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==Trigonometric functions==
{{Main article|Fourier series|Harmonic analysis}}
Several sets of orthogonal functions have become standard bases for approximating functions. For example, the sine functions {{nowrap|sin ''nx''}} and {{nowrap|sin ''mx''}} are orthogonal on the interval <math>x \in (-\pi, \pi)</math> when <math>m \neq n</math> and ''n'' and ''m'' are positive integers. For then
:<math>2 \sin (mx) \sin (nx) = \cos \left((m - n)x\right) - \cos\left((m+n) x\right), </math>
and the integral of the product of the two sine functions 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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