Orthogonal functions: Difference between revisions

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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 {{nowrap|sin ''nx''}} and {{nowrap|sin ''mx''}} are orthogonal on the interval {{nowrap|(−π, π)}} when {{nowrap|''m'' ≠ ''n''}}. 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]].