Uniform convergent expansions of integral transforms

Abstract:

Several convergent expansions are available for most of the special functions of the mathematical physics, as well as some asymptotic expansions [NIST Handbook of Mathematical Functions, 2010]. Usually, both type of expansions are given in terms of elementary functions; the convergent expansions provide a good approximation for small values of a certain variable, whereas the asymptotic expansions provide a good approximation for large values of that variable. Also, quite often, those expansions are not uniform: the convergent expansions fail for large values of the variable and the asymptotic expansions fail for small values. In recent papers [Bujanda & all, 2018-2019] we have designed new expansions of certain special functions, given in terms of elementary functions, that are uniform in certain variables, providing good approximations of those special functions in large regions of the variables, in particular for large and small values of the variables. The technique used in [Bujanda & all, 2018-2019] is based on a suitable integral representation of the special function. In this paper we face the problem of designing a general theory of uniform approximations of special functions based on their integral representations. Then, we consider the following integral transform of a function $g(t)$ with kernel $h(t,z)$, $F(z)\coloneq \int _0^1h(t,z)g(t)dt$. We require for the function $h(t,z)$ to be uniformly bounded for $z\in \mathcal {D}\subset \mathbb {C}$ by a function $H(t)$ integrable in $t\in [0,1]$, and for the function $g(t)$ to be analytic in an open region $\Omega$ that contains the open interval $(0,1)$. Then, we derive expansions of $F(z)$ in terms of the moments of the function $h$, $M[h(\cdot ,z),n]\coloneq \int _0^1h(t,z)t^ndt$, that are uniformly convergent for $z\in \mathcal {D}$. The convergence of the expansion is of exponential order $\mathcal {O}(a^{-n})$, $a>1$, when $[0,1]\in \Omega$ and of power order $\mathcal {O}(n^{-b})$, $b>0$, when $[0,1]\notin \Omega$. Most of the special functions $F(z)$ having an integral representation may be cast in this form, possibly after an appropriate change of the integration variable. Then, special interest has the case when the moments $M[h(\cdot ,z),n]$ are elementary functions of $z$, because in that case the uniformly convergent expansion derived for $F(z)$ is given in terms of elementary functions. We illustrate the theory with several examples of special functions different from those considered in [Bujanda & all, 2018-2019].