Fourier Transforms of Square-integrable Functions
Plancherel theorem allows the Fourier transform to be extended to a unitary operator on the Hilbert space of all square-integrable functions, i.e., all functions satisfying
Therefore it is invertible on L2.
In case f is a square-integrable periodic function on the interval, it has a Fourier series whose coefficients are
The Fourier inversion theorem might then say that
What kind of convergence is right? "Convergence in mean square" can be proved fairly easily:
What about convergence almost everywhere? That would say that if f is square-integrable, then for "almost every" value of x between 0 and 2π we have
This was not proved until 1966 in (Carleson, 1966).
For strictly finitary discrete Fourier transforms, these delicate questions of convergence are avoided.
Read more about this topic: Fourier Inversion Theorem
Famous quotes containing the words transforms and/or functions:
“Now, since our condition accommodates things to itself, and transforms them according to itself, we no longer know things in their reality; for nothing comes to us that is not altered and falsified by our Senses. When the compass, the square, and the rule are untrue, all the calculations drawn from them, all the buildings erected by their measure, are of necessity also defective and out of plumb. The uncertainty of our senses renders uncertain everything that they produce.”
—Michel de Montaigne (1533–1592)
“When Western people train the mind, the focus is generally on the left hemisphere of the cortex, which is the portion of the brain that is concerned with words and numbers. We enhance the logical, bounded, linear functions of the mind. In the East, exercises of this sort are for the purpose of getting in tune with the unconscious—to get rid of boundaries, not to create them.”
—Edward T. Hall (b. 1914)