In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Weyl integral (named after
Hermann Weyl
Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is assoc ...
) is an operator defined, as an example of
fractional calculus, on functions ''f'' on the
unit circle having integral 0 and a
Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
. 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
Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is assoc ...
(1917).
See also
*
Sobolev space
References
*{{springer, first=P.I., last=Lizorkin, id=f/f041230, title=Fractional integration and differentiation
Fourier series
Fractional calculus