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

${\displaystyle \sum _{n=-\infty }^{\infty }a_{n}e^{in\theta }}${\displaystyle \sum _{n=-\infty }^{\infty }a_{n}e^{in\theta }}

with a₀ = 0.

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

${\displaystyle \sum _{n=-\infty }^{\infty }(in)^{s}a_{n}e^{in\theta }}${\displaystyle \sum _{n=-\infty }^{\infty }(in)^{s}a_{n}e^{in\theta }}

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).

• Sobolev space

• Lizorkin, P.I. (2001) [1994], "Fractional integration and differentiation", Encyclopedia of Mathematics , EMS Press

• Fourier series
• Fractional calculus

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