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We prove an integral representation result for variational functionals in the space $BV^{\mathcal{B}}$ of functions with bounded $\mathcal{B}$-variation where $\mathcal{B}$ denotes a $k$-th order, $\mathbb{C}$-elliptic, linear homogeneous…

A set of complex numbers $\Lambda=\{\lambda_n,\mu_n\}_{n=1}^{\infty}$ with multiple terms \[ \{\lambda_n,\mu_n\}_{n=1}^{\infty}:= \{\underbrace{\lambda_1,\lambda_1,\dots,\lambda_1}_{\mu_1 - times},…

We introduce a family of differential-reflection operators $\Lambda_{A, \varepsilon}$ acting on smooth functions defined on $\mathbb R.$ Here $A$ is a Strum-Liouville function with additional hypotheses and $\varepsilon\in \mathbb R.$ For…

Consider an elliptic self-adjoint pseudodifferential operator $A$ acting on $m$-columns of half-densities on a closed manifold $M$, whose principal symbol is assumed to have simple eigenvalues. We show existence and uniqueness of $m$…

In this paper, we study the shift on the space of uniformly bounded continuous functions band-limited in a given compact interval with the standard topology of tempered distributions. We give a constructive proof of the existence of minimal…

Given two intervals $I, J \subset \mathbb{R}$, we ask whether it is possible to reconstruct a real-valued function $f \in L^2(I)$ from knowing its Hilbert transform $Hf$ on $J$. When neither interval is fully contained in the other, this…

In a Hadamard manifold $M$, it is proved that if $u$ is a $\lambda$-eigenfunction of the Laplacian that belongs to $L^p(M)$ for some $p \ge 2$, then $u$ is bounded and $\|u\|_{\infty} \le C \|u\|_p,$ where $C$ depends only on $p$, $\lambda$…

The Strang-Fix conditions are necessary and sufficient to reproduce spaces of algebraic polynomials up to some degree by integer shifts of compactly supported functions. W. Dahmen and Ch. Micchelli (Linear Algebra Appl. 52/3:217-234, 1983)…

Let $\Omega \subset \mathbb{R}^d$ be bounded open and connected. Suppose that $W^{1,2}(\Omega) \subset L^r(\Omega)$ for some $r > 2$. Let $A$ be a pure second-order elliptic differential operator with bounded real measurable coefficients on…

Let $M$ be a complete Riemannian manifold. Let $P_{x,y}(M)$ be the space of continuous paths on $M$ with fixed starting point $x$ and ending point $y$. Assume that $x$ and $y$ is close enough such that the minimal geodesic $c_{xy}$ between…

We obtain a representation formula for the derivative of the spectral shift function $\xi(\lambda; B, \epsilon)$ related to the operators $H_0(B,\epsilon) = (D_x - By)^2 + D_y^2 + \epsilon x$ and $H(B, \epsilon) = H_0(B, \epsilon) + V(x,y),…

We show that standard deviation $\s$ satisfies the Leibniz inequality $\s(fg) \leq \s(f)\|g\| + \|f\|\s(g)$ for bounded functions f, g on a probability space, where the norm is the supremum norm. A related inequality that we refer to as…

Let $(X, \mathscr{L}, \lambda)$ and $(Y, \mathscr{M}, \mu)$ be finite measure spaces for which there exist $A \in \mathscr{L}$ and $B \in \mathscr{M}$ with either $0 < \lambda(A) < 1 < \lambda(X)$ and $0 < \mu(B) < \mu(Y)$, or the other way…

Finite dimensional subspaces spanned by exponential functions in the space of square integrable functions on a finite interval of the real line are considered. Their limiting positions are studied and described in terms of expo-polynomials.

We show that if a separable space X has a meager open subset containing a copy of the Cantor set 2^\omega, then X has $\frak{c}$ types of countable dense subsets. We suggest a generalization of the \lambda-set for non-separable spaces. Let…

Let $\Lambda$ be the limit set of an infinite conformal iterated function system and let $F$ denote the set of fixed points of the maps. We prove that the box dimension of $\Lambda$ exists if and only if \[ \overline{\dim}_{\mathrm B} F\leq…

We provide an abstract framework for a Logvinenko-Sereda type theorem, where the classical compactness assumption on the support of the Fourier transform is replaced by the assumption that the functions under consideration belong to a…

We estimate the $1$-level density of low-lying zeros of $L(s,\chi)$ with $\chi$ ranging over primitive Dirichlet characters of conductor $\in [Q/2,Q]$ and for test functions whose Fourier transform is supported in $[- 2 - 50/1093, 2 +…

We prove that Hilbert space is distortable and, in fact, arbitrarily distortable. This means that for all lambda >1 there exists an equivalent norm |.| on l_2 such that for all infinite dimensional subspaces Y of l_2 there exist x,y in Y…

We show that there exist $\mathcal L_\infty$-subspaces of separable isomorphically polyhedral Lindenstrauss spaces that cannot be renormed to be a Lindenstrauss space.