Discretization of Continuous Frames by Quasi-Monte Carlo Methods

2025/07/01 by Zimmermann, Jan, Klotz, Andreas, Holighaus, Nicki
#11K38 #42C15 #46E22 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2507.00618

Abstract

We introduce a discretization scheme for continuous localized frames using quasi-Monte Carlo integration and discrepancy theory. By generalizing classical concepts, we define a discrepancy measure on the entire phase space ℝ² and establish a corresponding Koksma-Hlawka inequality. This approach enables control over the density of the discretized frame and ensures the universality of the sampling set, relying only on the discrepancy of the sampling set and on the Sobolev-type seminorm of an iterated kernel rather than on specific frame properties.

Citations

• Rotated time-frequency lattices are sets of stable sampling for continuous wavelet systems
• Numerical performance of optimized Frolov lattices in tensor product reproducing kernel Sobolev spaces
• The discretization problem for continuous frames
• Continuous Frames, Function Spaces, and the Discretization Problem

Related

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• Construction and Monte Carlo estimation of wavelet frames generated by a\n reproducing kernel
• Monte Carlo wavelets: a randomized approach to frame discretization
• Continuous Frames, Function Spaces, and the Discretization Problem
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