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We characterize the $L^p(\sigma)\to L^q(\omega)$ boundedness of positive dyadic operators of the form $ <br>T(f\sigma)=\sum_{Q\in\mathscr{D}}\lambda_Q\int_Q f\,\mathrm{d}\sigma\cdot 1_Q, $ and the $L^{p_1}(\sigma_1)\times L^{p_2}(\sigma_2)\to L^q(\omega)$ boundedness of their bilinear analogues, for arbitrary locally f... |
In recent work on the area of approximation methods for the solution of nonlinear differential equations, it has been suggested that the so-called generalized Taylor series approach is equivalent to the homotopy analysis method. In the present paper, we demonstrate that such a view is only valid in very special cases,... |
This work is a continuation of what was done in a previous paper and strongly connected to the recent work of U. Abel and I. Rasa [<a href="https://arxiv.org/abs/1707.00127" data-arxiv-id="1707.00127" class="link-https">arXiv:1707.00127</a>] |
We give a very simple proof of the positivity and unimodality of the Green function for the killed fractional Laplacian on the periodic domain. The argument relies on the Jacobi triple product and a probabilistic representation of the Green function. |
In this note functions that transform open segments of a linear space into open segments of another linear space are studied and characterized. Assuming that the range is non-collinear, it is proved that such a map can always be expressed as the ratio of two affine functions. |
Gowers introduced the notion of uniformity norm $\|f\|_{U^k(G)}$ of a bounded function $f:G\rightarrow\mathbb{R}$ on an abelian group $G$ in order to provide a Fourier-theoretic proof of Szemeredi's Theorem, that is, that a subset of the integers of positive upper density contains arbitrarily long arithmetic progre... |
Over the last years, considerable attention has been paid to the role of the quaternion differential equations (QDEs) which extend the ordinary differential equations. The theory of QDEs was recently well established and it has wide applications in physics and life science. |
Time-frequency analysis, such as the Gabor transform, plays an important role in many signal processing applications. The redundancy of such representations is often directly related to the computational load of any algorithm operating in the transform domain. |
Elementary problems like the evaluation of repeated derivatives of ordinary transcendent functions can usefully be treated by the use of special polynomials and of a formalism borrowed from combinatorial analysis. Motivated by previous researches in this field, we review the results obtained by other authors and devel... |
We consider a class of Hill equations where the periodic coefficient is the squared solution of some Duffing equation plus a constant. We study the stability of the trivial solution of this Hill equation and we show that a criterion due to Burdina (V.I. Burdina, Boundedness of solutions of a system of differential equ... |
In the paper, two new identities involving the local fractional integrals have been established. Using these two identities, we obtain some generalized Hermite-Hadamard type integral inequalities for the local differentiable generalized convex functions. |
The ranges of a certain type of second order differential operator, on a Sobolev subspace of the Lebesgue space $L^2$ of the circle group, can be characterised by the vanishing of the Fourier coefficients at (generally) two integers that are the zeros of the multiplier of the operator. It is proved here that the range... |
We generalize the well-known inequality that the limit of the $L^p$ norm of a function as $p\rightarrow\infty$ is the $L^\infty$ norm to the scale of Orlicz spaces. |
Integrals involving the kernel function $sech (\pi x)$ over a semi-infinite range are of general interest in the study of Riemann's function $\zeta(s)$ and Hurwitz' function $\zeta(s,a)$. Such integrals that include the $arctan$ and $log$ functions in the integrand are evaluated here in terms of $\zeta(s,a)$, ... |
Pull-back transformations between Heun and Gauss hypergeometric equations give useful expressions of Heun functions in terms of better understood hypergeometric functions. This article classifies, up to Mobius automorphisms, the coverings P1-to-P1 that yield pull-back transformations from hypergeometric to Heun equati... |
In this paper we improve and complement a result by Móricz and Siddiqi \cite{Mor}. In particular, we prove that their inequality of the Nörlund means with respect to the Walsh system holds also without their additional condition. |
In this paper we investigate the $L^p$ boundedness of the lacunary maximal function $ M_{\Ha}^{lac} $ associated to the spherical means $ A_r f$ taken over Koranyi spheres on the Heisenberg group. Closely following an approach used by M. Lacey in the Euclidean case, we obtain sparse bounds for these maximal functions ... |
We continue the development, by reduction to a first order system for the conormal gradient, of $L^2$ \textit{a priori} estimates and solvability for boundary value problems of Dirichlet, regularity, Neumann type for divergence form second order, complex, elliptic systems. We work here on the unit ball and more genera... |
Discrete analogs of the classical Mehler-Fock transforms are introduced and investigated. It involves series with the associated Legendre function $P^\mu_{in-1/2}(x), x > 1,\ {\rm Re} \mu < 1/2, \ n \in \mathbb{N}, i $ is the imaginary unit, and the so-called incomplete Legendre integrals. |
The main aim of this paper is to investigate weighted maximal operators of partial sums of Vilenkin-Fourier series. We also use our results to prove approximation and strong convergence theorems on the martingale Hardy spaces $H_{p},$ when $0<p\leq 1.$ |
Let $M$ be the Hardy-Littlewood maximal function. Denote by $M_b$ and $[b,M]$ the maximal and the nonlinear commutators of $M$ with a function $b$. |
Asymptotic expansions for the Bateman and Havelock functions defined respectively by the integrals \[\frac{2}{\pi}\int_0^{\pi/2} \! \! |
In this paper, we completely classify the raional solutions of the Noumi and Yamada system of type A_4^{(1)}, which is a generalization of the forth Painlevé equation. The rational solutions are classified to three types by the Bäcklund transformation group. |
Let ${\Bbb F}_q$ be a finite field of order $q. $ We prove that if $d\ge 2$ is even and $E \subset {\Bbb F}_q^d$ with $|E| \ge 9q^{\frac{d}{2}}$ then $$ {\Bbb F}_q=\frac{\Delta(E)}{\Delta(E)}=\left\{ \frac{a}{b}: a \in \Delta(E), b \in \Delta(E) \backslash \{0\} \right\},$$ where $$ \Delta(E)=\{||x-y||: x,y \in E\}, \ ... |
The Matrix Bochner Problem aims to classify which weight matrices have their sequence of orthogonal polynomials as eigenfunctions of a second-order differential operator. Casper and Yakimov, in [4], demonstrated that, under certain hypotheses, all solutions to the Matrix Bochner Problem are noncommutative bispectral D... |
A system of 4 nonlinearly-coupled Ordinary Differential Equations has been recently introduced to investigate the evolution of human respiratory virus epidemics. In this paper we point out that some explicit solutions of that system can be obtained by algebraic operations, provided the parameters of the model satisfy ... |
Let Y be a Calabi-Yau complete intersection in a weighted projective space. We show that the space of quadratic invariants of the hypergeometric group associated with the twisted I-function is one-dimensional, and spanned by the Gram matrix of a split-generator of the derived category of coherent sheaves on Y with res... |
The hyperoctahedral group is the Weyl group of type B and is associated with a two-parameter family of differential-difference operators T_i, i=1,. .,N (the dimension of the underlying Euclidean space). |
We prove boundedness of Calderón-Zygmund operators acting in Banach functions spaces on domains, defined by the $L_1$ Carleson functional and $L_q$ ($1<q<\infty$) Whitney averages. For such bounds to hold, we assume that the operator maps towards the boundary of the domain. |
Being inspired by phasor analysis in linear circuit theory, and its algebraic counterpart - the AC-(operational)-calculus for sinusoids developed by W. Marten and W. Mathis - we define a complex structure on several spaces of real-valued elementary functions. This is used to algebraize inhomogeneous linear ordinary di... |
We obtain exact solutions of a problem arising from queueing theory using properties of Kapteyn series. |
For point vortices in the plane, we consider the correlation coefficient of Ovchinnikov and Sigal. Generalising a result by Esposito, we show that it vanishes for all vortex equilibria. |
This paper is devoted to study the existence of solutions and the monotone method of second-order periodic boundary value problems when the lower and upper solutions $\alpha$ and $\beta$ violate the boundary conditions $ \alpha(0)=\alpha(2\pi)$ and $\beta(0)=\beta(2\pi)$. We present several comparison results. |
In the framework of Clifford analysis, a chain of harmonic and monogenic potentials in the upper half of Euclidean space $\mR^{m+1}$ was recently constructed, including a higher dimensional analogue of the logarithmic function in the complex plane. In this construction the distributional limits of these potentials at ... |
We study the continuous solutions of several classical functional equations by using the properties of the spaces of continuous functions which are invariant under some elementary linear trans-formations. Concretely, we use that the sets of continuous solutions of certain equations are closed vector subspaces of $C(\m... |
We prove a Hörmander type multiplier theorem for multilinear Fourier multipiers with multiple weights. We also give weighted estimates for their commutators with vector $BMO$ functions. |
The elementary geometric properties of Jacob's ladders of the second order lead to a class of new asymptotic formulae for short and microscopic parts of the Hardy-Littlewood integral of $|\zeta(1/2+it)|^4$. These formulae cannot be obtained by methods of Balasubramanian, Heath-Brown and Ivic. |
We discuss quadratic transformations for orthogonal polynomials in one and two variables. In the one-variable case we list many (or all) quadratic transformations between families in the Askey scheme or $q$-Askey scheme. |
Based on the popular Caputo fractional derivative of order $\beta$ in $(0,1)$, we define the censored fractional derivative on the positive half-line $\mathbb R_+$. This derivative proves to be the Feller generator of the censored (or resurrected) decreasing $\beta$-stable process in $\mathbb R_+$. |
We study the uncertainty principle <br>$$\lVert\widehat{\mu}(\xi) |\xi|^\beta\rVert_\infty^{\alpha} <br>\left(\int |x|^\alpha d \mu\right)^{\beta} \geq C(\alpha,\beta,d){\lVert{\mu}\rVert_{TV}^{\alpha+\beta}}$$ for finite non-negative measures on $\mathbb{R}^d $. We prove that $C(\alpha,\beta,d)>0$ for all $\alpha,... |
Stars are gravitationally stabilized fusion reactors changing their chemical composition while transforming light atomic nuclei into heavy ones. The atomic nuclei are supposed to be in thermal equilibrium with the ambient plasma. |
The distribution of the poles of branches of the Painleve' VI transcendents associated to semi-simple Frobenius manifolds is determined close to a critical point. It is shown that the poles accumulate at the critical point, asymptotically along two rays. |
We represent Matérn functions in terms of Schoenberg's integrals which ensure the positive definiteness and prove the systems of translates of Matérn functions form Riesz sequences in $L^2(\R^n)$ or Sobolev spaces. Our approach is based on a new class of integral transforms that generalize Fourier transforms for r... |
It is general knowledge that the harmonic mean $H(x,y)=\frac2{\frac1x+\frac1y}$ and that the geometric mean $G(x,y)=\sqrt{xy}\,$, where $x$ and $y$ are two positive numbers. In the paper, the authors show by several approaches that the harmonic mean $H_{x,y}(t)=H(x+t,y+t)$ and the geometric mean $G_{x,y}(t)=G(x+t,y+t)... |
In this note, we compute the reproducing kernel for the RKHS of functions on $\mathbb{R}^n$ in a sufficiently high Sobolev norm. |
We provide improved sufficient assumptions on sequences of Fucik eigenvalues of the one-dimensional Dirichlet Laplacian which guarantee that the corresponding Fucik eigenfunctions form a Riesz basis in $L^2(0,\pi)$. For that purpose, we introduce a criterion for a sequence in a Hilbert space to be a Riesz basis. |
In this paper we evaluate sums and integrals of products of Fubini polynomials and have new explicit formulas for Fubini polynomials and numbers. As a consequence of these results new explicit formulas for p-Bernoulli numbers and Apostol-Bernoulli functions are given. |
We define and study symmetrized and antisymmetrized multivariate exponential functions. They are defined as determinants and antideterminants of matrices whose entries are exponential functions of one variable. |
In this paper we use the Vandermonde matrices and their properties to give a new proof of the classical result of Karl Weierstrass about the approximation of continuous functions $f$ on closed intervals, using a sequence of polynomials. The proof solves linear systems of equations using that the Vandermonde matrices h... |
We deal with the boundedness of the multilinear fractional integral operator $I_{\gamma,m}$ from a product of weighted Lebesgue spaces into adequate weighted Lipschitz spaces. Our results generalize some previous estimates not only for the linear case but also for the unweighted problem in the multilinear context. |
Let $\mu$ be a positive measure on $R^d$. It is known that if the space $L^2(\mu)$ has a frame of exponentials then the measure $\mu$ must be of "pure type": it is either discrete, absolutely continuous or singular continuous. |
In this paper, the main aim is to demonstrate the boundedness for commutators of (fractional) maximal function and sharp maximal function in the slice spaces, where the symbols of the commutators belong to the BMO space, whereby some new characterizations for BMO spaces are given. |
In this paper we obtain precise estimates for the $L^2$ norm of the $s$-dimensional Riesz transforms on very general measures supported on Cantor sets in $\mathbb R^d$, with $d-1<s<d$. From these estimates we infer that, for the so called uniformly disconnected compact sets, the capacity $\gamma_s$ associated wi... |
We prove an integral representation and a power series expansion for the function $\det(A)^{-1}$ in a small neighborhood of the identity matrix. Both results are closely linked to the formula for the change of coordinates of the Dirac delta distribution in $\mathbb{R}^m$. |
A class of Laplace transforms is examined to show that particular cases of this class are associated with production-destruction and reaction-diffusion problems in physics, study of differences of independently distributed random variables and the concept of Laplacianness in statistics, alpha-Laplace and Mittag-Leffler... |
Let $\Omega$ be homogeneous of degree zero, have mean value zero and integrable on the unit sphere, and $\mathcal{M}_{\Omega}$ be the higher-dimensional Marcinkiewicz integral associated with $\Omega$. In this paper, the authors proved that if $\Omega\in L^q(S^{n-1})$ for some $q\in (1,\,\infty]$, then for $p\in (q... |
We present a Pfaffian identity involving elliptic functions, whose rational limit gives a generalization of Schur's Pfaffian identity for Pf ((x_j - x_i)/(x_j + x_i)). This identity is regarded as a Pfaffian counterpart of Frobenius identity, which is an elliptic generalization of Cauchy's determinant identity... |
Suppose that G is a finite group and A is a subset of G such that 1_A has algebra norm at most M. Then 1_A is a plus/minus sum of at most L cosets of subgroups of G, and L can be taken to be triply tower in O(M). This is a quantitative version of the non-abelian idempotent theorem. |
In this paper, we use two different approaches to introduce $q$-analogs of Riemann's zeta function and prove that their values at even integers are related to the $q$-Bernoulli and $q$ Euler's numbers introduced by Ismail and Mansour [Analysis and Applications, {\bf{17}}, 6, 2019, 853--895]. |
Let $\alpha_{n1},\dots,\alpha_{nn}$ be the zeros of the $n$th Bessel polynomial $y_n(z)$ and let $a_{nk}=1-\alpha_{nk}/2$, $b_{nk}=1+\alpha_{nk}/2$ $(k=1,\dots,n)$. We propose the new formula \[z f'(z)\approx \sum_{k=1}^n \big(f(a_{nk} z)-f(b_{nk} z)\big)\] for numerical differentiation of analytic functions $f(z)... |
Formal Laurent-Puiseux series are important in many branches of mathematics. This paper presents a {\it Mathematica} implementation of algorithms developed by the author for converting between certain classes of functions and their equivalent representing series. |
We study the number of nodal components (connected components of the set of zeroes) of functions in the ensemble of arithmetic random waves, that is, random eigenfunctions of the Laplacian on the flat $d$-dimensional torus $\mathbb{T}^{d}$ ($d\ge2$). Let $f_{L}$ be a random solution to $\Delta f+4\pi^{2}L^{2}f=0$ on $... |
A projective structure on a compact Riemann surface X of genus g is given by an atlas with transition functions in PGL(2,C). Equivalently, a projective structure is given by a projective sl(2,C)-bundle over X equipped with a section s and a foliation F which is both transversal to the fibers and the section s. |
This is the third revision. We study bases of Pfaffian systems for $A$-hypergeometric system. |
In the present article we define the Jain type modification of the generalized Szasz-Mirakjan operators that preserve constant and exponential mappings. Moments, recurrence formulas, and other identities are established for these operators. |
We demonstrate how the asymptotics for large $|z|$ of the generalised Bessel function \[{}_0\Psi_1(z)=\sum_{n=0}^\infty\frac{z^n}{\Gamma(an+b) n! },\] where $a>-1$ and $b$ is any number (real or complex), may be obtained by exploiting the well-established asymptotic theory of the generalised Wright function ${}_p\Ps... |
We consider the Rubio de Francia's Littlewood--Paley square function associated with an arbitrary family of intervals in $\mathbb{R}$ with finite overlapping. Quantitative weighted estimates are obtained for this operator. |
Simple upper and lower bounds are established for the integral $\int_0^x\mathrm{e}^{-\beta t}t^\nu \mathbf{L}_\nu(t)\,\mathrm{d}t$, where $x>0$, $\nu>-1$, $0<\beta<1$ and $\mathbf{L}_\nu(x)$ is the modified Struve function of the first kind. These bounds complement and improve on existing results, through ... |
We give a new computational method to obtain symmetries of ordinary differential equations. The proposed approach appears as an extension of a recent algorithm to compute variational symmetries of optimal control problems [Comput. Methods Appl. Math. 5 (2005), no. 4, pp. 387-409], and is based on the resolution of a f... |
In this note, we study both the Riesz and reverse Riesz transforms on broken line. This model can be described by $(-\infty, -1] \cup [1,\infty)$ equipped with the measure $d\mu = |r|^{d_{1}-1}dr$ for $r \le -1$ and $d\mu = r^{d_{2}-1}dr$ for $r\ge 1$, where $d_{1}, d_{2} >1$. |
We repurpose tools from the theory of quantitative rectifiability to study the qualitative rectifiability of measures in $\Bbb{R}^n$, $n\geq 2$. To each locally finite Borel measure $\mu$, we associate a function $\widetilde J_2(\mu, x)$ which uses a weighted sum to record how closely the mass of $\mu$ is concentrated... |
The study deals with a minimal energy problem over noncompact classes of infinite dimensional vector measures in a locally compact space. The components are positive measures (charges) satisfying certain normalizing assumptions and supported by given closed sets (plates) with the sign +1 or -1 prescribed such that opp... |
We propose an affine version of the Schwarz map for the hypergeometric differential equation, and study its image when the monodromy group is finite. |
In this paper our aim is to deduce some complete monotonicity properties and functional inequalities for the Bickley function. The key tools in our proofs are the classical integral inequalities, like Chebyshev, Hölder-Rogers, Cauchy-Schwarz, Carlson and Grüss inequalities, as well as the monotone form of l'Hospit... |
Given a positive definite kernel in a locally compact space, we study a minimal energy problem in the presence of an external field over the class of all nonnegative Radon measures that are supported by a given closed noncompact set, satisfy certain normalizing assumptions, and do not exceed a fixed measure serving as ... |
For a class of orthogonal polynomials related to the $q$-Meixner polynomials corresponding to an indeterminate moment problem we give a one-parameter family of orthogonality measures. For these measures we complement the orthogonal polynomials to an orthogonal basis for the corresponding weighted $L^2$-space explicitl... |
We first study the linear eigenvalue problem for a planar Dirac system in the open half-line and describe the nodal properties of its solution by means of the rotation number. We then give a global bifurcation result for a planar nonlinear Dirac system in the open half-line. |
In any quasi-metric space of homogeneous type, Auscher and Hytönen recently gave a construction of orthonormal wavelets with Hölder-continuity exponent $\eta>0$. However, even in a metric space, their exponent is in general quite small. |
In this paper, we study in prime fields the exceptional set estimates, which can be viewed as a refinement of Marstrand's orthogonal projection theorem. Additionally, we address a Furstenberg-type problem, which is closely related. |
The multiresolution analysis of Alpert is considered. Explicit formulas for the entries in the matrix coefficients of the refinement equation are given in terms of hypergeometric functions. |
We prove that a typical compact set does not contain any similar copy of a given pattern. We also prove that a typical compact set of $[0,1]^{d} (d\geq 2)$ intersects any $(d-1)$-dimensional plane in at most $d$ points. |
In this paper, we solve completely the $L^2\to L^r$ extension conjecture for the zero radius sphere over finite fields. We also obtain the sharp $L^p\to L^4$ extension estimate for non-zero radii spheres over finite fields, which improves significantly the previous result of the first and second listed authors. |
We study the surjectivity of, and the existence of right inverses for, the asymptotic Borel map in Carleman-Roumieu ultraholomorphic classes defined by regular sequences in the sense of E. M. Dyn'kin. We extend previous results by J. Schmets and M. Valdivia, by V. Thilliez, and by the authors, and show the promine... |
We consider spectral projectors associated to the Euclidean Laplacian on the two-dimensional torus, in the case where the spectral window is narrow. Bounds for their L2 to Lp operator norm are derived, extending the classical result of Sogge; a new question on the convolution kernel of the projector is introduced. |
This paper is devoted to a complete parametric study of Liouvillian solutions of the general trace-free second order differential equation with a Laurent polynomial coefficient. This family of equations, for fixed orders at $0$ and $\infty$ of the Laurent polynomial, is seen as an affine algebraic variety. |
A set $\Omega \subset \mathbb{R}^d$ is said to be spectral if the space $L^2(\Omega)$ admits an orthogonal basis of exponential functions. Fuglede (1974) conjectured that $\Omega$ is spectral if and only if it can tile the space by translations. |
The singularity structure of solutions of a class of Hamiltonian systems of ordinary differential equations in two dependent variables is studied. It is shown that for any solution, all movable singularities, obtained by analytic continuation along a rectifiable curve, are at most algebraic branch points. |
In this article, we study maximal functions related to hypersurfaces with vanishing Gaussian curvature in $\mathbb{R}^3$. Firstly, we characterize the $L^p\rightarrow L^q$ boundedness of local maximal operators along homogeneous hypersurfaces. |
We study the notion of regular singularities for parameterized complex ordinary linear differential systems, prove an analogue of the Schlesinger theorem for systems with regular singularities and solve both a parameterized version of the weak Riemann-Hilbert Problem and a special case of the inverse problem in paramet... |
In this paper, we determine the connection coefficients for Okubo's canonical solution matrix of types ${\rm I}$, ${\rm I}^*$, ${\rm II}$ and ${\rm III}$ in Yokoyama's <a href="http://list.To" rel="external noopener nofollow" class="link-external link-http">this http URL</a> solve these problems, we investigate... |
In this paper we study the spectrum of a fundamental differential operator on a Hilbert-Pólya space. A number is an eigenvalue of this differential operator if and only if it is a nontrivial zero of the Riemann zeta function. |
It is known that local zeta functions associated with real analytic functions can be analytically continued as meromorphic functions to the whole complex plane. In this paper, the case of specific (non-real analytic) smooth functions is precisely investigated. |
We consider univariate real polynomials with all roots real and with two sign changes in the sequence of their coefficients which are all non-vanishing. One of the changes is between the linear and the constant term. |
Fix an integer $ n$ and number $d$, $ 0< d\neq n-1 \leq n$, and two weights $ w$ and $ \sigma $ on $ \mathbb R ^{n}$. We two extra conditions (1) no common point masses and (2) the two weights separately are not concentrated on a set of codimension one, uniformly over locations and scales. |
The classical table of integrals by I. S. Gradshteyn and I. M. Ryzhik contains many definite integrals where the integrand is the product of a rational function times the logarithm of another rational function. We begin the systematic evaluation of these integrals. |
The main objective of this work is to introduce the generalized convolution with trigonometric weighted $\gamma=\sin y$ involving the Fourier cosine-sine and Kontorovich-Lebedev transforms, and to study its fundamental results. We establish boundedness properties in a two-parametric family of Lebesgue spaces for this ... |
The change of the precession angle is studied analytically and numerically for the integrable tops of Kovalevskaya and Goryachev-Chaplygin. Based on the known results on the topology of Liouville foliations for these systems, we find initial conditions for which the average change of the precession angle is zero or ca... |
The first purpose of this paper is to provide new finite field extension theorems for paraboloids and spheres. By using the unusual good Fourier transform of the zero sphere in some specific dimensions, which has been discovered recently in the work of Iosevich, Lee, Shen, and the first and second listed authors (2018... |
In this paper, we consider the realization of configuration of limit cycles of piecewise linear systems on the plane. We show that any configuration of Jordan curves can be realized by a discontinuous piecewise linear system with two zones separated by a continuous curve. |
In this paper we improve classical Hardy-Littlewood exponent $1/2$ by about $16.6\%$ 62 years after the original result. This result is the first step to prove the Selberg's hypothesis (1942). |
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