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In [Temme N.M., Special functions. An introduction to the classical functions of mathematical physics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1996, Section 11.3.3.1] a uniform asymptotic expansion for the…
A class of distortions termed functional Bregman divergences is defined, which includes squared error and relative entropy. A functional Bregman divergence acts on functions or distributions, and generalizes the standard Bregman divergence…
We compare possibilities of extension of bounded and unbounded Baire-one functions from subspaces of topological spaces.
We study discrete probabilistic programs with potentially unbounded looping behaviors over an infinite state space. We present, to the best of our knowledge, the first decidability result for the problem of determining whether such a…
We study the distance in the Zygmund class $\Lambda_{\ast}$ to the subspace $\operatorname{I}(\operatorname{BMO})$ of functions with distributional derivative with bounded mean oscillation. In particular, we describe the closure of…
We prove that every nonnegative continuous real-valued function on a given compact metric space is the uniform limit of some increasing sequence of nonnegative simple functions being linear combinations of indicators of open sets; here the…
We provide sufficient conditions on a family of functions $(\phi_\alpha)_{\alpha\in A}:\mathbb{R}^d\to\mathbb{R}$ for sampling of multivariate bandlimited functions at certain nonuniform sequences of points in $\mathbb{R}^d$. We consider…
We consider a class of "box-like" statistically self-affine functions, and compute the almost-sure box-counting dimension of their graphs. Furthermore, we consider the differentiability of our functions, and prove that, depending on an…
In this paper, using the tools from the lineability theory, we distinguish certain subsets of $p$-adic differentiable functions. Specifically, we show that the following sets of functions are large enough to contain an infinite dimensional…
Nearest neighbor cells in $R^d,d\in\mathbb{N}$, are used to define coefficients of divergence ($\phi$-divergences) between continuous multivariate samples. For large sample sizes, such distances are shown to be asymptotically normal with a…
Generalization under distribution shift remains a core challenge in modern machine learning, yet existing learning bound theory is limited to narrow, idealized settings and is non-estimable from samples. In this paper, we bridge the gap…
We study the property of global-local mixing for full-branched expanding maps of either the half-line or the interval, with one indifferent fixed point. Global-local mixing expresses the decorrelation of global vs local observables w.r.t.…
Discretization of the uniform norm of functions from a given finite dimensional subspace of continuous functions is studied. We pay special attention to the case of trigonometric polynomials with frequencies from an arbitrary finite set…
Within the study of uncertain dynamical systems, iterated random functions are a key tool. There, one samples a family of functions according to a stationary distribution. Here, we introduce an extension, where one sample functions…
A space $G(M, \varPhi)$ of infinitely differentiable functions in ${\mathbb R}^n$ constructed with a help of a family $\varPhi=\{\varphi_m\}_{m=1}^{\infty}$ of real-valued functions $\varphi_m \in~C({\mathbb R}^n)$ and a logarithmically…
Maximum Mean Discrepancy (MMD) has been widely used in the areas of machine learning and statistics to quantify the distance between two distributions in the $p$-dimensional Euclidean space. The asymptotic property of the sample MMD has…
Let us assume that $f$ is a continuous function defined on the unit ball of $\mathbb R^d$, of the form $f(x) = g (A x)$, where $A$ is a $k \times d$ matrix and $g$ is a function of $k$ variables for $k \ll d$. We are given a budget $m \in…
In this paper, we are concerned with a fractional differential inequality containing a lower order fractional derivative and a polynomial source term in the right hand side. A non-existence of non-trivial global solutions result is proved…
In this paper, we introduce and investigate a class P of continuous and periodic functions on R. The class P is defined so that second-order central differences of a function satisfy some concavity-type estimate. Although this definition…
We define a generalized likelihood function based on uncertainty measures and show that maximizing such a likelihood function for different measures induces different types of classifiers. In the probabilistic framework, we obtain…