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Fourier transform of self-affine measuresMar 22 2019Suppose $F$ is a self-affine set on $\mathbb{R}^d$, $d\geq 2$, which is not a singleton, associated to affine contractions $f_j = A_j + b_j$, $A_j \in \mathrm{GL}(d,\mathbb{R})$, $b_j \in \mathbb{R}^d$, $j \in \mathcal{A}$, for some finite $\mathcal{A}$. ... More

Dirichlet-to-Neumann maps on TreesMar 22 2019In this paper we study the Dirichlet-to-Neumann map for solutions to mean value formulas on trees. We give two alternative definition of the Dirichlet-to-Neumann map. For the first definition (that involves the product of a "gradient" with a "normal vector") ... More

Bernoulli and Euler numbers from divergent seriesMar 21 2019The aim of this note is to provide a simple proof of some well-known identities and recurrences relating classical Bernoulli and Euler numbers by using the Abel sum of the divergent series $\sum_{n=0}^\infty (-1)^{n} (n+1)^k$, $k$ a positive integers. ... More

Wavelets on compact abelian groupsMar 21 2019Multiresolution analysis (MRA) on compact abelian group $G$ has been constructed with epimorphism as a dilation operator. We show a characterization of scaling sequences of an MRA on $L^p(G)$, $1\le p<\infty$. With the help of this scaling sequence we ... More

Roots of trigonometric polynomials and the Erdős-Turán theoremMar 21 2019We prove, informally put, that it is not a coincidence that $\cos{(n \theta)} + 1 \geq 0$ and that the roots of $z^n + 1 =0$ are uniformly distributed in angle -- a version of the statement holds for all trigonometric polynomials with `few' real roots. ... More

Existence and uniqueness of positive periodic solutions for nonlinear fractional mixed problemsMar 21 2019This paper is devoted to study the existence and uniqueness of solutions of a one parameter family of nonlinear fractional differential equation with mixed boundary value conditions. Riemann-Liouville fractional derivative is considered. An exhaustive ... More

Condensers with infinitely many touching Borel plates and minimum energy problemsMar 21 2019Defining a condenser in a locally compact space as a locally finite, countable collection of Borel sets $A_i$, $i\in I$, with the sign $s_i=\pm1$ prescribed such that $A_i\cap A_j=\varnothing$ whenever $s_is_j=-1$, we consider a minimum energy problem ... More

Implementing zonal harmonics with the Fueter principleMar 21 2019By exploiting the Fueter theorem, we give new formulas to compute zonal harmonic functions in any dimension. We first give a representation of them as a result of a suitable ladder operator acting on the constant function equal to one. Then, inspired ... More

A certain vector-valued function space and its applications to multilinear operatorsMar 21 2019In this paper we present several (quasi-)norm equivalences involving $L^p(l^q)$ norm of a certain vector-valued functions and extend the equivalences to $p=\infty$ and $0<q<\infty$ in the scale of Triebel-Lizorkin spaces, motivated by Fraizer, Jawerth. ... More

Differential equation and recurrence relations of the Sheffer-Appell polynomial sequence: A matrix approachMar 21 2019Motivated by the effective impact of the Pascal functional and the Wronskian matrices, we investigate several identities and differential equation for the Sheffer-Appell polynomial sequence by using matrix algebra. The matrix approach, which we have used ... More

Three Convolution Inequalities on the Real Line with Connections to Additive CombinatoricsMar 20 2019We discuss three convolution inequalities that are connected to additive combinatorics. Cloninger and the second author showed that for nonnegative $f \in L^1(-1/4, 1/4)$, $$ \max_{-1/2 \leq t \leq 1/2} \int_{\mathbb{R}}{f(t-x) f(x) dx} \geq 1.28 \left( ... More

Box-constrained monotone $L_\infty$-approximations and Lipschitz-continuous regularized functionsMar 20 2019Let $f:[0,1]\to[0,1]$ be a nondecreasing function. The main goal of this work is to provide a regularized version, say $\tilde f_L$, of $f$. Our choice will be a best $L_\infty$-approximation to $f$ in the set of functions $h:[0,1]\to[0,1]$ which are ... More

On the asymptotics of the rescaled Appell polynomialsMar 20 2019We introduce a new representation for the rescaled Appell polynomials and discuss how their asymptotics can be derived from it. This representation generically consists of a finite sum and an integral over a universal contour (i.e. independent of the ... More

On integrals involving quotients of hyperbolic functionsMar 20 2019We evaluate some integrals over $[0,\infty)$ of quotients of powers of the hyperbolic functions $\sinh x$ and $\cosh x$ using a hypergeometric approach. Some of these results appear to be new but several verify the entries in the table of integrals of ... More

Littlewood-Paley Characterization for Musielak-Orlicz-Hardy Spaces Associated with OperatorsMar 20 2019Let $X$ be a space of homogeneous type. Assume that $L$ is an non-negative second-order self-adjoint operator on $L^2\left(X\right)$ with (heart) kernel associated to the semigroup $e^{ - tL}$ that satisfies the Gaussian upper bound. In this paper, the ... More

Inequalities related to some types of entropies and divergencesMar 20 2019The aim of this paper is to discuss new results concerning some kinds of parametric extended entropies and divergences. As a sereis of our studies for mathematical properties on entropy and divergence, we give new bounds for Tsallis quasilinear entropy ... More

Generalized-hypergeometric solutions of the biconfluent Heun equationMar 19 2019We examine the power-series solutions and the series solutions in terms of the Hermite functions for the biconfluent Heun equation. The conditions under which the solutions reduce to finite sums as well as the cases when the coefficients of power-series ... More

Maximal estimates for the bilinear spherical averages and the bilinear Bochner-Riesz operatorsMar 19 2019We study the maximal estimates for the bilinear spherical average and the bilinear Bochner-Riesz operator. Firstly, we obtain $L^p\times L^q \to L^r$ estimates for the bilinear spherical maximal function on the sharp range. Thus, except some of the endpoint ... More

On a class of linear functional equations without range conditionMar 19 2019The main purpose of this work is to provide the general solutions of a class of linear functional equations. Let $n\geq 2$ be an arbitrarily fixed integer, let further $X$ and $Y$ be linear spaces over the field $\mathbb{K}$ and let $\alpha_{i}, \beta_{i}\in ... More

Sum-Product Type Estimates over Finite FieldsMar 19 2019Let $\mathbb{F}_q$ denote the finite field with $q$ elements where $q=p^l$ is a prime power. Using Fourier analytic tools with a third moment method, we obtain sum-product type estimates for subsets of $\mathbb{F}_q$. In particular, we prove that if $A\subset ... More

Parabolic arcs for time-dependent perturbations of the Kepler problemMar 19 2019We prove the existence of parabolic arcs with prescribed asymptotic direction for the equation \begin{equation*} \ddot{x} = - \dfrac{x}{\lvert x \rvert^{3}} + \nabla W(t,x), \qquad x \in \mathbb{R}^{d}, \end{equation*} where $d \geq 2$ and $W$ is a (possibly ... More

Almost sure Assouad-like Dimensions of Complementary setsMar 19 2019Given a non-negative, decreasing sequence $a$ with sum $1$, we consider all the closed subsets of $[0,1]$ such that the lengths of their complementary open intervals are given by the terms of $a$, the so-called complementary sets. In this paper we determine ... More

Existence of solutions for a class of multivalued functional integral equations of Volterra type via the measure of nonequicontinuity on the Fréchet space ${\bf C(Ω,E)}$Mar 18 2019The existence of continuous not necessarily bounded solutions of nonlinear functional Volterra integral inclusions in infinite dimensional setting is shown with the aid of the measure of nonequicontinuity. New abstract topological fixed point results ... More

The Maslov and Morse Indices for Sturm-Liouville Systems on the Half-LineMar 18 2019We show that for Sturm-Liouville Systems on the half-line $[0,\infty)$, the Morse index can be expressed in terms of the Maslov index and an additional term associated with the boundary conditions at $x = 0$. Relations are given both for the case in which ... More

A family of entire functions connecting the Bessel function $J_1$ and the Lambert $W$ functionMar 18 2019Motivated by the problem of determining the values of $\alpha>0$ for which $f_\alpha(x)=e^\alpha - (1+1/x)^{\alpha x},\ x>0$ is a completely monotonic function, we combine Fourier analysis with complex analysis to find a family $\varphi_\alpha$, $\alpha>0$, ... More

Quantum walks: Schur functions meet symmetry protected topological phasesMar 18 2019This paper uncovers and exploits a link between a central object in harmonic analysis, the so-called Schur functions, and the very hot topic of symmetry protected topological phases of quantum matter. This connection is found in the setting of quantum ... More

Applications of generalized trigonometric functions with two parametersMar 18 2019Mar 19 2019Generalized trigonometric functions (GTFs) are simple generalization of the classical trigonometric functions. GTFs are deeply related to the $p$-Laplacian, which is known as a typical nonlinear differential operator, and there are a lot of works on GTFs ... More

New bounds of Weyl sumsMar 18 2019We augment the method of Wooley (2015) by some new ideas and among a series of other bounds, we show that for a set of $(u_2, \ldots, u_{d-1})\in [0,1)^{d-2}$, $d\ge 3$, of full Lebesgue measure, one has $$ \sup_{(u_1, u_d)\in [0,1)^2}\left | \sum_{n=1}^N ... More

Intermediate Assouad-like dimensionsMar 17 2019We introduce and study bi-Lipschitz-invariant dimensions that range between the box and Assouad dimensions. The quasi-Assouad dimensions and $\theta$-spectrum are other special examples of these intermediate dimensions. These dimensions are localized, ... More

Mountain pass solutions to Euler-Lagrange equations with general anisotropic operatorMar 17 2019Using the Mountain Pass Theorem we show that the problem \begin{equation*} \begin{cases} \frac{d}{dt}\mathcal{L}_v(t,u(t),\dot u(t))=\mathcal{L}_x(t,u(t),\dot u(t))\quad \text{ for a.e. }t\in[a,b]\\ u(a)=u(b)=0 \end{cases} \end{equation*} has a solution ... More

An Alternative Proof of Steinhaus TheoremMar 17 2019In measure theory, Steinhaus theorem is a result that deals with a property of the difference between two sets of positive measure. We give a simple elementary proof of the result.

Transfer operators, atomic decomposition and the BestiaryMar 16 2019Arbieto and S. recently used atomic decomposition to study transfer operators. We give a long list of old and new expanding dynamical systems for which those results can be applied, obtaining the quasi-compactness of transfer operator acting on Besov ... More

Transfer operators and atomic decompositionMar 16 2019We use the method of atomic decomposition and a new family of Banach spaces to study the action of transfer operators associated to piecewise-defined maps. It turns out that these transfer operators are quasi-compact even when the associated potential, ... More

Classic and exotic Besov spaces induced by good gridsMar 16 2019In a previous work we introduced Besov spaces $\mathcal{B}^s_{p,q}$ defined on a measure spaces with a good grid, with $p\in [1,\infty)$, $q\in [1,\infty]$ and $0< s< 1/p$. Here we show that classical Besov spaces on compact homogeneous spaces are examples ... More

Besov-ish spaces through atomic decompositionMar 16 2019We use the method of atomic decomposition to build new families of function spaces, similar to Besov spaces, in measure spaces with grids, a very mild assumption. Besov spaces with low regularity are considered in measure spaces with good grids, and results ... More

Unimodular multipliers on $α$-modulation spaces: A revisit with new method under weaker conditionsMar 16 2019By a new method derived from Nicola--Primo--Tabacco[24], we study the boundedness on $\alpha$-modulation spaces of unimodular multipliers with symbol $e^{i\mu(\xi)}$. Comparing with the previous results, the boundedness result is established for a larger ... More

An Extremal Property of the Hexagonal LatticeMar 16 2019We describe an extremal property of the hexagonal lattice $\Lambda \subset \mathbb{R}^2$. Let $p$ denote the circumcenter of its fundamental triangle (a so-called deep hole) and let $A_r$ denote the set of lattice points that are at distance $r$ from ... More

A Universality Law For Sign Correlations of Eigenfunctions of Differential OperatorsMar 15 2019We establish a universality law for sequences of functions $\{w_n\}_{n \in \mathbb{N}}$ satisfying a form of WKB approximation on compact intervals. This includes eigenfunctions of generic Schr\"odinger operators, as well as Laguerre and Chebyshev polynomials. ... More

Real Zeros of Random Sums with I.I.D. CoefficientsMar 15 2019Let $\{f_k\}$ be a sequence of entire functions that are real valued on the real-line. We study the expected number of real zeros of random sums of the form $P_n(z)=\sum_{k=0}^n\eta_k f_k(z)$, where $\{\eta_k\}$ are real valued i.i.d. random variables. ... More

Weighted estimates for maximal functions associated to skeletonsMar 14 2019We provide quantitative weighted estimates for the $L^p(w)$ norm of a maximal operator associated to cube skeletons in $\mathbb{R}^n$. The method of proof differs from the usual in the area of weighted inequalities since there are no covering arguments ... More

De Branges canonical systems with finite logarithmic integralMar 13 2019Krein-de Branges spectral theory establishes a correspondence between the class of differential operators called canonical Hamiltonian systems and measures on the real line with finite Poisson integral. We further develop this area by giving a description ... More

Optimal arithmetic structure in exponential Riesz sequencesMar 13 2019We consider exponential systems $E\left(\Lambda\right)=\left\{ e^{i\lambda t}\right\} _{\lambda\in\Lambda}$ for $\Lambda\subset\mathbb{Z}$. It has been shown by Londner and Olevskii in [9] that there exists a subset of the circle, of positive Lebesgue ... More

Local uncertainty principles for the two-sided Gabor quaternion Fourier transformMar 13 2019By the important applications of Gabor transform in time-frequency analysis and signal analysis, in this paper, we consider the Gabor quaternion Fourier transform (GQFT), and we prove of it a version Benedicks-type uncertainty principle for GQFT and some ... More

Sampling expansions associated with quaternion difference equationsMar 13 2019Starting with a quaternion difference equation with boundary conditions, a parameterized sequence which is complete in finite dimensional quaternion Hilbert space is deduced. By employing the parameterized sequence as the kernel of discrete transform, ... More

Inverse problem of the spectral analysis for the Sturm-Liouville operator with non-separated boundary conditions and spectral parameter in the boundary conditionMar 13 2019This work deals with an inverse problem for the Sturm-Liouville operator with non-separated boundary conditions, one of which linearly depends on a spectral parameter. Uniqueness theorem is proved, solution algorithm is constructed and sufficient conditions ... More

Geometric properties of a certain class of functions related to the Fox-Wright functionsMar 12 2019The purpose of this paper is to provide a set of sufficient conditions so that the normalized form of the Fox-Wright functions have certain geometric properties like close-to-convexity, univalency, convexity and starlikeness inside the unit disc. In particular, ... More

Norm-variation of cubic ergodic averagesMar 11 2019In this paper we study cubic averages with respect to $d$ general commuting transformations and prove quantitative results on their convergence in the norm. The approach we are using is based on estimates for certain entangled multilinear singular integral ... More

Doubling coverings via resolution of singularities and preparationMar 11 2019In this paper we provide asymptotic upper bounds on the complexity in two (closely related) situations. We confirm for the total doubling coverings and not only for the chains the expected bounds of the form $$ \kappa({\mathcal U}) \le K_1(\log ({1}/{\delta}))^{K_2} ... More

Restriction estimates to complex hypersurfacesMar 11 2019Mar 12 2019The restriction problem is better understood for hypersurfaces and recent progresses have been made by bilinear and multilinear approaches and most recently polynomial partitioning method which is combined with those estimates. However, for surfaces with ... More

Restriction estimates to complex hyper-surfacesMar 11 2019The restriction problem is better understood for hyper-surfaces and recent progresses have been made by bilinear and multilinear approaches and most recently polynomial partitioning method which is combined with those estimates. However, for surfaces ... More

Extension theorems for Hamming varieties over finite fieldsMar 10 2019We study the finite field extension estimates for Hamming varieties $H_j, j\in \mathbb F_q^*,$ defined by $H_j=\{x\in \mathbb F_q^d: \prod_{k=1}^d x_k=j\},$ where $\mathbb F_q^d$ denotes the $d$-dimensional vector space over a finite field $\mathbb F_q$ ... More

Multivariate Bell Polynomials and Derivatives of Composed FunctionsMar 10 2019How do we take repeated derivatives of composed multivariate functions? for one-dimensional functions, the common tools consist of the Fa\'a di Bruno formula with Bell polynomials; while there are extensions of the Fa\'a di Bruno formula, there are no ... More

Sharp Bounds for the Arc Lemniscate Sine FunctionMar 10 2019The arc lemniscate sine function is given by $$ \mbox{arcsl}(x)=\int_0^x \frac{1}{\sqrt{1-t^4}}dt. $$ In 2017, Mahmoud and Agarwal presented bounds for $\mbox{arcsl}$ in terms of the Lerch zeta function $$ \Phi(z,s,a)=\sum_{k=0}^\infty \frac {z^k}{(k+a)^s}. ... More

Recurrent Trajectories and Finite Critical Trajectories of Quadratic Differentials on the Riemann SphereMar 08 2019In this paper, the focus will be on both the existence and non-existence respectively of finite critical trajectories and recurrent trajectories of a quadratic differential on the Riemann sphere. We show the connection between these two items. More precisely, ... More

New proofs for several $q$-extensions of Dougall's $_2H_2$-series identityMar 08 2019In terms of the analytic continuation arguement, we give new proofs for several $q$-extensions of Dougall's $_2H_2$-series identity.

A sharp Lorentz-invariant Strichartz norm expansion for the cubic wave equation in $\mathbb{R}^{1+3}$Mar 07 2019We provide an asymptotic formula for the maximal Strichartz norm of small solutions to the cubic wave equation in Minkowski space. The leading coefficient is given by Foschi's sharp constant for the linear Strichartz estimate. We calculate the constant ... More

Finite trees inside thin subsets of ${\Bbb R}^d$Mar 06 2019Bennett, Iosevich and Taylor proved that compact subsets of ${\Bbb R}^d$, $d \ge 2$, of Hausdorff dimensions greater than $\frac{d+1}{2}$ contain chains of arbitrary length with gaps in a non-trivial interval. In this paper we generalize this result to ... More

A weighted estimate for generalized harmonic extensionsMar 06 2019We prove some weighted $L_p$ estimates for generalized harmonic extensions in the half-space.

Generalized Fourier series by double trigonometric systemMar 06 2019Necessary and sufficient conditions are obtained on the function $M$ such that $\{ M(x,y) e^{i kx}e^{i my}: (k,m)\in \Omega \}$ is complete and minimal in $L^{p}(\mathbb{T}^{2})$ when $\Omega^{c}=\{(0,0)\}$ and $\Omega^{c} = 0\times\mathbb{Z}$. If $\Omega^{c} ... More

An introduction to fractal uncertainty principleMar 06 2019This article provides a broad review of recent developments on the fractal uncertainty principle and their applications to quantum chaos.

A comparison of box and Carleson conditions on bi-treesMar 06 2019In this note we give an example of measure satisfying the box condition on certain sub-bi-trees (see below) but not satisfying Carleson condition on those sub-bi-trees. This can be considered as a certain counterexample for two weight bi-parameter embedding ... More

A comparison of box and Carleson conditions on bi-treesMar 06 2019Mar 18 2019In this note we give an example of measure satisfying the box condition on certain sub-bi-trees (see below) but not satisfying Carleson condition on those sub-bi-trees. This can be considered as a certain counterexample for two weight bi-parameter embedding ... More

A comparison of box and Carleson conditions on bi-treesMar 06 2019Mar 07 2019In this note we give an example of measure satisfying the box condition on certain sub-bi-trees (see below) but not satisfying Carleson condition on those sub-bi-trees. This can be considered as a certain counterexample for two weight bi-parameter embedding ... More

Heun's differential equation and its q-deformationMar 06 2019The $q$-Heun equation is a $q$-difference analogue of Heun's differential equation. We review several solutions of Heun's differential equation and investigate polynomial-type solutions of $q$-Heun equation. The limit $q\to 1$ corresponding to Heun's ... More

Pointwise convergence along restricted directions for the fractional Schrödinger equationMar 06 2019We consider the pointwise convergence problem for the solution of Schr\"odinger-type equations along directions determined by a given compact subset of the real line. This problem contains Carleson's problem as the most simple case and was studied in ... More

Solution and asymptotic analysis of a boundary value problem in the spring-mass model of runningMar 06 2019We consider the classic spring-mass model of running which is built upon an inverted elastic pendulum. In a natural way, there arises an interesting boundary value problem for the governing system of two nonlinear ordinary differential equations. It requires ... More

Carleson measure spaces with variable exponents and their applicationsMar 06 2019In this paper, we introduce the Carleson measure spaces with variable exponents $CMO^{p(\cdot)}$. By using discrete Littlewood$-$Paley$-$Stein analysis as well as Frazier and Jawerth's $\varphi-$transform in the variable exponent settings, we show that ... More

Multilinear fractional type operators and their commutators on Hardy spaces with variable exponentsMar 06 2019In this article, we show that multilinear fractional type operators are bounded from product Hardy spaces with variable exponents into Lebesgue or Hardy spaces with variable exponents via the atomic decomposition theory. We also study continuity properties ... More

Multilinear fractional type operators and their commutators on Hardy spaces with variable exponentsMar 06 2019Mar 11 2019In this article, we show that multilinear fractional type operators are bounded from product Hardy spaces with variable exponents into Lebesgue or Hardy spaces with variable exponents via the atomic decomposition theory. We also study continuity properties ... More

On a uniqueness theorem of E. B. VulMar 05 2019We recall a uniqueness theorem of E. B. Vul pertaining to a version of the cosine transform originating in spectral theory. Then we point out an application to the Bernstein approximation problem with non-symmetric weights: a theorem of Volberg is proved ... More

A Comment on the Sums $\sum_{n \in \mathbb{Z}} \frac{(-1)^{nk}}{(an+1)^k}$Mar 05 2019We recall a proof of Euler's identity $\sum_{n=1}^{\infty} \frac{1}{n^2}=\frac{\pi^2}{6}$ involving the evaluation of a double integral. We extend the method to find Hurwitz Zeta series of the form $S(k,a)=\sum_{n \in \mathbb{Z}} \frac{(-1)^{nk}}{(an+1)^k},$ ... More

Classifying Four-Body Convex Central ConfigurationsMar 05 2019We classify the full set of convex central configurations in the Newtonian four-body problem. Particular attention is given to configurations possessing some type of symmetry or defining geometric property. Special cases considered include kite, trapezoidal, ... More

Multilinear fractional Calderón-Zygmund operators on weighted Hardy spacesMar 04 2019We prove norm estimates for multilinear fractional integrals acting on weighted and variable Hardy spaces. In the weighted case we develop ideas we used for multilinear singular integrals [7]. For the variable exponent case, a key element of our proof ... More

Limited polynomialsMar 04 2019In this paper we study a particular class of polynomials. We study the distribution of their zeros, including the zeros of their derivatives as well as the interaction between this two. We prove a weak variant of the sendov conjecture in the case the ... More

Supercritical Regime for the Kissing PolynomialsMar 03 2019We study a family of polynomials which are orthogonal with respect to the varying, highly oscillatory complex weight function $e^{ni\lambda z}$ on $[-1,1]$, where $\lambda$ is a positive parameter. This family of polynomials has appeared in the literature ... More

Criterion for robust global asymptotic stability of the linear time-varying systemsMar 03 2019In this paper we prove a new criterion for robust global asymptotic stability of the zero solution of LTV system $\dot x=A(t)x.$ To prove the result, a logarithmic norm of the system matrix $A(t)$ will be used under which the stability becomes a topological ... More

Exact cubature rules for symmetric functionsMar 03 2019We employ a multivariate extension of the Gauss quadrature formula, originally due to Berens, Schmid and Xu [BSX95], so as to derive cubature rules for the integration of symmetric functions over hypercubes (or infinite limiting degenerations thereof) ... More

Solutions of convex Bethe Ansatz equations and the zeros of (basic) hypergeometric orthogonal polynomialsMar 03 2019Via the solutions of systems of algebraic equations of Bethe Ansatz type, we arrive at bounds for the zeros of orthogonal (basic) hypergeometric polynomials belonging to the Askey-Wilson, Wilson and continuous Hahn families.

On fractional calculus with general analytic kernelsMar 01 2019Many possible definitions have been proposed for fractional derivatives and integrals, starting from the classical Riemann-Liouville formula and its generalisations and modifying it by replacing the power function kernel with other kernel functions. We ... More

Smoothness of functions vs. smoothness of approximation processesMar 01 2019We provide a comprehensive study of interrelations between different measures of smoothness of functions on various domains and smoothness properties of approximation processes. Two general approaches to this problem have been developed: the first based ... More

Discrete Uncertainty Principle in Quaternion Setting and Application in Signal ReconstructionMar 01 2019In this paper, the uncertainty principle of discrete signals associated with Quaternion Fourier transform is investigated. It suggests how sparsity helps in the recovery of missing frequency.

On a property of random walk polynomials involving Christoffel functionsFeb 28 2019Discrete-time birth-death processes may or may not have certain properties known as asymptotic aperiodicity and the strong ratio limit property. In all cases known to us a suitably normalized process having one property also possesses the other, suggesting ... More

Hypergeometric identities arising from the elephant random walkFeb 28 2019A probabilistic approach is provided to establish new hypergeometric identities. It is based on the calculation of moments of the limiting distribution of the position of the elephant random walk in the superdiffusive regime.

A characterization of the uniform strong type $(1,1)$ bounds for averaging operatorsFeb 28 2019We prove that in a metric measure space $(X, d, \mu)$, the averaging operators $A_{r, \mu }$ satisfy a uniform strong type $(1,1)$ bound $\sup_{r, \mu} \|A_{r, \mu }\|_{L^1\to L^1} < \infty$ if and only if $X$ satisfies a certain geometric condition, ... More

Quadrature rules from finite orthogonality relations for Bernstein-Szego polynomialsFeb 28 2019We glue two families of Bernstein-Szego polynomials to construct the eigenbasis of an associated finite-dimensional Jacobi matrix. This gives rise to finite orthogonality relations for this composite eigenbasis of Bernstein-Szego polynomials. As an application, ... More

Singular stochastic integral operatorsFeb 27 2019In this paper we introduce Calder\'on-Zygmund theory for singular stochastic integrals with operator-valued kernel. In particular, we prove $L^p$-extrapolation results under a H\"ormander condition on the kernel. Sparse domination and sharp weighted bounds ... More

A Simple Master Theorem For Discrete Divide And Conquer RecurrencesFeb 27 2019The aim of this note is to provide a Master Theorem for some discrete divide and conquer recurrences: $$X_n = a_n+\sum_{j=1}^m b_j X_{\lfloor p_j n\rfloor},$$ where the $p_i$'s belong to $(0,1)$. The main novelty of this work is there is no assumption ... More

Improved fractional Poincaré type inequalities on John domainsFeb 27 2019We obtain improved fractional Poincar\'e inequalities in John domains of a metric space $(X, d)$ endowed with a doubling measure $\mu$ under some mild regularity conditions on the measure $\mu$. We also give sufficient conditions on a bounded domain to ... More

Modulation invariant operatorsFeb 27 2019The first three results in this thesis are motivated by a far-reaching conjecture on boundedness of singular Brascamp-Lieb forms. Firstly, we improve over the trivial estimate for their truncations, thus excluding potential trivial counterexamples. Secondly, ... More

Convergence of lacunary SU(1,1)-valued trigonometric productsFeb 27 2019This note attempts to study lacunary trigonometric products with values in the matrix group SU(1,1) in analogy with lacunary trigonometric series. The central questions are the characterization of their convergence in an appropriately define $L^p$-metric ... More

T(1) theorem for dyadic singular integral forms associated with hypergraphsFeb 27 2019This paper studies dyadic singular integral forms associated with $r$-partite $r$-uniform hypergraphs such that all their connected components are complete. We characterize their $L^p$ boundedness by T(1)-type conditions in two different ways. We also ... More

Extending functions from Nikolskii-Besov spaces of mixed smoothness beyond a cubeFeb 27 2019The article examines Nikolskii and Besov spaces with norms defined using "$L_p$-averaged" mixed moduli of continuity of functions of appropriate orders, instead of mixed moduli of continuity of known orders for certain mixed derivative functions. The ... More

Oscillatory integrals and periodic homogenization of Robin boundary value problemsFeb 27 2019In this paper, we consider a family of second-order elliptic systems subject to a periodically oscillating Robin boundary condition. We establish the qualitative homogenization theorem on any Lipschitz domains satisfying a non-resonance condition. We ... More

Sharp Constants of Approximation Theory. III. Polynomial Inequalities of Different Metrics on Convex SetsFeb 26 2019Let $V\subset\R^m$ be a centrally symmetric convex body and let $V^*\subset\R^m$ be its polar. We prove limit relations between the sharp constants in the multivariate Markov-Bernstein-Nikolskii type inequalities for algebraic polynomials on $V^*$ and ... More

Summation formula for generalized discrete $q$-Hermite II polynomialsFeb 26 2019In this paper, we provide a family of generalized discrete $q$-Hermite II polynomials denoted by $\tilde{h}_{n,\alpha}(x,y|q)$. An explicit relations connecting them with the $q$-Laguerre and Stieltjes-Wigert polynomials are obtained. Summation formula ... More

On a novel class of polyanalytic Hermite polynomialsFeb 26 2019We carry out some algebraic and analytic properties of a new class of orthogonal polyanalytic polynomials, including their operational formulas, recurrence relations, generating functions, integral representations and different orthogonality identities. ... More

Floquet Theory for Quaternion-valued Differential EquationsFeb 26 2019This paper describes the Floquet theory for quaternion-valued differential equations (QDEs). The Floquet normal form of fundamental matrix for linear QDEs with periodic coefficients is presented and the stability of quaternionic periodic systems is accordingly ... More

Grassmann convexity and multiplicative Sturm theory, revisitedFeb 26 2019In this paper we settle a special case of the Grassmann convexity conjecture formulated earlier by B.and M.Shapiro. We present a conjectural formula for the maximal total number of real zeros of the consecutive Wronskians of an arbitrary fundamental solution ... More

Rotational hypersurfaces of prescribed mean curvatureFeb 25 2019We use a phase space analysis to give some classification results for rotational hypersurfaces in $\mathbb{R}^{n+1}$ whose mean curvature is given as a prescribed function of its Gauss map. For the case where the prescribed function is an even function ... More

Liouville propertiesFeb 25 2019The classical Liouville theorem states that a bounded harmonic function on all of $\RR^n$ must be constant. In the early 1970s, S.T. Yau vastly generalized this, showing that it holds for manifolds with nonnegative Ricci curvature. Moreover, he conjectured ... More