# §30.15(i) Scaled Spheroidal Wave Functions

Let $\tau$ $(>0)$ and $\sigma$ $(>0)$ be given. Set $\gamma=\tau\sigma$ and define

 30.15.1 $\phi_{n}(t)=\sqrt{\frac{2n+1}{2\tau}}\sqrt{\Lambda_{n}}\mathop{\mathsf{Ps}^{0}% _{n}\/}\nolimits\!\left(\frac{t}{\tau},\gamma^{2}\right),$ $n=0,1,2,\dots$, Symbols: $\mathop{\mathsf{Ps}^{m}_{n}\/}\nolimits\!\left(x,\gamma^{2}\right)$: spheroidal wave function of the first kind, $n\geq m$: integer degree, $\tau>0$: parameter, $\phi_{n}(t)$: function, $\Lambda$ and $\gamma^{2}$: real parameter Permalink: http://dlmf.nist.gov/30.15.E1 Encodings: TeX, pMML, png
 30.15.2 $\Lambda_{n}=\frac{2\gamma}{\pi}\left(K_{n}^{0}(\gamma)A_{n}^{0}(\gamma^{2})% \right)^{2};$ Symbols: $n\geq m$: integer degree, $A_{n}^{m}(\gamma^{2})$, $K_{n}^{m}(\gamma)$: connection coefficient, $\Lambda$ and $\gamma^{2}$: real parameter Permalink: http://dlmf.nist.gov/30.15.E2 Encodings: TeX, pMML, png

see §30.11(v).

# §30.15(ii) Integral Equation

 30.15.3 $\int_{-\tau}^{\tau}\frac{\mathop{\sin\/}\nolimits\sigma(t-s)}{\pi(t-s)}\phi_{n% }(s)ds=\Lambda_{n}\phi_{n}(t).$

# §30.15(iii) Fourier Transform

 30.15.4 $\int_{-\infty}^{\infty}e^{-it\omega}\phi_{n}(t)dt=(-i)^{n}\sqrt{\frac{2\pi\tau% }{\sigma\Lambda_{n}}}\phi_{n}\left(\frac{\tau}{\sigma}\omega\right)\chi_{% \sigma}(\omega),$
 30.15.5 $\int_{-\tau}^{\tau}e^{-it\omega}\phi_{n}(t)dt=(-i)^{n}\sqrt{\frac{2\pi\tau% \Lambda_{n}}{\sigma}}\phi_{n}\left(\frac{\tau}{\sigma}\omega\right),$

where

 30.15.6 $\chi_{\sigma}(\omega)=\begin{cases}1,&\mbox{|\omega|\leq\sigma},\\ 0,&\mbox{|\omega|>\sigma}.\end{cases}$ Symbols: $\sigma>0$: parameter Referenced by: §30.15(iii) Permalink: http://dlmf.nist.gov/30.15.E6 Encodings: TeX, pMML, png

Equations (30.15.4) and (30.15.6) show that the functions $\phi_{n}$ are $\sigma$-bandlimited, that is, their Fourier transform vanishes outside the interval $[-\sigma,\sigma]$.

# §30.15(iv) Orthogonality

 30.15.7 $\displaystyle\int_{-\tau}^{\tau}\phi_{k}(t)\phi_{n}(t)dt$ $\displaystyle=\Lambda_{n}\delta_{k,n},$ 30.15.8 $\displaystyle\int_{-\infty}^{\infty}\phi_{k}(t)\phi_{n}(t)dt$ $\displaystyle=\delta_{k,n}.$

The sequence $\phi_{n}$, $n=0,1,2,\dots$ forms an orthonormal basis in the space of $\sigma$-bandlimited functions, and, after normalization, an orthonormal basis in $L^{2}(-\tau,\tau)$.

# §30.15(v) Extremal Properties

The maximum (or least upper bound) $\mathrm{B}$ of all numbers

 30.15.9 $\beta=\frac{1}{2\pi}\int_{-\sigma}^{\sigma}\left|\int_{-\infty}^{\infty}e^{-it% \omega}f(t)dt\right|^{2}d\omega$

taken over all $f\in L^{2}(-\infty,\infty)$ subject to

 30.15.10 $\displaystyle\int_{-\infty}^{\infty}|f(t)|^{2}dt$ $\displaystyle=1,$ $\displaystyle\int_{-\tau}^{\tau}|f(t)|^{2}dt$ $\displaystyle=\alpha,$

for (fixed) $\Lambda_{0}<\alpha\leq 1$, is given by

 30.15.11 $\mathop{\mathrm{arccos}\/}\nolimits\sqrt{\mathrm{B}}+\mathop{\mathrm{arccos}\/% }\nolimits\sqrt{\alpha}=\mathop{\mathrm{arccos}\/}\nolimits\sqrt{\Lambda_{0}},$

or equivalently,

 30.15.12 $\mathrm{B}=\left(\sqrt{\Lambda_{0}\alpha}+\sqrt{1-\Lambda_{0}}\sqrt{1-\alpha}% \right)^{2}.$ Symbols: $\Lambda$ and $\mathrm{B}$: maximum bound Permalink: http://dlmf.nist.gov/30.15.E12 Encodings: TeX, pMML, png

The corresponding function $f$ is given by

 30.15.13 $\displaystyle f(t)$ $\displaystyle=a\phi_{0}(t)\chi_{\tau}(t)+b\phi_{0}(t)(1-\chi_{\tau}(t)),$ $\displaystyle a$ $\displaystyle=\sqrt{\frac{\alpha}{\Lambda_{0}}},$ $\displaystyle b$ $\displaystyle=\sqrt{\frac{1-\alpha}{1-\Lambda_{0}}}.$

If $0<\alpha\leq\Lambda_{0}$, then $\mathrm{B}=1$.

For further information see Frieden (1971), Lyman and Edmonson (2001), Papoulis (1977, Chapter 6), Slepian (1983), and Slepian and Pollak (1961).