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A Localized Core-Tail Fourier-Laguerre Frame Method with Adaptive Frequency Detection for Unbounded-Domain Problems

Chenyang Wang, Zhenyu Zhao, Tinggang Zhao

arXiv:2609.27910v1math.NAcs.NA

TL;DR

Truncating unbounded domains can make approximation accuracy sensitive to boundary choices, while global systems may be inefficient for localized oscillations and irregularities. The paper combines local Fourier extension in a finite core with modulated Laguerre frames in the tails, using adaptive indicators and a separated error analysis. Experiments support accuracy across oscillatory, multi-frequency, and derivative-discontinuous cases, and illustrate a differential-operator discretization with interface constraints.

  • Problem

    Truncated-domain methods can be sensitive to truncation size and boundary treatment, while global frame approximations can be inefficient for localized oscillations or piecewise-smooth structures.

    Source: §1 · pp. 1–2

  • Method

    The method uses local Fourier extension in a finite core and modulated Laguerre frames in the tails, with LFE-based indicators for edge detection and modulation-frequency selection.

    Source: §1 · p. 3

  • Results

    Numerical experiments support the method’s advantages for oscillatory and multi-frequency tails, localized core structures, and derivative-discontinuous functions; a decaying model problem demonstrates differential-operator discretization with interface continuity constraints.

    Source: §7 · p. 31

  • Takeaways & Limitations

    Tail approximation depends chiefly on how well modulation keeps residual phase moderate, rather than on recovering exact carrier frequencies.

    Source: §7 · p. 31

    A detected breakpoint with location error ρ introduces an additional O(ρ) term for Lipschitz functions with a first-derivative jump.

    Source: §4.1 · p. 12

Abstract

from arXiv · show

We propose a localized core--tail Fourier--Laguerre frame method for approximation and model problems on the real line. The domain is decomposed into a finite core and two semi-infinite tails. Local Fourier extension is used in the core to resolve nonperiodic, oscillatory, and locally nonsmooth structures, while modulated Laguerre frames are used in the tails so that the Laguerre functions approximate only slowly varying decaying envelopes. The local Fourier extension component also provides two data-driven mechanisms: coefficient-energy indicators for internal edge detection and local frequency indicators for selecting tail modulation centers. We derive error estimates that separate the core approximation error, the modulated Laguerre envelope error, the effect of frequency mismatch, and the finite-tail truncation error. The analysis shows that the tail complexity is governed mainly by the residual phase after modulation rather than by the original carrier frequency. Numerical experiments demonstrate high accuracy for oscillatory, multi-frequency, and derivative-discontinuous functions, and a decaying model problem illustrates that the representation can be combined with differential operators through exact interface constraints.

1. Introduction

Unbounded-domain approximations can be sensitive to artificial boundary placement, while global spectral and frame methods may struggle with localized oscillations or nonsmooth structure. The paper proposes a localized core–tail representation that pairs local Fourier extension with modulated Laguerre frames and adaptive indicators.

Motivation

  • Accuracy of truncated-domain methods may depend sensitively on truncation size and boundary treatment, especially for slowly decaying, oscillatory, or nonlocal solutions.Global frame approximations can also be inefficient for localized oscillations, endpoint layers, or piecewise-smooth structures.Source: §1 · pp. 1–2

Core–tail method

  • The method combines local Fourier extension in a finite core with modulated Laguerre frames in the two tails, matching the representation to their different numerical features.Local Fourier extension also supplies frequency information for tail modulation and coefficient-energy indicators for internal edge detection.Source: §1 · p. 3

Contributions

  • The paper’s error framework separates core approximation, Laguerre envelope approximation, frequency-mismatch residual phase, finite-tail truncation, and TSVD regularization contributions.The work also introduces adaptive tail-frequency selection and coefficient-energy-based internal edge indicators.Source: §1 · p. 3

Practical advantages

  • The method uses local least-squares systems and can work with uniform samples or existing data, avoiding problem-dependent Gaussian quadrature nodes.Its localized systems are smaller than a global spectral system, with complexity growing mainly with the number of local subintervals.Source: §1 · p. 3

Validation

  • Numerical experiments cover smooth and oscillatory approximation, derivative-jump examples, Hermite comparisons, and a decaying model problem on the real line.The introduction lists these as validation settings but does not report their outcomes.Source: §1 · pp. 3–4

2. Local Fourier extension approximation and adaptive frequency detection

Local Fourier extension (LFE) provides the core approximation and structural and frequency information for the core–tail method. Subdivision reduces local frequency scales, while coefficient-energy and interface-window indicators support edge detection and adaptive tail-frequency selection.

2. Local Fourier extension approximation and adaptive frequency detection

  • LFE is used for bounded-interval approximation, internal low-regularity detection, and determining the frequency range for tail modulation.The interface-adjacent terminal LFE scale informs the frequency-detection range.Source: §2 · p. 4

2.1. Fourier extension on a finite interval

  • Fourier extension represents a nonperiodic function on a finite interval using a Fourier series on an enlarged interval; TSVD regularizes its potentially ill-conditioned system.The redundant representation accommodates nonperiodic data, while TSVD suppresses components associated with small singular values.Source: §2.1 · p. 5

2.2. Local Fourier extension in the core region

  • On a local interval of length h_k, the reference-variable frequency scales with ωh_k, so subdivision can make high physical frequencies moderate locally.Adaptive residual-based partitioning creates smaller intervals where oscillation is stronger or regularity is lower.Source: §2.2 · pp. 5–6

2.3. Internal edge detection in the core region

  • A local coefficient-energy outlier can flag a window containing an internal edge; refining or aligning the partition then lets LFE approximate one-sided smooth pieces.The present method uses this mechanism as a lightweight structural tool for continuous functions that are not globally smooth in the core.Source: §2.3 · p. 6

2.4. Adaptive frequency detection near the interfaces

  • Near each core–tail interface, local frequency indicators identify candidate modulation centers, and moderate frequency mismatch is treated as residual oscillation in the tail envelope.A smaller terminal LFE element usually indicates a more oscillatory or less regular interface neighborhood; detection cost is comparable with one LFE block.Source: §2.4 · pp. 7–8

3. Modulated Laguerre frame approximation on semi-infinite tails

The tail method maps each semi-infinite region to a half-line and uses modulated Laguerre functions to represent oscillatory carriers separately from decaying envelopes. It extends this construction to jointly fitted frequency-sparse components and discrete least-squares systems.

3. Modulated Laguerre frame approximation on semi-infinite tails

  • Modulated Laguerre frames represent oscillatory tail carriers through sine and cosine factors, leaving Laguerre functions to approximate slowly varying decaying envelopes.This avoids requiring the Laguerre expansion to resolve the full oscillatory tail.Source: §3 · p. 8, §3.2 · p. 9

3.1. Laguerre functions and tail mappings

  • The left and right tails are mapped to s ≥ 0, with scaling parameters adapting the Laguerre variable to each tail envelope’s physical decay scale.The Laguerre functions are defined using the exponentially weighted Laguerre polynomials.Source: §3.1 · p. 9

3.2. Single-frequency modulation

  • A mismatch between the true carrier frequency and modulation center appears as residual envelope oscillation, which a Laguerre approximation can compensate when the mismatch is moderate.In the scaled variable, that residual oscillation has frequency |δ|/α_R.Source: §3.2 · p. 9

3.3. Multi-frequency modulated Laguerre frames

  • For frequency-sparse tails, selected modulation centers are fitted jointly, allowing the least-squares problem to distribute the data among modulated components.The tail system remains moderate in size when the number q of selected centers is small.Source: §3.3 · p. 10

3.4. Discrete least-squares formulation

  • The discrete right-tail system uses samples on a finite window [b, b + L_R], chosen so the remaining far-field magnitude is negligible or irrelevant to the task.Tail coefficients are computed with TSVD regularization; the left-tail system is constructed analogously.Source: §3.4 · pp. 10–11

4. Approximation properties of the core–tail representation

The section separates approximation error into core, modulated-tail, frequency-mismatch, finite-window, and TSVD contributions. Its bounds link core accuracy to local smoothness and frequency-length resolution, and tail accuracy to envelope approximation after modulation.

4.1. Core approximation on smooth pieces

  • On smooth core pieces, fixed-order LFE accuracy depends on the local frequency-length product; aligning derivative breaks restores piecewise estimates, while breakpoint error ρ adds O(ρ) for Lipschitz data.Source: §4.1 · p. 12

4.2. Tail approximation with exact modulation centers

  • With exact modulation centers, Laguerre error is governed by envelope approximability, not carrier frequency; for exponentially decaying tails, truncation error decreases exponentially with window length LR.Source: §4.2 · p. 13

4.3. Frequency mismatch and residual phase

  • Frequency mismatch enters through residual parameters |δ_j|/α rather than original frequencies; close modes may share a center if the Laguerre approximation resolves their residual beat.Source: §4.3 · pp. 13–14

4.4. A combined error estimate

  • With aligned breakpoints and frequency-sparse decaying tails, the total error combines core, envelope, mismatch, truncation, TSVD, and O(ρ) edge terms for Lipschitz data with derivative jumps.The edge term reflects breakpoint location error ρ.Source: §4.4 · p. 14

5. Parameter selection and complexity analysis

This section selects tail scaling and sampling parameters, then relates residual phase and local refinement to approximation cost. Its design choices keep Laguerre orders moderate while maintaining accuracy and controlling computational cost.

5.1. Scaling rule for the Laguerre tail approximation

5.2. Influence of the residual modulation frequency

  • The required Laguerre order grows approximately linearly with accumulated residual phase and is governed by |ω − κ|L, not the full carrier phase.Source: §5.2 · p. 17, Figure 2 · p. 17

5.3. Choice of the sampling ratio

  • Approximation error falls rapidly as γ rises from values near one, then improves much less after γ ≈ 3–4; too little sampling risks instability or under-resolution.Source: §5.3 · p. 18

5.4. Computational complexity

  • With fixed local basis sizes and small q, refinement cost is essentially linear in K; edge and frequency detection remain local preprocessing, and M = 25–30 keeps tail systems smaller than a global basis.Source: §5.4 · pp. 19–20

6. Numerical experiments

The experiments test frequency detection, compare adaptive core–tail approximation with fixed-scaling Hermite approximation, and apply the framework to a decaying real-line model problem. Results show advantages on oscillatory and localized structures, accurate edge-aligned approximation, and errors below 10^-12 for the representative constrained model solve.

6.1. Tail frequency detection

  • The reduced detection system has a 201 × (2Ndet + 1) matrix and is small relative to the final approximation problems while sufficient to select effective modulation centers.Well-separated local maxima of the LFE coefficient indicator are retained as candidate centers for multi-frequency tails.Source: §6.1 · pp. 21–22
  • For g1–g3, detected peaks lie near ω = 50; weak phase modulation broadens g4’s peak, while the dominant components of g5–g6 are separated.Accumulated residual phase is below 3 × 10^-2 for g1–g3, and all dominant components are detected with small residual phases for g5–g6.Source: §6.1 · pp. 22–23, Figure 4 · p. 22, Figure 5 · p. 23

6.2. Comparison with Hermite approximation

  • After edge alignment, the derivative-jump example recovers spectral accuracy to roundoff, whereas the unaligned local Fourier error stagnates around 10^-4.The coefficient-energy indicator peaks near the low-regularity point, and one-sided LFE models estimate its location with subgrid accuracy.Source: §6.2 · p. 26, Figure 7 · p. 26, Table 2 · p. 27
  • The localized construction’s runtime grows more slowly with approximation size because it solves several small local systems instead of repeatedly forming one large global system.The comparison includes frequency detection as a preprocessing cost.Source: §6.2 · p. 27, Figure 9 · p. 28
  • The core–tail method benefits on oscillatory tails, multiple frequencies, localized structures, and internal derivative jumps, while Hermite remains efficient for globally smooth Gaussian-type functions.The comparison is made using approximation degrees of freedom and maximum error.Source: §6.2 · pp. 24, 27, Figure 6 · p. 25

6.3. A decaying model problem on the real line

  • Figure 11 shows error below 10^-12 across the domain at K = 20 with 828 degrees of freedom; Figure 10 reaches about 10^-13 maximum error around K = 20.Further improvement is limited by TSVD truncation and floating-point round-off.Source: §6.3 · p. 29, Figure 11 · p. 30, Figure 10 · p. 30

7. Conclusion and remarks

The method assigns local Fourier extension to the core and modulated Laguerre frames to the tails, separating their approximation tasks. Experiments support its use for oscillatory and multi-frequency tails, complex core structures, and derivative-discontinuous functions.

  • Local Fourier extension resolves core structures and supplies coefficient-energy indicators for detecting internal derivative jumps, while modulated Laguerre frames approximate slowly varying tail envelopes.Source: §7 · p. 31
  • Core error depends on local smoothness and frequency-length products, while tail error depends on residual-envelope approximability and moderate residual phase after modulation.Source: §7 · p. 31
  • Experiments support reliable modulation-center detection and advantages over fixed-scaling Hermite for oscillatory and multi-frequency tails, localized core structures, and derivative-discontinuous functions.Source: §7 · p. 31
  • A decaying model problem combines the representation with differential operators by applying them analytically to basis functions and enforcing interface continuity constraints.Source: §7 · p. 31
  • Future work includes unified automatic refinement and edge alignment, higher-dimensional versions, and faster local Fourier extension and reduced-order tail solvers.Source: §7 · p. 31
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