Biorthogonal Spline-Wavelet Transfers: An L 2 Criterion, a Rational Construction, and a Minimal-Mask Obstruction
Abstract
A 2009 preprint [1] proposed transferring a cardinal B-spline factor between the members of a biorthogonal scaling pair. The inverse box convolution is unbounded on L 2 and requires an input condition. For compactly supported data, we prove equivalent Fourier, Sobolev, and local-moment tests for its L 2 existence and a finite inverse formula. For a certified biorthogonal seed, we derive translate Riesz bounds. We then give an explicit rational order-four spline seed with a 19-position dual filter. An exact Bernstein certificate proves the dual scaling function belongs to H 1 ; a finite correlation rank witness establishes functional biorthogonality, and rational Gram bounds yield the transferred pair's squared-norm translate constants 1/48 and 48. A strict Fourier-decay bound and the Cohen-Daubechies-Feauveau cross-scale criterion prove dual Riesz wavelet bases of L 2 (R). This coefficient choice repairs a failure of the degree-minimal symmetric complement: we prove its unique algebraic mask has no compactly supported L 2 dual for every spline order m ≥ 4. The order-four filters alone therefore give a counterexample to inferring functions from perfect reconstruction. Daubechies D8 and D24 seeds give further certified transfers, including full-scale wavelet bases. The spline transfer, Bézout construction, and general cross-scale criterion have precedents in He and Cohen-Daubechies-Feauveau; our claims concern the stated inverse-domain characterization, explicit rational design and certificates, and uniform obstruction within the specified minimal class.
Related
- Three-Body Earth-Moon Transfers with Different Departure/Arrival Orbital Altitudes: New Phenomenon and Diffusion Model-Augmented Construction
- Analytic Construction of Rational Curves on Fano Manifolds
- Positive biorthogonal curvature on S2× T2
- Biorthogonal Time-Dependent Variational Principle for Non-Hermitian Systems
- High order biorthogonal functions for the discrete de Rham complex on simplices*
BibTeX
@misc{ersoy2026biorthogonal,
title = {{Biorthogonal Spline-Wavelet Transfers: An L 2 Criterion, a Rational Construction, and a Minimal-Mask Obstruction}},
author = {Mehmet Ersoy},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.science/hal-05803266v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Biorthogonal Spline-Wavelet Transfers: An L 2 Criterion, a Rational Construction, and a Minimal-Mask Obstruction
AU - Mehmet Ersoy
PY - 2026
DA - 2026/10/16
PB - HAL
UR - https://hal.science/hal-05803266v1
AB - A 2009 preprint [1] proposed transferring a cardinal B-spline factor between the members of a biorthogonal scaling pair. The inverse box convolution is unbounded on L 2 and requires an input condition. For compactly supported data, we prove equivalent Fourier, Sobolev, and local-moment tests for its L 2 existence and a finite inverse formula. For a certified biorthogonal seed, we derive translate Riesz bounds. We then give an explicit rational order-four spline seed with a 19-position dual filter. An exact Bernstein certificate proves the dual scaling function belongs to H 1 ; a finite correlation rank witness establishes functional biorthogonality, and rational Gram bounds yield the transferred pair's squared-norm translate constants 1/48 and 48. A strict Fourier-decay bound and the Cohen-Daubechies-Feauveau cross-scale criterion prove dual Riesz wavelet bases of L 2 (R). This coefficient choice repairs a failure of the degree-minimal symmetric complement: we prove its unique algebraic mask has no compactly supported L 2 dual for every spline order m ≥ 4. The order-four filters alone therefore give a counterexample to inferring functions from perfect reconstruction. Daubechies D8 and D24 seeds give further certified transfers, including full-scale wavelet bases. The spline transfer, Bézout construction, and general cross-scale criterion have precedents in He and Cohen-Daubechies-Feauveau; our claims concern the stated inverse-domain characterization, explicit rational design and certificates, and uniform obstruction within the specified minimal class.
ER -