Schematic showing continuous to discrete functions, periodic or not, and their Fourier transforms. The original of this figure was <a href="//commons.wikimedia.org/wiki/File:From_Continuous_To_Discrete_Fourier_Transform.gif" title="File:From Continuous To Discrete Fourier Transform.gif">put into the public domain</a> by <a href="//commons.wikimedia.org/wiki/User:Sbyrnes321" title="User:Sbyrnes321">Steve Byrnes</a>, but is here completely regenerated <i>with labels</i>, new functions, and shading. "Even functions" (symmetric about the origin) have been chosen so that the Fourier transform coefficients are real rather than complex. 
<p>The yellow shaded regions show the limited data regions (when finite) needed to generate the plots, from either the direct time/space coordinate (top) or the reciprocal frequency coefficient (bottom) side.
</p><p>The horizontal axis on the top row is in units of space or time, while the horizontal "frequency" axis on the bottom row is in cycles per unit space or time. The vertical displacement/amplitude scales are normalized to keep the scales similar as well.
</p>
Discrete sampling (in direct and/or reciprocal space) matches the realities of digital sampling, and opens the door to rapid "Fast Fourier Transform" calculations, but entails the often artificial assumption of periodicity (in reciprocal and/or direct space respectively). This assumption is therefore something that we are learning to live with, at least as a useful approximation.
<a href="//commons.wikimedia.org/wiki/User:Unitsphere" title="User:Unitsphere">P. Fraundorf</a>
CC BY-SA 4.0
<span class="int-own-work" lang="en">Own work</span>