Weyl integral

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

[math]\displaystyle{ \sum_{n=-\infty}^{\infty} a_n e^{in \theta} }[/math]

with a₀ = 0.

Then the Weyl integral operator of order s is defined on Fourier series by

[math]\displaystyle{ \sum_{n=-\infty}^{\infty} (in)^s a_n e^{in\theta} }[/math]

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 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

• Hazewinkel, Michiel, ed. (2001), "Fractional integration and differentiation", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=f/f041230