Weil Differential

In mathematics, the Weyl differential integral is an operator defined on the integrable functions f of the unit circle ( $2\pi$ ${\ displaystyle 2 \ pi}$ $2 \ pi$ - periodic) with zero mean (i.e., the integral of f over the period is 0). In other words, the function f can be expanded in a Fourier series :

f(\varphi )=\sum _{n=-\infty }^{\infty }a_{n}e^{in\varphi }

{\ displaystyle f (\ varphi) = \ sum _ {n = - \ infty} ^ {\ infty} a_ {n} e ^ {in \ varphi}}

{\ displaystyle f (\ varphi) = \ sum _ {n = - \ infty} ^ {\ infty} a_ {n} e ^ {in \ varphi}}

Where $a_{0}=0$ ${\ displaystyle a_ {0} = 0}$ ${\ displaystyle a_ {0} = 0}$ , or:

f(\varphi )=\sum '_{n}a_{n}e^{in\varphi }

{\ displaystyle f (\ varphi) = \ sum '_ {n} a_ {n} e ^ {in \ varphi}}

{\ displaystyle f (\ varphi) = \ sum '_ {n} a_ {n} e ^ {in \ varphi}}

,

where is the symbol $\sum '_{n}$ ${\ displaystyle \ sum '_ {n}}$ ${\ displaystyle \ sum '_ {n}}$ denotes summation over all natural $n$ ${\ displaystyle n}$ $n$ except 0.

Weyl integral of order $\alpha >0$ ${\ displaystyle \ alpha> 0}$ $\ alpha> 0$ is determined by expansion in a Fourier series as:

f^{\alpha }(\varphi )=\sum '_{n}{\frac {a_{n}e^{in\varphi }}{(in)^{\alpha }}}

{\ displaystyle f ^ {\ alpha} (\ varphi) = \ sum '_ {n} {\ frac {a_ {n} e ^ {in \ varphi}} {(in) ^ {\ alpha}}}}

{\ displaystyle f ^ {\ alpha} (\ varphi) = \ sum '_ {n} {\ frac {a_ {n} e ^ {in \ varphi}} {(in) ^ {\ alpha}}}}

,

and the Weil derivative of order $\beta >0$ ${\ displaystyle \ beta> 0}$ ${\ displaystyle \ beta> 0}$ defined as:

f_{\beta }(\varphi )={\frac {\partial ^{n}}{\partial \varphi ^{n}}}f_{n-\beta }(\varphi )

{\ displaystyle f _ {\ beta} (\ varphi) = {\ frac {\ partial ^ {n}} {\ partial \ varphi ^ {n}}} f_ {n- \ beta} (\ varphi)}

{\ displaystyle f _ {\ beta} (\ varphi) = {\ frac {\ partial ^ {n}} {\ partial \ varphi ^ {n}}} f_ {n- \ beta} (\ varphi)}

.

Thus, the Weyl differential integral is completely defined.

Condition $a_{0}=0$ ${\ displaystyle a_ {0} = 0}$ ${\ displaystyle a_ {0} = 0}$ necessary in these definitions, since otherwise division by 0 would occur.

This definition was introduced by Hermann Weil in 1917.

Links

Lizorkin, PI (2001), "Fractional integration and differentiation", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104

Weil Differential

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