The generalized Fourier series of a square-integrable function f: →, with respect to Φ, is then
where the coefficients are given by
If Φ is a complete set, i.e., an orthonormal basis of the space of all square-integrable functions on, as opposed to a smaller orthonormal set, the relation becomes equality in the L² sense, more precisely modulo |·|w (not necessarily pointwise, nor almost everywhere).
Read more about Generalized Fourier Series: Example (Fourier–Legendre Series)
Famous quotes containing the words generalized and/or series:
“One is conscious of no brave and noble earnestness in it, of no generalized passion for intellectual and spiritual adventure, of no organized determination to think things out. What is there is a highly self-conscious and insipid correctness, a bloodless respectability submergence of matter in mannerin brief, what is there is the feeble, uninspiring quality of German painting and English music.”
—H.L. (Henry Lewis)
“I look on trade and every mechanical craft as education also. But let me discriminate what is precious herein. There is in each of these works an act of invention, an intellectual step, or short series of steps taken; that act or step is the spiritual act; all the rest is mere repetition of the same a thousand times.”
—Ralph Waldo Emerson (18031882)