Abstract We show that a sufficient density condition for Gabor systems with Hermite functions over lattices is not sufficient in general. This follows from a result on how zeros of the Zak transform determine the frame property of integer over-sampled Gabor systems.

Abstract This paper outlines a methodology for constructing a geometrically smooth interpolatory curve in $$\mathbb{R}^n$$ , applicable to any given set of ordered points in $$\mathbb{R}^n, n\ge 2$$ . The construction involves four essential components: local functions, blending functions, redistributing functions, and gluing functions. The resulting curve possesses favorable attributes, including $$G^2$$ geometric smoothness, locality, the absence of cusps, and no self-intersections. Numerical examples show that the curve interpolates the given points without overshooting or undershooting. Moreover, the algorithm is adaptable to various scenarios, such as preserving convexity, interpolating sharp corners, and ensuring sphere preservation. This paper substantiates the efficacy of the proposed method through the presentation of numerous examples, offering a practical demonstration of its capabilities.

Abstract Motivated by classical harmonic analysis results characterizing Hölder spaces in terms of the decay of their wavelet coefficients, we consider wavelet methods for computing $$s$$ -Wasserstein type distances. Previous work by Sheory (né Shirdhonkar) and Jacobs showed that, for $$0 < s\leqslant1$$ , the $$s$$ -Wasserstein distance $$W_s$$ between certain probability measures on Euclidean space is equivalent to a weighted $$\ell^1$$ difference of their wavelet coefficients. We demonstrate that the original statement of this equivalence is incorrect in a few aspects and, furthermore, fails to capture key properties of the $$W_s$$ distance, such as its behavior under translations of probability measures. Inspired by this, we consider a variant of the previous wavelet distance formula for which equivalence (up to an arbitrarily small error) does hold for $$0 < s < 1$$ . We analyze the properties of this distance, one of which is that it provides a natural embedding of the $$s$$ -Wasserstein space into a linear space. We conclude with several numerical simulations. Even though our theoretical result merely ensures that the new wavelet $$s$$ -Wasserstein distance is equivalent to the classical $$W_s$$ distance (up to an error), our numerical simulations show that the new wavelet distance succeeds in capturing the behavior of the exact $$W_s$$ distance under translations and dilations of probability measures.

Abstract In order to better understand manifold neural networks (MNNs), we introduce Manifold Filter-Combine Networks (MFCNs). Our filter-combine framework parallels the popular aggregate-combine paradigm for graph neural networks (GNNs) and naturally suggests many interesting families of MNNs which can be interpreted as manifold analogues of various popular GNNs. We propose a method for implementing MFCNs on high-dimensional point clouds that relies on approximating an underlying manifold by a sparse graph. We then prove that our method is consistent in the sense that it converges to a continuum limit as the number of data points tends to infinity, and we numerically demonstrate its effectiveness on real-world and synthetic data sets.