Wavelet Expansion Method Overview

• Wavelet expansion method is a technique that decomposes functions into localized scaling components and wavelet details indexed by both position and scale.
• It facilitates multiresolution analysis, supports adaptive statistical estimation, and yields sparse numerical discretizations of complex, inhomogeneous data.
• Its diverse applications in quantum mechanics, image processing, and inverse problems demonstrate its power in improving computational efficiency and accuracy.

Searching arXiv for the specified paper and closely related wavelet-expansion literature. Wavelet expansion method denotes a family of representations in which a function, field, distribution, or stochastic process is decomposed into scaling components and wavelet details indexed by location and resolution. In a standard discrete multiresolution formulation, a function $\alpha_l\in L_2(\mathbb R)$ is written as

$$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$

with $\phi$ a scaling function and $\psi$ a mother wavelet; in a continuous formulation, a local field $\phi(x)$ is replaced by scale-dependent coefficients $\phi_a(b)$ together with an exact reconstruction formula. The common principle is localization in both position and scale, which makes wavelet expansions suitable for functions with inhomogeneous smoothness, multiscale operators, nonlocal field formulations, and sparse numerical discretizations [(Sousa, 2022); (Altaisky et al., 2013); (Benhaddou et al., 2012)].

1. Multiresolution structure and basic formulas

The discrete wavelet expansion is built from a multiresolution analysis. Given a scaling function $\phi$ and a mother wavelet $\psi$, the atoms are

$$\phi_{j,k}(x)=2^{j/2}\phi(2^j x-k),\qquad \psi_{j,k}(x)=2^{j/2}\psi(2^j x-k).$$

The associated approximation and detail spaces satisfy

$$V_j=\operatorname{span}\{\phi_{j,k}\},\qquad W_j=\operatorname{span}\{\psi_{j,k}\},\qquad V_j\subset V_{j+1},\qquad V_{j+1}=V_j\oplus W_j,$$

and hence

$$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$0

for any fixed coarse scale $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$1. In quantum-mechanical notation, a wavefunction $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$2 is expanded as

$$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$3

with coefficients given by the standard inner products $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$4 and $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$5 (Chawhan et al., 2020).

For aggregated functional data, the same structure is used componentwise. If $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$6 denotes the $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$7-th component curve, then the expansion separates low-frequency approximation coefficients $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$8 from detail coefficients $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$9. The cited application emphasizes that jumps, spikes, and rapid oscillations correspond to a small number of large detail coefficients, while smooth regions are represented by many small coefficients (Sousa, 2022).

High-dimensional and nonhomogeneous variants replace the single-resolution starting point by a coarse-scale family $\phi$0 and a wavelet family $\phi$1. For an expansive matrix $\phi$2, a nonhomogeneous system at scale $\phi$3 is

$\phi$4

and every $\phi$5 has a dual-frame expansion in terms of coarse-scale scaling functions and all finer-scale wavelets. As $\phi$6, the low-pass part vanishes and a purely homogeneous expansion is recovered (Han, 2010).

2. Continuous wavelet expansion and asymptotic analysis

The continuous wavelet transform replaces dyadic indexing by a continuous scale $\phi$7 and translation $\phi$8. With a mother wavelet $\phi$9 satisfying the admissibility condition

$\psi$0

one defines

$\psi$1

and the wavelet coefficients of a scalar field $\psi$2 by

$\psi$3

The reconstruction formula is

$\psi$4

In momentum space, the scale decomposition reads $\psi$5, so each scale component acts as a frequency-localized filter (Altaisky et al., 2013).

This continuous formulation supports asymptotic expansions of the wavelet transform itself. For

$\psi$6

the small-$\psi$7 regime can be analyzed by Mellin-convolution methods. Under expansions of $\psi$8 near the origin and decay assumptions on $\psi$9, one obtains a series

$\phi(x)$0

with $\phi(x)$1 as $\phi(x)$2. The cited derivations work out explicit cases for Morlet, Mexican-hat, and Haar wavelets (Pathak et al., 2014). A complementary large-$\phi(x)$3 expansion has the form

$\phi(x)$4

with $\phi(x)$5 under the stated hypotheses (Pathak et al., 2014).

These results make clear that the wavelet expansion method is not only a representation device but also an asymptotic calculus for scale-dependent integral transforms.

3. Statistical estimation, shrinkage, and inverse problems

In aggregated functional data analysis, one observes

$\phi(x)$6

at $\phi(x)$7 grid points. Applying the orthonormal discrete wavelet transform matrix $\phi(x)$8 yields

$\phi(x)$9

where $\phi_a(b)$0. The proposed estimator shrinks each empirical coefficient $\phi_a(b)$1 under the model

$\phi_a(b)$2

using the prior

$\phi_a(b)$3

Under squared-error loss the Bayes estimator is the posterior mean $\phi_a(b)$4, implemented through the integral formula given in the paper. After shrinkage of the aggregated coefficients, the componentwise detail coefficients are recovered by

$\phi_a(b)$5

and the curves are reconstructed via $\phi_a(b)$6. In simulation studies, the averaged mean-squared error

$\phi_a(b)$7

was used. For functions with local features, the wavelet/logistic-shrinkage method dramatically outperformed B-splines in AMSE, often by orders of magnitude; for completely smooth functions, cubic B-splines slightly beat wavelets, though wavelets still gave very low AMSE (Sousa, 2022).

A distinct statistical use appears in adaptive nonparametric empirical Bayes estimation. There the Bayes rule

$\phi_a(b)$8

is approximated by a finite wavelet series, reparametrized in practice as a scaling-only expansion

$\phi_a(b)$9

Coefficient estimation reduces to a sparse linear system

$\phi$0

with adaptive resolution selected by Lepski’s method. The resulting estimator achieves the minimax rate $\phi$1 up to a $\phi$2 factor, and the adaptive version satisfies

$\phi$3

uniformly over Hölder classes of order $\phi$4 (Benhaddou et al., 2012).

Wavelet expansion has also been adapted to local differential privacy for numerical distribution estimation. In that setting, a density on $\phi$5 is expanded in the Haar basis,

$\phi$6

and low-order coefficients are prioritized by splitting users across resolution levels. With optimal allocation $\phi$7 and $\phi$8, the expected Wasserstein error satisfies

$\phi$9

for small $\psi$0, and $\psi$1 for moderate or large $\psi$2 (Zhao et al., 24 Sep 2025).

4. Bases, frames, approximate duals, and nonstandard geometries

Wavelet expansion does not require exact orthonormal duality. In Hardy spaces $\psi$3, $\psi$4, one defines the mixed frame operator

$\psi$5

The pair $\psi$6 is called an approximate dual if $\psi$7 is bounded and

$\psi$8

Then $\psi$9 is invertible by Neumann series, and every $$\phi_{j,k}(x)=2^{j/2}\phi(2^j x-k),\qquad \psi_{j,k}(x)=2^{j/2}\psi(2^j x-k).$$0 has the convergent expansion

$$\phi_{j,k}(x)=2^{j/2}\phi(2^j x-k),\qquad \psi_{j,k}(x)=2^{j/2}\psi(2^j x-k).$$1

with convergence in the $$\phi_{j,k}(x)=2^{j/2}\phi(2^j x-k),\qquad \psi_{j,k}(x)=2^{j/2}\psi(2^j x-k).$$2-quasi-norm and in $$\phi_{j,k}(x)=2^{j/2}\phi(2^j x-k),\qquad \psi_{j,k}(x)=2^{j/2}\psi(2^j x-k).$$3. The construction works for all $$\phi_{j,k}(x)=2^{j/2}\phi(2^j x-k),\qquad \psi_{j,k}(x)=2^{j/2}\psi(2^j x-k).$$4, does not assume exact duals, and provides an explicit bound on the frame error. In the Mexican-hat example, $$\phi_{j,k}(x)=2^{j/2}\phi(2^j x-k),\qquad \psi_{j,k}(x)=2^{j/2}\psi(2^j x-k).$$5, the overlap quantities $$\phi_{j,k}(x)=2^{j/2}\phi(2^j x-k),\qquad \psi_{j,k}(x)=2^{j/2}\psi(2^j x-k).$$6 are all $$\phi_{j,k}(x)=2^{j/2}\phi(2^j x-k),\qquad \psi_{j,k}(x)=2^{j/2}\psi(2^j x-k).$$7, and with $$\phi_{j,k}(x)=2^{j/2}\phi(2^j x-k),\qquad \psi_{j,k}(x)=2^{j/2}\psi(2^j x-k).$$8, $$\phi_{j,k}(x)=2^{j/2}\phi(2^j x-k),\qquad \psi_{j,k}(x)=2^{j/2}\psi(2^j x-k).$$9 one checks $$V_j=\operatorname{span}\{\phi_{j,k}\},\qquad W_j=\operatorname{span}\{\psi_{j,k}\},\qquad V_j\subset V_{j+1},\qquad V_{j+1}=V_j\oplus W_j,$$0 for every $$V_j=\operatorname{span}\{\phi_{j,k}\},\qquad W_j=\operatorname{span}\{\psi_{j,k}\},\qquad V_j\subset V_{j+1},\qquad V_{j+1}=V_j\oplus W_j,$$1 (Hur et al., 2023).

In high dimensions, nonhomogeneous systems clarify the relation between refinement structure and generator counts. If a nonhomogeneous system $$V_j=\operatorname{span}\{\phi_{j,k}\},\qquad W_j=\operatorname{span}\{\psi_{j,k}\},\qquad V_j\subset V_{j+1},\qquad V_{j+1}=V_j\oplus W_j,$$2 is a Riesz basis, then

$$V_j=\operatorname{span}\{\phi_{j,k}\},\qquad W_j=\operatorname{span}\{\psi_{j,k}\},\qquad V_j\subset V_{j+1},\qquad V_{j+1}=V_j\oplus W_j,$$3

Hence a one-generator nonhomogeneous orthonormal basis requires $$V_j=\operatorname{span}\{\phi_{j,k}\},\qquad W_j=\operatorname{span}\{\psi_{j,k}\},\qquad V_j\subset V_{j+1},\qquad V_{j+1}=V_j\oplus W_j,$$4. By contrast, for every expansive matrix $$V_j=\operatorname{span}\{\phi_{j,k}\},\qquad W_j=\operatorname{span}\{\psi_{j,k}\},\qquad V_j\subset V_{j+1},\qquad V_{j+1}=V_j\oplus W_j,$$5 there exists a nonhomogeneous smooth tight $$V_j=\operatorname{span}\{\phi_{j,k}\},\qquad W_j=\operatorname{span}\{\psi_{j,k}\},\qquad V_j\subset V_{j+1},\qquad V_{j+1}=V_j\oplus W_j,$$6-wavelet frame in $$V_j=\operatorname{span}\{\phi_{j,k}\},\qquad W_j=\operatorname{span}\{\psi_{j,k}\},\qquad V_j\subset V_{j+1},\qquad V_{j+1}=V_j\oplus W_j,$$7 with a single wavelet generator whose Fourier transform is a compactly supported $$V_j=\operatorname{span}\{\phi_{j,k}\},\qquad W_j=\operatorname{span}\{\psi_{j,k}\},\qquad V_j\subset V_{j+1},\qquad V_{j+1}=V_j\oplus W_j,$$8 function (Han, 2010).

The geometry of wavelet expansion can also be highly anisotropic. A two-dimensional orthonormal basis of $$V_j=\operatorname{span}\{\phi_{j,k}\},\qquad W_j=\operatorname{span}\{\psi_{j,k}\},\qquad V_j\subset V_{j+1},\qquad V_{j+1}=V_j\oplus W_j,$$9 can be constructed by applying the standard one-dimensional wavelet action only to the first variable,

$$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$00

provided the generating function is chosen so that the missing action on $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$01 is compensated in phase space. The associated tiling lives in $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$02 and consists of hyperboloid-type blocks $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$03 of constant Liouville measure $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$04 (Nowak et al., 2016).

A recurrent misconception is that wavelet expansion is synonymous with an orthonormal dyadic basis. The cited literature shows a broader picture: continuous transforms, approximate duals, nonhomogeneous tight frames, and one-variable affine actions all fit within the same methodological family.

5. Physics and stochastic-process formulations

In quantum mechanics, the expansion of a wavefunction in a Daubechies basis produces a Hamiltonian matrix with block structure

$$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$05

where $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$06. Because $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$07 and $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$08 have compact support, the matrix is sparse, kinetic-energy terms couple only near-neighbor translations at the same scale, and the off-diagonal blocks represent couplings between different resolutions. Truncation is performed in both volume and resolution. For the one-dimensional harmonic oscillator, with $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$09, the ground-state energy improves from $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$10 at $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$11 to $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$12 at $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$13 and $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$14 at $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$15, while excited states similarly approach the exact values $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$16 (Chawhan et al., 2020).

In Euclidean quantum field theory, continuous wavelet expansion replaces the local field $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$17 by finite-resolution fields $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$18. Substituting the reconstruction formula into the $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$19 generating functional yields a genuinely nonlocal action $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$20 whose quadratic kernel is

$$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$21

The associated region-causality principle forbids internal scales smaller than the minimal external scale

$$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$22

so the connected Green functions $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$23 are finite for any given set of regions. This makes the cutoff a physical resolution parameter rather than an external momentum parameter $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$24 (Altaisky et al., 2013).

Wavelet-type random series also provide multiscale representations of self-similar stochastic processes. For generalized Hermite processes of order $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$25, the process is decomposed into a scaling part $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$26 and a detail part $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$27, both expressed through tensorized Meyer scaling functions and wavelets. The remainder

$$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$28

satisfies the almost-sure uniform bound

$$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$29

with $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$30 almost surely finite. This yields uniform almost-sure convergence with an explicit rate, extending earlier wavelet expansions for fractional Brownian motion and the Rosenblatt process to arbitrary integer order $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$31 (Ayache et al., 2023).

6. Numerical analysis, multidimensional constructions, and method-specific trade-offs

Wavelet Galerkin discretization uses a wavelet expansion as the trial and test representation of a variational problem. For the 2D Helmholtz equation on $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$32, boundary-adapted spline biorthogonal wavelets on $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$33 are tensorized to form a normalized basis

$$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$34

which is a Riesz basis of

$$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$35

Expanding $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$36 leads to a linear system $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$37 with $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$38. Because the basis is a Riesz basis in $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$39, the stiffness matrix has uniformly bounded condition numbers: $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$40 In contrast, for the standard FEM nodal basis one has $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$41. The cited comparison reports that, when an iterative scheme is applied to the wavelet coefficient matrix, much fewer iterations are needed for the relative residuals to be within a tolerance level, and the number of required iterations is practically independent of the size of the matrix for a given bounded variable wavenumber (Han et al., 2023).

Multidimensional expansions need not be tensor-product-symmetric. Orbital wavelets construct a 2D anti-symmetric wavelet from two 1D anti-symmetric wavelets: $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$42 The resulting family is orthonormal, complete in $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$43, and localized in both space and frequency. This supports image decompositions that analyze two distinct scales simultaneously (Oliveira et al., 2020).

The major computational and modeling consequences can be summarized briefly.

Setting	Expansion object	Reported property
Aggregated functional data	Component curves and wavelet coefficients	Local features preserved; AMSE favorable for Bumps, Blocks, Doppler, Heavisine
Quantum field theory	Scale-dependent fields $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$44	Green functions finite with cutoff $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$45
Helmholtz discretization	2D Riesz wavelet basis in $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$46	Sparse systems with uniformly bounded condition numbers

The method is therefore highly effective when the target exhibits discontinuities, spikes, oscillations, multiscale roughness, or nonlocal interactions. The same sources also delimit its trade-offs. In globally smooth functional data, cubic B-splines can slightly beat wavelets in AMSE (Sousa, 2022). In nonredundant nonhomogeneous orthonormal constructions, single-generator expansions are generally impossible unless $$\alpha_l(t)=\sum_k c_{J_0,k}^{(l)}\,\phi_{J_0,k}(t)+\sum_{j=J_0}^{J-1}\sum_k \gamma_{j,k}^{(l)}\,\psi_{j,k}(t),$$47 (Han, 2010). These constraints do not diminish the generality of wavelet expansion; rather, they specify the regimes in which localization, sparse representation, and scale separation are the decisive structural advantages.