arXiv subjects
Publications and source records attributed to Hera Yun.
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.