arXiv subjects
Publications and source records attributed to Yong-Kum Cho.
It is proved by the method of partial fraction expansions and Sturm's oscillation theory that the zeros of certain Hankel transforms are all real and distributed regularly between consecutive zeros of Bessel functions. As an application, the sufficient or necessary conditions on parameters for which ${}_1F_2$ hypergeometric functions belong to the Laguerre-Pólya class are investigated in a constructive manner.
In consideration of the integral transform whose kernel arises as an oscillatory solution of certain second-order linear differential equation, its positivity is investigated on the basis of Sturm's theory. As applications, positivity criteria are obtained for Hankel transforms as well as trigonometric integrals defined on the positive real line.
As for the Fourier transforms of positive and integrable functions supported in the unit interval, we make a list of improvements for Pólya's results on the distribution of their positive zeros and give new sufficient conditions under which those zeros are simple and regularly distributed. As an application, we take the two-parameter family of beta probability density functions defined by \begin{equation*} f(t)= \frac{1}{B(α, β)}\, (1-t)^{α-1} t^{β-1},\quad 0 0,\,β>0,$ and specify the distribution of zeros of the associated Fourier transforms for some region of $(α, β)$ in the first quadrant which turns out to be much larger than the region where Pólya's results are applicable.
As for the ${}_2F_3$ hypergeometric function of the form \begin{equation*} {}_2F_3\left[\begin{array}{c} a_1, a_2\\ b_1, b_2, b_3\end{array}\biggr| -x^2\right]\qquad(x>0), \end{equation*} where all of parameters are assumed to be positive, we give sufficient conditions on $(b_1, b_2, b_3)$ for its positivity in terms of Newton polyhedra with vertices consisting of permutations of $\,(a_2, a_1+1/2, 2a_1)\,$ or $\,(a_1, a_2+1/2, 2a_2).$ As an application, we obtain an extensive validity region of $(α, λ, μ)$ for the inequality \begin{equation*} \int_0^x (x-t)^λ\, t^μ J_α(t)\, dt \ge 0\qquad(x>0). \end{equation*}
We present a rational extension of Newton diagram for the positivity of ${}_1F_2$ generalized hypergeometric functions. As an application, we give upper and lower bounds for the transcendental roots $β(α)$ of \begin{align*} \int_0^{j_{α, 2}} t^{-β} J_α(t) dt = 0\qquad(-1<α\le 1/2), \end{align*} where $j_{α, 2}$ denotes the second positive zero of Bessel function $J_α$.
As to the Bessel integrals of type \begin{equation*} \int_0^x \left(x^μ-t^μ\right)^λt^αJ_β(t)dt\qquad(x>0), \end{equation*} we improve known positivity results by making use of new positivity criteria for ${}_1F_2$ and ${}_2F_3$ generalized hypergeometric functions. As an application, we extend Buhmann's class of compactly supported radial basis functions.
As for the positivity of ${}_1F_2$ generalized hypergeometric functions, we present a list of necessary and sufficient conditions in terms of parameters and determine the region of positivity by certain Newton diagram.
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 radial functions. We also consider inverse multi-quadrics and obtain similar results.
We exhibit three classes of compactly supported functions which provide reproducing kernels for the Sobolev spaces $H^δ(\R^d)$ of arbitrary order $\,δ>d/2.\,$ Our method of construction is based on a new class of oscillatory integral transforms that incorporate radial Fourier transforms and Hankel transforms.
We characterize probability measure with finite moment of any order in terms of the symmetric difference operators of their Fourier transforms. By using our new characterization, we prove the continuity $f(t,v)\in C((0, \infty),L^1_{2k-2 +α})$, where $f(t, v)$ stands for the density of unique measure-valued solution $(F_t)_{t\ge0}$ of the Cauchy problem for the homogeneous non-cutoff Boltzmann equation, with Maxwellian molecules, corresponding to a probability measure initial datum $F_0$ satisfying \[ \int |v|^{2k-2+α} dF_0(v) < \infty, 0\leq α< 2,k= 2, 3, 4,\cdots \] provided that $F_0$ is not a single Dirac mass.
We prove a local-in-time existence and uniqueness theorem for a smooth classical solution to the spatially homogeneous Boltzmann equation with cutoff soft potentials. Our proof is based on a series of bilinear estimates for the integrability and Sobolev regularity of the associated collision operator. While the global-in-time existence is left inconclusive, we give a lower bound of the maximal time of existence and a necessary condition for finite time extinction of existence.
As for the spatially homogeneous Boltzmann equation of Maxwellian molecules with the fractional Fokker-Planck diffusion term, we consider the Cauchy problem for its Fourier-transformed version, which can be viewed as a kinetic model for the stochastic time-evolution of characteristic functions associated with the symmetric stable Levy process and the Maxwellian collision dynamics. Under a non-cutoff assumption on the kernel, we establish a global existence theorem with maximum growth estimate, uniqueness and stability of solutions.
We present a family of explicit formulae for evaluating absolute moments of probability measures on $\mathbb{R}^d$ in terms of Fourier transforms. As to the space of probability measures possessing finite absolute moments of an arbitrary order, we exploit our formulae to characterize its Fourier image and construct Fourier-based probability metrics which make the space complete. As applications, we compute absolute moments of those probability measures whose characteristic functions belong to the Scheonberg classes, estimate absolute moments of convolutions and investigate the asymptotic behavior of solutions to the heat-diffusion equations from a probability view-point.