Uniform Chebyshev approximations of functions satisfying a variation-type condition: reconstruction of time-varying signals on graphs
Abstract
We prove uniform approximation theorems for Chebyshev expansions of continuous functions of two variables. Under a variation-type condition on the square [−1,1]2, a continuous function admits a uniformly convergent Chebyshev expansion in the first variable whose coefficients are continuous functions of the second variable. These results are applied to the spectral graph wavelet transform of time-varying signals on finite weighted graphs: the scaling and wavelet kernels are approximated by expansions with time-varying coefficients whose degrees do not depend on time, the composition of the approximate transform with its adjoint admits the same explicit coefficient formulas as in the time-independent case, and reconstruction by the pseudoinverse is stable, with explicit bounds in terms of the uniform kernel errors. Numerical experiments on a sensor network confirm the convergence and stability estimates. In a denoising problem, soft thresholding of the graph wavelet coefficients with a time-varying transform parameter improves over its time-independent, fixed-parameter counterpart.
Related
- Poisson representations of Dunkl harmonic functions satisfying a Carleson condition
- Space-time finite element interpolation of nonsmooth functions satisfying boundary conditions
- Modular L-functions and Chebyshev systems
- Chebyshev-Exact Acceleration under Hessian Variation, II: Computer-Assisted Variational Lower Bound
- Chebyshev-Exact Acceleration under Hessian Variation, I: Sine-Jacobi Method
BibTeX
@misc{bianchi2026uniform,
title = {{Uniform Chebyshev approximations of functions satisfying a variation-type condition: reconstruction of time-varying signals on graphs}},
author = {Davide Bianchi and Sandra Saliani and Dimitrios Vavitsas},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09792},
eprint = {2609.09792},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09792},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Uniform Chebyshev approximations of functions satisfying a variation-type condition: reconstruction of time-varying signals on graphs
AU - Davide Bianchi
AU - Sandra Saliani
AU - Dimitrios Vavitsas
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.09792
UR - https://arxiv.org/abs/2609.09792
AB - We prove uniform approximation theorems for Chebyshev expansions of continuous functions of two variables. Under a variation-type condition on the square [−1,1]^2, a continuous function admits a uniformly convergent Chebyshev expansion in the first variable whose coefficients are continuous functions of the second variable. These results are applied to the spectral graph wavelet transform of time-varying signals on finite weighted graphs: the scaling and wavelet kernels are approximated by expansions with time-varying coefficients whose degrees do not depend on time, the composition of the approximate transform with its adjoint admits the same explicit coefficient formulas as in the time-independent case, and reconstruction by the pseudoinverse is stable, with explicit bounds in terms of the uniform kernel errors. Numerical experiments on a sensor network confirm the convergence and stability estimates. In a denoising problem, soft thresholding of the graph wavelet coefficients with a time-varying transform parameter improves over its time-independent, fixed-parameter counterpart.
ER -