Uniform convergent expansions of integral transforms,Mathematics of Computation

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Uniform convergent expansions of integral transforms
Mathematics of Computation ( IF 2 ) Pub Date : 2021-01-26 , DOI: 10.1090/mcom/3601
José L. López , Pablo Palacios , Pedro J. Pagola

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

with kernel

, $F(z)\coloneq \int _0^1h(t,z)g(t)dt$ . We require for the function

to be uniformly bounded for $z\in \mathcal {D}\subset \mathbb{C}$ by a function

integrable in $t\in [0,1]$ , and for the function

to be analytic in an open region $\Omega$ that contains the open interval

. Then, we derive expansions of

in terms of the moments of the function

, $M[h(\cdot ,z),n]\coloneq \int _0^1h(t,z)t^ndt$ , that are uniformly convergent for $z\in \mathcal {D}$ . The convergence of the expansion is of exponential order $\mathcal {O}(a^{-n})$ ,

, when $[0,1]\in \Omega$ and of power order $\mathcal {O}(n^{-b})$ ,

, when $[0,1]\notin \Omega$ . Most of the special functions

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

, because in that case the uniformly convergent expansion derived for

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].

中文翻译：

积分变换的一致收敛展开

摘要：对于数学物理学的大多数特殊功能，以及一些渐近扩展，可以使用几个收敛展开式[NIST手册，数学函数，2010]。通常，两种扩展都是根据基本函数给出的。对于某些变量的较小值，收敛展开提供良好的近似，而对于该变量的较大值，渐近展开提供良好的近似。同样，这些扩展经常不是统一的：对于较大的变量值，收敛的扩展会失败；对于较小的值，渐近的扩展会失败。在最近的论文[Bujanda＆all，2018-2019]中，我们设计了某些特殊功能的新扩展，这些扩展以基本函数的形式给出，这些基本函数在某些变量中是一致的，在变量的大区域中提供这些特殊函数的良好近似，尤其是对于变量的大和小值。[Bujanda＆all，2018-2019]中使用的技术基于特殊功能的适当积分表示。在本文中，我们面临基于特殊积分的整体表示设计特殊函数的统一逼近的一般理论的问题。然后，我们考虑以下函数的积分变换在本文中，我们面临基于特殊积分的整体表示设计特殊函数的统一逼近的一般理论的问题。然后，我们考虑以下函数的积分变换在本文中，我们面临基于特殊积分的整体表示设计特殊函数的统一逼近的一般理论的问题。然后，我们考虑以下函数的积分变换

与内核

，。我们要求函数必须由可积分的函数统一界定，并且要求函数在包含开放区间的开放区域中进行分析。于是，我们得出的扩展在功能的时刻而言，即是一致收敛。膨胀的收敛是指数级的，当功率级的和，当。大多数特殊功能 $F（z）\ coloneq \ int _0 ^ 1h（t，z）g（t）dt$

$z \ in \ mathcal {D} \ subset \ mathbb {C}$

$t \ in [0,1]$

$\ Omega$

$M [h（\ cdot，z），n] \ coloneq \ int _0 ^ 1h（t，z）t ^ ndt$ $z \ in \ mathcal {D}$ $\ mathcal {O}（a ^ {-n}）$

$[0,1] \ in \ Omega$ $\ mathcal {O}（n ^ {-b}）$

$[0,1] \ notin \ Omega$

可以在适当更改积分变量之后以这种形式强制转换具有整数表示形式的具有整数表示形式的值。然后，当矩是的基本函数时，就会特别感兴趣，因为在这种情况下，针对基本函数给出了一致收敛的展开式。我们用几个特殊功能示例来说明该理论，这些示例不同于[Bujanda＆all，2018-2019]中考虑的那些特殊功能。 $M [h（\ cdot，z），n]$

更新日期：2021-03-21

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