§28.30 Expansions in Series of Eigenfunctions

§28.30(i) Real Variable

Let $\widehat{\lambda}_{m}$, $m=0,1,2,\dots$, be the set of characteristic values (28.29.16) and (28.29.17), arranged in their natural order (see (28.29.18)), and let $w_{m}(x)$, $m=0,1,2,\dots$, be the eigenfunctions, that is, an orthonormal set of $2\pi$-periodic solutions; thus

 28.30.1 $\displaystyle w_{m}^{\prime\prime}+(\widehat{\lambda}_{m}+Q(x))w_{m}$ $\displaystyle=0,$ ⓘ Symbols: $m$: integer, $x$: real variable and $w(z)$: Mathieu’s equation solution Permalink: http://dlmf.nist.gov/28.30.E1 Encodings: TeX, pMML, png See also: Annotations for 28.30(i), 28.30 and 28 28.30.2 $\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi}w_{m}(x)w_{n}(x)\mathrm{d}x$ $\displaystyle=\delta_{m,n}.$

Then every continuous $2\pi$-periodic function $f(x)$ whose second derivative is square-integrable over the interval $[0,2\pi]$ can be expanded in a uniformly and absolutely convergent series

 28.30.3 $f(x)=\sum_{m=0}^{\infty}f_{m}w_{m}(x),$ ⓘ Symbols: $m$: integer, $x$: real variable and $w(z)$: Mathieu’s equation solution Permalink: http://dlmf.nist.gov/28.30.E3 Encodings: TeX, pMML, png See also: Annotations for 28.30(i), 28.30 and 28

where

 28.30.4 $f_{m}=\frac{1}{2\pi}\int_{0}^{2\pi}f(x)w_{m}(x)\mathrm{d}x.$

§28.30(ii) Complex Variable

For analogous results to those of §28.19, see Schäfke (1960, 1961b), and Meixner et al. (1980, §1.1.11).