Author

Laura Weidensager

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Preprint Jul 2026

Approximating the Fourier Transform from Non-equispaced Discrete Samples

We study the approximation of the Fourier transform of a function from finitely many samples. Departing from the equispaced setting, we sample the function at deterministic non-equispaced nodes obtained by transforming equispaced points on $[0,1]$ through the inverse cumulative distribution function of a probability density. This change of variables compactifies the real line, so that no truncation of the space domain is necessary. Combining an exact aliasing identity for the midpoint rule with stationary-phase estimates for the transformed oscillatory integrals, we derive deterministic error bounds for every frequency and in $L_p$. For functions with polynomial decay in space and frequency, an explicitly optimized density recovers the equispaced convergence rate, while an additional variance parameter substantially reduces the pre-asymptotic error constants. For (sub-)exponentially decaying functions a polynomially decaying density yields (sub-)exponential rates. Numerical experiments confirm the theory and demonstrate the reduced pre-asymptotic error compared with optimally scaled equispaced sampling.

Daniel Potts, Laura Weidensager · 0 citations

#machine learning Preprint Sep 2026

A Weighted Kernel Method for Approximation that Adapts to Learned Multivariable Structure

Approximating the input-output behavior of a multivariable black-box function from limited data is challenging when blind to the importance of its inputs and their interactions. We introduce total sensitivity kernels (TSKs), a method based on families of weighted ANOVA kernels that learn and adapt to this multivariable structure. TSKs parameterize the weights on each multivariable component of the target function by factors for each input. We propose learning these factors directly from function evaluations by selecting the reproducing kernel Hilbert space (RKHS) in which the target function has minimum norm. Under suitable conditions, we show that this norm-minimization problem admits a unique solution, and we establish consistency of a finite-data formulation based on minimum-norm interpolation. The learned TSK factors characterize the participation of individual inputs across interactions and main effects, providing a kernel-dependent notion of input sensitivity related to total Sobol indices. Numerical experiments demonstrate that adapting the kernel to learned multivariable structure can substantially improve approximation accuracy over a standard product kernel.

John Darges, Laura Weidensager · 0 citations