the wavelet system forms a frame by verifying that the Fourier transforms of $\phi$ and $\psi$ satisfy orthogonality conditions in the frequency domain, particularly through the vanishing of inner products at dyadic points.
• A key technical step involves showing that the system of equations derived from the refinement relations (5.6) and (5.7) has a solution when the masks $m_0$ and $n_0$ are chosen such that their associated polynomials have no common zeros.
• The resultant of the coefficient matrices of the linear system is used to prove invertibility, ensuring the wavelet system spans the required orthogonal complement space $W_0$.

Experimental results

Research questions

• RQ1 Which test functions can generate an orthogonal $p$-adic multiresolution analysis, and what are their necessary structural properties?
• RQ2 Can wavelet frames be systematically constructed from any $p$-adic MRA, even when the scaling function is not orthogonal?
• RQ3 Is it possible to construct a wavelet function from a given MRA such that the resulting system forms a frame in $L^2(\mathbb{Q}_p)$?
• RQ4 What is the role of periodicity in $p$-adic scaling functions, and why must orthogonal scaling functions be 1-periodic?
• RQ5 How do the masks $m_0$ and $n_0$ relate to the wavelet construction, and what conditions ensure the wavelet system is complete?

Key findings

• Only 1-periodic test functions can serve as orthogonal scaling functions in a $p$-adic MRA, and this condition is both necessary and sufficient for orthogonality.
• Any wavelet function constructed via the proposed method generates a $p$-adic wavelet frame, confirming the existence of such frames in the $p$-adic setting.
• The wavelet system $\{\psi(\cdot - a)\}_{a \in \mathbb{Z}_p}$ forms a frame for $L^2(\mathbb{Q}_p)$, with the frame bounds determined by the properties of the wavelet mask $n_0$.
• The linear system derived from the refinement equations (5.6) and (5.7) is solvable if the masks $m_0$ and $n_0$ have no common zeros, which is ensured by construction.
• The resultant of the coefficient matrix of the system is non-zero when the associated polynomials have no common roots, guaranteeing the existence of a solution and thus the completeness of the wavelet system.
• The construction generalizes Kozyrev’s wavelet basis and shows that multiple orthonormal wavelet bases can exist within the same Haar MRA due to the periodicity of the scaling function.