Let f be a bandlimited function with the support of its Fourier transform being contained in the interval [-\(\sigma\),\(\sigma\) ]. A standard proof for the representation of f by its Shannon sampling series is to replace f by its \(2\pi\)-periodic Fourier series in the Fourier inversion integral and afterwards interchange summation and integration.
In the present paper almost periodic extensions of the Fourier transform f are used instead of the periodic one. The theory of almost periodic functions including Fourier series expansions and Parseval’s equation leads to generalized sampling theorems with samples which are not equidistantly spaced. Examples which are carried out for the present approach include the classical sampling series, sampling expansions with bunched samples and also the multichannel sampling theorem.