在数学中，长球波函数由一个限时、限频、与第二个限时的函数相乘而成。假定表示一个切截时间的运算器，且，则x必为有限时间区间的函数，当x在的区间内。同理，假定表示一个理想的低频滤波器，且，则x必为有限带宽区间的函数，当x在的区间内。透过组合上述运算子，使得转变成线性、有界且自伴的运算式。对于，我们假设为第n项的本征函数，定义下列函式

其中为对应的本征值。此时限函数即为长球波函数(PSWFs).在此领域中，几个非常重要的先驱文章由Slepian and Pollak, Landau and Pollak, and Slepian.所提出。

In mathematics, the prolate spheroidal wave functions (PSWF) are a set of functions derived by timelimiting and lowpassing, and a second timelimit operation. Let denote the time truncation operator, such that if and only if (iff) x is timelimited within . Similarly, let denote an ideal low-pass filtering operator, such that iff x is bandlimited within . The operator turns out to be linear, bounded and self-adjoint. For we denote with the n-th eigenfunction, defined as