The generalization of the Koopman operator to systems with control input and the derivation of a nonlinear fundamental lemma are two open problems that play a key role in the development of data-driven control methods for nonlinear systems. Both problems hinge on the construction of observable or basis functions and their corresponding Hilbert space that enable an infinite-dimensional, linear system representation. In this paper we derive a novel solution to these problems based on orthonormal expansion in a product Hilbert space constructed as the tensor product between the Hilbert spaces of the state and input observable functions, respectively. We prove that there exists an infinite-dimensional linear operator, i.e. the generalized Koopman operator, from the constructed product Hilbert space to the Hilbert space corresponding to the lifted state propagated forward in time. A scalable data-driven method for computing finite-dimensional approximations of generalized Koopman operators and several choices of observable functions are also presented. Moreover, we derive a nonlinear fundamental lemma by exploiting the bilinear structure of the infinite-dimensional generalized Koopman model. The effectiveness of the developed generalized Koopman embedding is illustrated on the Van der Pol oscillator.

We develop sampling formulas for high-dimensional functions in reproducing kernel Hilbert spaces, where we rely on irregular samples that are taken at determining sequences of data points. We place particular emphasis on sampling formulas for tensor product kernels, where we show that determining irregular samples in lower dimensions can be composed to obtain a tensor of determining irregular samples in higher dimensions. This in turn reduces the computational complexity of sampling formulas for high-dimensional functions quite significantly.

Recently, samplets have been introduced as localized discrete signed measures which are tailored to an underlying data set. Samplets exhibit vanishing moments, i.e., their measure integrals vanish for all polynomials up to a certain degree, which allows for feature detection and data compression. In the present article, we extend the different construction steps of samplets to functionals in Banach spaces more general than point evaluations. To obtain stable representations, we assume that these functionals form frames with square-summable coefficients or even Riesz bases with square-summable coefficients. In either case, the corresponding analysis operator is injective and we obtain samplet bases with the desired properties by means of constructing an isometry of the analysis operator's image. Making the assumption that the dual of the Banach space under consideration is imbedded into the space of compactly supported distributions, the multilevel hierarchy for the generalized samplet construction is obtained by spectral clustering of a similarity graph for the functionals' supports. Based on this multilevel hierarchy, generalized samplets exhibit vanishing moments with respect to a given set of primitives within the Banach space. We derive an abstract localization result for the generalized samplet coefficients with respect to the samplets' support sizes and the approximability of the Banach space elements by the chosen primitives. Finally, we present three examples showcasing the generalized samplet framework.

We investigate perturbations of orthonormal bases of $L^2$ via a composition operator $C_h$ induced by a mapping $h$. We provide a comprehensive characterization of the mapping $h$ required for the perturbed sequence to form an orthonormal or Riesz basis. Restricting our analysis to differentiable mappings, we reveal that all Riesz bases of the given form are induced by bi-Lipschitz mappings. In addition, we discuss implications of these results for approximation theory, highlighting the potential of using bijective neural networks to construct complete sequences with favorable approximation properties.

Implicit Neural Representations (INRs) have gained popularity for encoding signals as compact, differentiable entities. While commonly using techniques like Fourier positional encodings or non-traditional activation functions (e.g., Gaussian, sinusoid, or wavelets) to capture high-frequency content, their properties lack exploration within a unified theoretical framework. Addressing this gap, we conduct a comprehensive analysis of these activations from a sampling theory perspective. Our investigation reveals that sinc activations, previously unused in conjunction with INRs, are theoretically optimal for signal encoding. Additionally, we establish a connection between dynamical systems and INRs, leveraging sampling theory to bridge these two paradigms.

Modelling dynamical systems is an integral component for understanding the natural world. To this end, neural networks are becoming an increasingly popular candidate owing to their ability to learn complex functions from large amounts of data. Despite this recent progress, there has not been an adequate discussion on the architectural regularization that neural networks offer when learning such systems, hindering their efficient usage. In this paper, we initiate a discussion in this direction using coordinate networks as a test bed. We interpret dynamical systems and coordinate networks from a signal processing lens, and show that simple coordinate networks with few layers can be used to solve multiple problems in modelling dynamical systems, without any explicit regularizers.