Quantitative estimates for the absolute convergence of wavelet-type series

Abstract: We establish new quantitative estimates for general systems of functions with wavelet-type dyadic structure. These estimates are applied to obtain the optimal growth of various types of Weyl multipliers for certain wavelet-type systems. Some of our results are sufficiently general to allow the orthogonality assumption to be removed. In particular, as a consequence of these estimates we show that the condition \begin{equation*} \sum_{n=1}^\infty\frac{1}{nw(n)}<\infty \end{equation*} is necessary and sufficient for an increasing sequence $w(n)$ to be an almost everywhere unconditional convergence Weyl multiplier for an arbitrary wavelet-type system. We also prove that $\log n$ is an almost everywhere convergence Weyl multiplier for any rearranged wavelet-type system, and that this bound is optimal.

• The paper establishes sharp quantitative L2 estimates and necessary multiplier growth conditions ensuring absolute convergence in wavelet-type series.
• It extends classical convergence results from orthonormal bases to general, non-orthogonal wavelet-like structures using advanced tree decomposition methods.
• It employs novel combinatorial techniques and explicit counterexamples to prove the optimality of the log n Weyl multiplier condition for both RC and UC convergences.

Quantitative Estimates for the Absolute Convergence of Wavelet-Type Series

Context and Motivation

The study of series expansions in function systems with wavelet-type dyadic structure lies at the core of modern harmonic analysis and approximation theory. Of particular interest is the absolute and unconditional convergence of such series, typically characterized via Weyl multipliers. For classical orthonormal systems—such as the trigonometric, Haar, Walsh, and Franklin systems—the growth conditions on multipliers ensuring almost everywhere (a.e.) absolute or unconditional convergence have been clarified in much of the 20th-century literature, with sharp results in many cases. However, extending these characterizations optimally to broader classes of wavelet-like systems, potentially without orthogonality, has remained an active domain. This paper delivers a suite of quantitative $L^2$-estimates, yielding sharp necessary and sufficient conditions for absolute convergence in series expansions over general (possibly non-orthogonal) wavelet-type systems.

Key Definitions and Preliminaries

Let $\Phi = \{\phi_n : n \geq 1\} \subset L^2(a, b)$ be a sequence of functions, possibly orthonormal, indexed over an interval $(a, b)$. An increasing sequence $w(n) \nearrow \infty$ is a Weyl multiplier for $\Phi$ if, for any sequence of coefficients $\{a_n\}$ with $\sum a_n^2 w(n) < \infty$, the associated series $\sum a_n \phi_n(x)$ converges almost everywhere. The RC (rearrangement convergence) multiplier property requires convergence under arbitrary rearrangements; the UC (unconditional convergence) multiplier property requires a.e. convergence of $\sum a_n \phi_n(x)$ after any permutation of terms.

The classical Menshov-Rademacher theorem shows that $w(n) = (\log n)^2$ is a Weyl multiplier for every orthonormal system, with the bound being sharp. For specific canonical systems (e.g., Haar, Franklin, wavelets), the optimal growth profile and distinctions between RC/UC multipliers are subtler, especially when relaxing orthogonality.

Wavelet-type systems on intervals are typically structured as families $\Phi = \{\phi_n : n \geq 1\} \subset L^2(a, b)$0, with $\Phi = \{\phi_n : n \geq 1\} \subset L^2(a, b)$1 supported on appropriately scaled and shifted dyadic intervals, satisfying normalization, zero mean, decay, and regularity properties indexed by parameters $\Phi = \{\phi_n : n \geq 1\} \subset L^2(a, b)$2 and $\Phi = \{\phi_n : n \geq 1\} \subset L^2(a, b)$3.

Main Technical Results

Novel Quantitative $\Phi = \{\phi_n : n \geq 1\} \subset L^2(a, b)$4-Estimates

The authors derive explicit bounds for the $\Phi = \{\phi_n : n \geq 1\} \subset L^2(a, b)$5-norm of linear combinations of tidily shifted dilations of a mother function $\Phi = \{\phi_n : n \geq 1\} \subset L^2(a, b)$6: $\Phi = \{\phi_n : n \geq 1\} \subset L^2(a, b)$7 This logarithmic bound is crucial: it encodes the degree of non-orthogonality yet recovers classical orthonormal results for Haar and other canonical dyadic systems.

Optimal Characterizations of RC and UC Multipliers

The analysis is extended to arbitrary wavelet-type systems (not necessarily orthonormal). The main characterization is as follows:

• For a wavelet-type system $\Phi = \{\phi_n : n \geq 1\} \subset L^2(a, b)$8, an increasing sequence $\Phi = \{\phi_n : n \geq 1\} \subset L^2(a, b)$9 is a UC-multiplier if and only if $(a, b)$0 (Ul’yanov’s condition).
• For RC-multipliers, the optimal growth is $(a, b)$1: this is always sufficient and (within this setting) necessary, even absent orthogonality.
• The result applies equally to all rearrangements and, crucially, does not require the system to be orthogonal for the UC-multiplier case.

Further, the paper shows that for wavelet-type systems, the value $(a, b)$2 is an a.e. convergence Weyl multiplier, and that the bound is sharp. For any $(a, b)$3, there exist sequences of coefficients and rearrangements for which divergence a.e. occurs, as constructed explicitly in the arguments.

Extension to Non-Orthogonal Systems

A significant contribution is the generality with which the orthogonality constraint is removed. The main theorems and corollaries establish optimal RC/UC multiplier growth for arbitrary wavelet-type systems satisfying standard decay and regularity estimates; the core techniques incorporate refined maximal function decompositions, tree systems, and sophisticated use of truncated kernels.

Methods

The analysis proceeds by intricate combinatorics over dyadic intervals, exploiting tree decompositions and partitioning strategies that facilitate control over local maxima and distributions of the shifted-dilated functions. The authors construct "tree systems" facilitating control over partial sum rearrangements, and engage good-$(a, b)$4 inequalities in $(a, b)$5. The sharpness of multiplier results leverages explicit counterexamples for suboptimal growth, using carefully structured coefficient sequences and partial sums whose rearrangements induce divergence. The necessity argument employs extensions of methods due to Nikishin, Ul’yanov, and others, adapted to the wavelet context.

Implications and Further Directions

These results close a long-standing problem in harmonic analysis: that of extending sharp RC and UC multiplier characterizations, previously resolved for specific orthonormal bases, to large classes of structurally similar but possibly non-orthogonal wavelet-type systems. The optimality of $(a, b)$6 and the summability condition over $(a, b)$7 for convergence are now established in their proper generality. The proofs and auxiliary lemmas suggest that further refinements—possibly to systems with even weaker structural assumptions, or for $(a, b)$8 settings with $(a, b)$9—may be within reach. Importantly, these results facilitate a more precise understanding of convergence phenomena in sparse coding, adaptive approximation, and other applications where non-classical wavelets or bases are employed. They are also informative for the development of numerical schemes in signal processing, where absolute convergence is essential for stability.

Conclusion

The paper offers a set of precise, optimal, and technically robust estimates for the absolute and unconditional convergence of series over wavelet-type systems, both orthogonal and non-orthogonal. The derived bounds on RC and UC multipliers resolve the exact growth conditions for convergence in the most general dyadic settings considered to date, providing both a consolidation of classical results and their extension to rich classes of function systems relevant in modern analysis (2604.25442).