Related papers: On optimality of mollifiers

In this note we find optimal one-sided majorants of exponential type for the signum function subject to certain monotonicity conditions. As an application, we use these special functions to obtain a simple Fourier analysis proof of the…

We study the $2k$-th moment of central values of the family of Dirichlet $L$-functions to a fixed prime modulus. We establish sharp lower bounds for all real $k \geq 0$ and sharp upper bounds for $k$ in the range $0 \leq k \leq 1$.

We give a new framework for proving the existence of low-degree, polynomial approximators for Boolean functions with respect to broad classes of non-product distributions. Our proofs use techniques related to the classical moment problem…

Molecular dipole moment in liquid water is an intriguing property, partly due to the fact that there is no unique way to partition the total electron density into individual molecular contributions. The prevailing method to circumvent this…

We present an alternative formalism for calculating the maximally localized Wannier functions in crystalline solids, obtaining an expression which is extremely simple and general. In particular, our scheme is exactly invariant under…

Localized Wannier functions provide an efficient and intuitive means by which to compute dielectric properties from first principles. They are most commonly constructed in a post-processing step, following total-energy minimization.…

We extend the work of Fouvry, Kowalski and Michel on correlation between Hecke eigenvalues of modular forms and algebraic trace functions in order to establish an asymptotic formula for a generalized cubic moment of modular L-functions at…

We introduce mollified Christoffel-Darboux (CD) kernels on varieties, a systematic regularization of the classical CD kernel associated with a probability measure on a compact domain. The main motivations are twofold: first, to sharpen the…

We compute the second moment in the family of quadratic Dirichlet $L$-functions with prime conductors over $\mathbb{F}_q[x]$ when the degree of the discriminant goes to infinity, obtaining one of the lower order terms. We also obtain an…

Maximally-localized Wannier functions (MLWFs) are a powerful and broadly used tool to characterize the electronic structure of materials, from chemical bonding to dielectric response to topological properties. Most generally, one can…

In this paper, we will establish the best constants for certain classes of weighted Moser-Trudinger inequalities on the entire Euclidean spaces $\mathbb{R}^N$. We will also prove the existence of maximizers of these sharp weighted…

This survey synthesizes the current state of the art on the regularity theory for solutions to the optimal partition problem. Namely, we consider non-negative, vector-valued Sobolev functions whose components have mutually disjoint support,…

The approximation properties of the finite element method can often be substantially improved by choosing smooth high-order basis functions. It is extremely difficult to devise such basis functions for partitions consisting of arbitrarily…

Maximally localized Wannier functions (MLWFs) are widely used to construct first-principles tight-binding models that accurately reproduce the electronic structure of materials. Recently, robust and automated approaches to generate these…

The main objective of this article is to compute a first moment for product of Dirichlet and twisted self-dual $GL(3)$ $L$-functions. We discuss the possible simultaneous non vanishing at the central point. We use properties of symmetric…

We study the $2k$-th moment of central values of the family of Dirichlet $L$-functions to a fixed prime modulus and establish sharp upper bounds for all real $k \in [0,2]$.

We study the second moment of the L-function associated to a holomorphic primitive cusp form of even weight perturbed by a new family of mollifiers. This family is a natural extension of the mollifers considered by Conrey and by Bui, Conrey…

We study Dirichlet-type spaces $\mathfrak{D}_{\alpha}$ of analytic functions in the unit bidisk and their cyclic elements. These are the functions $f$ for which there exists a sequence $(p_n)_{n=1}^{\infty}$ of polynomials in two variables…

The purpose of this work is to study an optimal control problem for a semilinear elliptic partial differential equation with a linear combination of Dirac measures as a forcing term; the control variable corresponds to the amplitude of such…

We prove asymptotics for mollified first and second moments of subfamilies of Dirichlet $L$-functions given by shrinking angular restrictions on the root number. Using these moments, we prove that for even primitive characters with prime…