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{{Main article|Fourier series|Harmonic analysis}}
Several sets of orthogonal functions have become standard bases for approximating functions. For example, the sine functions, sin ''nx'' and sin ''mx'', are orthogonal on the interval (-π, π), if ''m'' ≠ ''n''. For then
:<math>2 \sin mx \sin nx = \cos (m - n)x - \cos (m+n) x, </math>
so that the integral of the product of the two sines vanishes.<ref>[[Antoni Zygmund]] (1935) ''Trigonometrical Series'', page 6, Mathematical Seminar, University of Warsaw</ref> Together with cosine functions, these orthogonal functions may be assembled into a [[trigonometric polynomial]] to approximate a given function on the interval with its [[Fourier series]].
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