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We discuss the analytic continuation of the Hadamard product of two holomorphic functions under assumptions pertaining to Ecalles Resurgence Theory, proving that if both factors are endlessly continuable with prescribed sets of singular points $A$ and $B$, then so is their Hadamard product with respect to the set ${0}cup A cdot B$. In this generalization of the classical Hadamard Theorem, all the branches of the multivalued analytic continuation of the Hadamard product are considered.
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We investigate polynomials, called m-polynomials, whose generator polynomial has coefficients that can be arranged as a matrix, where q is a positive integer greater than one. Orthogonality relations are established and coefficients are obtained for the expansion of a polynomial in terms of m-polynomials. We conclude this article by an implementation in MATHEMATICA of m-polynomials and the results obtained for them.
Entrywise powers of matrices have been well-studied in the literature, and have recently received renewed attention in the regularization of high-dimensional correlation matrices. In this paper, we study powers of positive semidefinite block matrices $(H_{st})_{s,t=1}^n$ with complex entries. We first characterize the powers $alphainmathbb{R}$ such that the blockwise power map $(H_{st}) mapsto (H_{st}^alpha)$ preserves Loewner positivity. The characterization is obtained by exploiting connections with the theory of matrix monotone functions developed by Loewner. Second, we revisit previous work by Choudhury [Proc. AMS 108] who had provided a lower bound on $alpha$ for preserving positivity when the blocks $H_{st}$ pairwise commute. We completely settle this problem by characterizing the full set of powers preserving positivity in this setting. Our characterizations generalize previous work by FitzGerald-Horn, Bhatia-Elsner, and Hiai from scalars to arbitrary block size, and in particular, generalize the Schur Product Theorem. Finally, a natural and unifying framework for studying the case of diagonalizable blocks consists of replacing real powers by general characters of the complex plane. We thus classify such characters, and generalize our results to this more general setting. In the course of our work, given $betainmathbb{Z}$, we provide lower and upper bounds for the threshold power $alpha >0$ above which the complex characters $re^{itheta}mapsto r^alpha e^{ibetatheta}$ preserve positivity when applied entrywise to positive semidefinite matrices. In particular, we completely resolve the $n=3$ case of a question raised in 2001 by Xingzhi Zhan. As an application, we extend previous work by de Pillis [Duke Math. J. 36] by classifying the characters $K$ of the complex plane for which the map $(H_{st})_{s,t=1}^n mapsto (K({rm tr}(H_{st})))_{s,t=1}^n$ preserves positivity.
Let $C$ be an arithmetic circuit of $poly(n)$ size given as input that computes a polynomial $finmathbb{F}[X]$, where $X={x_1,x_2,ldots,x_n}$ and $mathbb{F}$ is any field where the field arithmetic can be performed efficiently. We obtain new algorithms for the following two problems first studied by Koutis and Williams cite{Kou08, Wi09, KW16}. k-MLC: Compute the sum of the coefficients of all degree-$k$ multilinear monomials in the polynomial $f$. k-MMD: Test if there is a nonzero degree-$k$ multilinear monomial in the polynomial $f$. Our algorithms are based on the fact that the Hadamard product $fcirc S_{n,k}$, is the degree-$k$ multilinear part of $f$, where $S_{n,k}$ is the $k^{th}$ elementary symmetric polynomial. 1. For k-MLC problem, we give a deterministic algorithm of run time $O^*(n^{k/2+clog k})$ (where $c$ is a constant), answering an open question of Koutis and Williams cite[ICALP09]{KW16}. As corollaries, we show $O^*(binom{n}{downarrow k/2})$-time exact counting algorithms for several combinatorial problems: k-Tree, t-Dominating Set, m-Dimensional k-Matching. 2. For k-MMD problem, we give a randomized algorithm of run time $4.32^kcdot poly(n,k)$. Our algorithm uses only $poly(n,k)$ space. This matches the run time of a recent algorithm cite{BDH18} for $k-MMD$ which requires exponential (in $k$) space. Other results include fast deterministic algorithms for k-MLC and k-MMD problems for depth three circuits.
Using complex methods combined with Baires Theorem we show that one-sided extendability, extendability and real analyticity are rare phenomena on various spaces of functions in the topological sense. These considerations led us to introduce the p-continuous analytic capacity and variants of it, $p in { 0, 1, 2, cdots } cup { infty }$, for compact or closed sets in $mathbb{C}$. We use these capacities in order to characterize the removability of singularities of functions in the spaces $A^p$.
We investigate the frame set of regular multivariate Gaussian Gabor frames using methods from Kahler geometry such as Hormanders $dbar$-method, the Ohsawa--Takegoshi extension theorem and a Kahler-variant of the symplectic embedding theorem of McDuff-Polterovich for ellipsoids. Our approach is based on the well-known link between sets of interpolation for the Bargmann-Fock space and the frame set of multivariate Gaussian Gabor frames. We state sufficient conditions in terms of a certain extremal type Seshadri constant of the complex torus associated to a lattice to be a set of interpolation for the Bargmann-Fock space, and give also a condition in terms of the generalized Buser-Sarnak invariant of the lattice. Our results on Gaussian Gabor frames are in terms of the Sehsadri constant and the generalized Buser-Sarnak invariant of the associated symplectic dual lattice. The theory of Hormander estimates and the Ohsawa--Takegoshi extension theorem allow us to give estimates for the frame bounds in terms of the Buser-Sarnack invariant and in the one-dimensional case these bounds are sharp thanks to Faltings work on Green functions in Arakelov theory.
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