A = O^{t} D O

with

O

D = diag (ξ_{1}, \dots, ξ_{n})

. Part a and the tensor-product property of the Fourier transform give

F [e^{i D x \cdot x /2}] (λ) = \prod_{j = 1}^{n} (\frac{2 π}{∣ ξ _{j} ∣} e^{iπ sgn (ξ_{j}) /4} e^{- i λ_{j}^{2} / (2 ξ_{j})}) .

Thus

F [e^{i D x \cdot x /2}] (λ) = \frac{( 2 π ) ^{n}}{∣ d e t D ∣} exp [\frac{iπ}{4} sgn (D) - \frac{i}{2} D^{- 1} λ \cdot λ] .

Applying the pullback rule from part c to the orthogonal change of variables, for which

∣ det O ∣ = 1

, replaces

D

A

and

D^{- 1}

A^{- 1}

. Since determinant and signature are invariant under orthogonal conjugation,

[e^{i A x \cdot x /2}] (λ) = \frac{( 2 π ) ^{n}}{∣ det A ∣} exp [\frac{iπ}{4} sgn (A) - \frac{i}{2} (A^{- 1} λ) \cdot λ] .

 Ancestors (10)

 Discussion (0)

There are no discussions about this article yet.

There are currently no matching articles.