In mathematics, the Weyl integral (named after Hermann Weyl) is an operator defined, as an example of fractional calculus, on functions f on the unit circle having integral 0 and a Fourier series. In other words there is a Fourier series for f of the form

with a0=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 fromθ=0.

The condition a0=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

References