This paper presents a novel systematic methodology to obtain new simple and tight approximations, lower bounds, and upper bounds for the Gaussian Q-function, and functions thereof, in the form of a weighted sum of exponential functions. They are based on minimizing the maximum absolute or relative error, resulting in globally uniform error functions with equalized extrema. In particular, we construct sets of equations that describe the behaviour of the targeted error functions and solve them numerically in order to find the optimized sets of coefficients for the sum of exponentials. This also allows for establishing a trade-off between absolute and relative error by controlling weights assigned to the error functions' extrema. We further extend the proposed procedure to derive approximations and bounds for any polynomial of the Q-function, which in turn allows approximating and bounding many functions of the Q-function that meet the Taylor series conditions, and consider the integer powers of the Q-function as a special case. In the numerical results, other known approximations of the same and different forms as well as those obtained directly from quadrature rules are compared with the proposed approximations and bounds to demonstrate that they achieve increasingly better accuracy in terms of the global error, thus requiring significantly lower number of sum terms to achieve the same level of accuracy than any reference approach of the same form.

中文翻译：

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指数和的高斯 Q 函数的全局极小极大近似和边界

本文提出了一种新颖的系统方法，以指数函数的加权和的形式获得高斯 Q 函数及其函数的新的简单而严格的近似值、下界和上界。它们基于最小化最大绝对或相对误差，从而产生具有均衡极值的全局均匀误差函数。特别是，我们构建了描述目标误差函数行为的方程组，并对其进行了数值求解，以便找到指数总和的优化系数组。这还允许通过控制分配给误差函数极值的权重来在绝对误差和相对误差之间建立权衡。我们进一步扩展了所提出的程序，以推导出 Q 函数的任何多项式的近似值和界限，这反过来又允许逼近和限制满足泰勒级数条件的 Q 函数的许多函数，并将 Q 函数的整数幂视为一种特殊情况。在数值结果中，将其他已知的相同和不同形式的近似值以及直接从正交规则获得的近似值与所提出的近似值和界限进行比较，以证明它们在全局误差方面的精度越来越高，因此需要显着降低与任何相同形式的参考方法相比，达到相同精度水平的求和项数。