Differential operators like \ frac { d } { dx } are also linear transformations, this time from differentiable functions into integrable functions.
42.
Another convention reserves the term " continuous probability distribution " for Lebesgue integrable function \, f defined on the real numbers such that
43.
In the former case, one may regard the Gelfand representation as a far-reaching generalization of the Fourier transform of an integrable function.
44.
It remains to assume that the integral criterion holds for indicator functions and prove that it holds for general Riemann-integrable functions as well.
45.
In 1907, Frigyes Riesz and Ernst Sigismund Fischer independently proved that the space of square Lebesgue-integrable functions is a complete metric space.
46.
For the definition of the Fourier transform of a tempered distribution, let and be integrable functions, and let and be their Fourier transforms respectively.
47.
This may be done, by proving, that there exists no non-zero square integrable function on G orthogonal to all the irreducible characters.
48.
The theorem is true also for functions belonging only to the space of locally-integrable functions, therefore the theorem implies also the following result.
49.
For example, her 1936 paper proves a version of Rolle's theorem for Denjoy Perron integrable functions using different techniques from the standard proofs:
50.
For locally integrable functions that decay at infinity or are of exponential type, the integral can be understood to be a ( proper ) Lebesgue integral.
