Editing Orthogonal functions (section)

==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 \left(mx\right) \sin \left(nx\right) = \cos \left(\left(m - n\right)x\right) - \cos\left(\left(m+n\right) x\right), </math>
and the integral of the product of the two sine functions vanishes.<ref>[[Antoni Zygmund]] (1935) ''[[Trigonometric Series|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]].