arXiv subjects
Publications and source records attributed to Alex Iosevich.
Let $\Q=[0,1)^d$ denote the unit cube in $d$-dimensional Euclidean space \Rd and let \T be a discrete subset of \Rd. We show that the exponentials $e_t(x):=exp(i2πtx)$, $t\in\T$ form an othonormal basis for $L^2(\Q)$ if and only if the translates $\Q+t$, $t\in\T$ form a tiling of \Rd.
Let $D$ be a bounded domain in ${\Bbb R}^n$ whose boundary has a Minkowski dimension $α<n$. Suppose that $E_Λ= {\{e^{2 πi x \cdot λ}\}}_{λ\in Λ}$, $Λ$ an infinite discrete subset of ${\Bbb R}^n$, is a frame of exponentials for $L^2(D)$, with frame constants $A,B$, $A \leq B$. Then if $$ R \ge C{(\frac{{B|\partial D|}_α}{A|D|} )}^ {\frac{1}{n-α}},$$ where $C$ depends only on the ambient dimension $n$ and ${|\partial D|}_α$ denotes the Minkowski content, then every cube of sidelength $R$ contains at least one element of $Λ$. We give examples that illustrate the extent to which our estimates are sharp.
We prove some restriction theorems for flat homogeneous surfaces of codimension greater than one.
We prove nearly sharp Orlicz space estimates for maximal averages over flat radial hypersurfaces in ${\Bbb R}^d$, $d \ge 3$.
We show, using a Knapp-type homogeneity argument, that the $(L^p, L^2)$ restriction theorem implies a growth condition on the hypersurface in question. We further use this result to show that the optimal $(L^p, L^2)$ restriction theorem implies the sharp isotropic decay rate for the Fourier transform of the Lebesgue measure carried by compact convex finite hypersurfaces.
Let $Ω$ be a compact convex domain in the plane. We prove that $L^2(Ω)$ has an orthogonal basis of exponentials if and only if $Ω$ tiles the plane by translation.
We prove that no smooth symmetric convex body $Ω$ with at least one point of non-vanishing Gaussian curvature can admit an orthogonal basis of exponentials. (The non-symmetric case was proven by Kolountzakis). This is further evidence of Fuglede's conjecture, which states that such a basis is possible if and only if $Ω$ can tile $R^d$ by translations.