In
mathematics, the Weyl integral (named after
Hermann Weyl) is an operator defined, as an example of
fractional calculus
Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator D
:D f(x) = \frac f(x)\,,
and of the integration ...
, on functions ''f'' on the
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
having integral 0 and a
Fourier series. In other words there is a Fourier series for ''f'' of the form
:
with ''a''
0 = 0.
Then the Weyl integral operator of order ''s'' is defined on Fourier series by
:
where this is defined. Here ''s'' can take any real value, and for integer values ''k'' of ''s'' the series expansion is the expected ''k''-th derivative, if ''k'' > 0, or (−''k'')th indefinite integral normalized by integration from ''θ'' = 0.
The condition ''a''
0 = 0 here plays the obvious role of excluding the need to consider division by zero. The definition is due to
Hermann Weyl (1917).
See also
*
Sobolev space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...
References
*{{springer, first=P.I., last=Lizorkin, id=f/f041230, title=Fractional integration and differentiation
Fourier series
Fractional calculus