Related papers: Extended Gevrey regularity via the short-time Four…
We completely characterize the boundedness on $L^p$ spaces and on Wiener amalgam spaces of the short-time Fourier transform (STFT) and of a special class of pseudodifferential operators, called localization operators. Precisely, a…
Let $\phi$ be a smooth function on a compact interval $I$. Let $$\gamma(t)=\left (t,t^2,\cdots,t^{n-1},\phi(t)\right).$$ In this paper, we show that $$\left(\int_I \big|\hat f(\gamma(t))\big|^q \big|\phi^{(n)}(t)\big|^{\frac{2}{n(n+1)}}…
Motivated by applications in number theory, analysis, and fractal geometry, we consider regularity properties and dimensions of graphs associated with Fourier series of the form $F(t)=\sum_{n=1}^\infty f(n)e^{2\pi i nt}/n$, for a large…
In this paper, we consider a method for fast numerical computation of the Fourier transform of a slowly decaying function with given accuracy in given ranges of the frequency. In these decades, some useful formulas for the Fourier transform…
We study decoupling theory for functions on $\mathbb{R}$ with Fourier transform supported in a neighborhood of short Dirichlet sequences $\{\log n\}_{n=N+1}^{N+N^{1/2}}$, as well as sequences with similar convexity properties. We utilize…
This article is devoted to derivation of the Laplace transforms of the derivatives with respect to parameters of certain special functions, namely, the Mittag-Leffler type, Wright and Le Roy type functions. These formulas show…
Given a definable function $f: S \to \mathbb{R}$ on a definable set $S$, we study sublevel sets of the form $S^f_t \coloneqq \{x \in S: f(x) \leq t\}$ for all $t \in \mathbb{R}$. Using o-minimal structures, we prove that the Euler…
We study a general class of multiplicative functions by relating "short averages" to its "long average". More precisely, we estimate asymptotically the variance of such a class of functions in short intervals using Fourier analysis and…
While spike trains are obviously not band-limited, the theory of super-resolution tells us that perfect recovery of unknown spike locations and weights from low-pass Fourier transform measurements is possible provided that the minimum…
We study the problem of recovering an unknown compactly-supported multivariate function from samples of its Fourier transform that are acquired nonuniformly, i.e. not necessarily on a uniform Cartesian grid. Reconstruction problems of this…
Building on the univariate techniques developed by Ray and Schmidt-Hieber, we study the class $\mathcal{F}^s(\mathbb{R}^n)$ of multivariate nonnegative smooth functions that are sufficiently flat near their zeroes, which guarantees that…
In a recent short note the first author gave the first positive result on the higher order regularity of the discrete noncentered Hardy-Littlewood maximal function. In this article we conduct a thorough investigation of possible similar…
We consider sequences of random variables of the type $S_n= n^{-1/2} \sum_{k=1}^n \{f(X_k)-\E[f(X_k)]\}$, $n\geq 1$, where $X=(X_k)_{k\in \Z}$ is a $d$-dimensional Gaussian process and $f: \R^d \rightarrow \R$ is a measurable function. It…
We define a family of quaternary sequences over the residue class ring modulo $4$ of length $pq$, a product of two distinct odd primes, using the generalized cyclotomic classes modulo $pq$ and calculate the discrete Fourier transform (DFT)…
The paper is devoted to the study of the short rate equation of the form $$ dR(t)=F(R(t)) dt +\sum_{i=1}^{d}G(R(t-))dZ_i(t)$$ with deterministic functions $F,G_1,...,G_d$ and a multivariate L\'evy process $Z=(Z_1,...,Z_d)$ with possibly…
A new class of fractional-order stochastic evolution equations of the form $(\partial_t + A)^\gamma X(t) = \dot{W}^Q(t)$, $t\in[0,T]$, $\gamma \in (0,\infty)$, is introduced, where $-A$ generates a $C_0$-semigroup on a separable Hilbert…
The use of Fourier methods in wave-front reconstruction can significantly reduce the computation time for large telescopes with a high number of degrees of freedom. However, Fourier algorithms for discrete data require a rectangular data…
It has been shown that Paley-Wiener functions may be recovered from their values on a complete interpolating sequence. This paper explores the same phenomenon, and gives a sufficient condition on a function $\phi(x)$, called an…
Let $F$ be a local non archimedian field of characteristic $0$, and $G$ a non-connected reductive group over $F$. We denote $G^0$ the connected component of the identity and assume the quotient $G/G^0$ is abelian. For $f$ a locally constant…
The sharp Wolff-type decoupling estimates of Bourgain--Demeter are extended to the variable coefficient setting. These results are applied to obtain new sharp local smoothing estimates for wave equations on compact Riemannian manifolds,…