Derivatives of the Fourier transform #
In this file we compute the Fréchet derivative of the Fourier transform of f
, where f
is a
function such that both f
and v ↦ ‖v‖ * ‖f v‖
are integrable. Here the Fourier transform is
understood as an operator (V → E) → (W → E)
, where V
and W
are normed ℝ
-vector spaces
and the Fourier transform is taken with respect to a continuous ℝ
-bilinear
pairing L : V × W → ℝ
and a given reference measure μ
.
We also investigate higher derivatives: Assuming that ‖v‖^n * ‖f v‖
is integrable, we show
that the Fourier transform of f
is C^n
.
We give specialized versions of these results on inner product spaces (where L
is the scalar
product) and on the real line, where we express the one-dimensional derivative in more concrete
terms, as the Fourier transform of -2πI x * f x
(or (-2πI x)^n * f x
for higher derivatives).
Main definitions and results #
We introduce two convenience definitions:
VectorFourier.fourierSMulRight L f
: givenf : V → E
andL
a bilinear pairing betweenV
andW
, then this is the functionfun v ↦ -(2 * π * I) (L v ⬝) • f v
, fromV
toHom (W, E)
. This is essentiallyContinousLinearMap.smulRight
, up to the factor- 2πI
designed to make sure that the Fourier integral offourierSMulRight L f
is the derivative of the Fourier integral off
.VectorFourier.fourierPowSMulRight
is the higher order analogue for higher derivatives:fourierPowSMulRight L f v n
is informally(-(2 * π * I))^n (L v ⬝)^n • f v
, in the space of continuous multilinear mapsW [×n]→L[ℝ] E
.
With these definitions, the statements read as follows, first in a general context
(arbitrary L
and μ
):
VectorFourier.hasFDerivAt_fourierIntegral
: the Fourier integral off
is differentiable, with derivative the Fourier integral offourierSMulRight L f
.VectorFourier.differentiable_fourierIntegral
: the Fourier integral off
is differentiable.VectorFourier.fderiv_fourierIntegral
: formula for the derivative of the Fourier integral off
.VectorFourier.hasFTaylorSeriesUpTo_fourierIntegral
: under suitable integrability conditions, the Fourier integral off
has an explicit Taylor series up to orderN
, given by the Fourier integrals offun v ↦ fourierPowSMulRight L f v n
.VectorFourier.contDiff_fourierIntegral
: under suitable integrability conditions, the Fourier integral off
isC^n
.VectorFourier.iteratedFDeriv_fourierIntegral
: under suitable integrability conditions, explicit formula for then
-th derivative of the Fourier integral off
, as the Fourier integral offun v ↦ fourierPowSMulRight L f v n
.
These statements are then specialized to the case of the usual Fourier transform on
finite-dimensional inner product spaces with their canonical Lebesgue measure (covering in
particular the case of the real line), replacing the namespace VectorFourier
by
the namespace Real
in the above statements.
We also give specialized versions of the one-dimensional real derivative (and iterated derivative)
in Real.deriv_fourierIntegral
and Real.iteratedDeriv_fourierIntegral
.
Send a function f : V → E
to the function f : V → Hom (W, E)
given by
v ↦ (w ↦ -2 * π * I * L (v, w) • f v)
. This is designed so that the Fourier transform of
fourierSMulRight L f
is the derivative of the Fourier transform of f
.
Equations
- VectorFourier.fourierSMulRight L f v = -(2 * ↑Real.pi * Complex.I) • ContinuousLinearMap.smulRight (L v) (f v)
Instances For
The w
-derivative of the Fourier transform integrand.
Main theorem of this section: if both f
and x ↦ ‖x‖ * ‖f x‖
are integrable, then the
Fourier transform of f
has a Fréchet derivative (everywhere in its domain) and its derivative is
the Fourier transform of smulRight L f
.
The formal multilinear series whose n
-th term is
(w₁, ..., wₙ) ↦ (-2πI)^n * L v w₁ * ... * L v wₙ • f v
, as a continuous multilinear map in
the space W [×n]→L[ℝ] E
.
This is designed so that the Fourier transform of v ↦ fourierPowSMulRight L f v n
is the
n
-th derivative of the Fourier transform of f
.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Decomposing fourierPowSMulRight L f v n
as a composition of continuous bilinear and
multilinear maps, to deduce easily its continuity and differentiability properties.
If ‖v‖^n * ‖f v‖
is integrable for all n ≤ N
, then the Fourier transform of f
is C^N
.
If ‖v‖^n * ‖f v‖
is integrable for all n ≤ N
, then the n
-th derivative of the Fourier
transform of f
is the Fourier transform of fourierPowSMulRight L f v n
,
i.e., (L v ⬝) ^ n • f v
.
The Fréchet derivative of the Fourier transform of f
is the Fourier transform of
fun v ↦ -2 * π * I ⟪v, ⬝⟫ f v
.
The Fréchet derivative of the Fourier transform of f
is the Fourier transform of
fun v ↦ -2 * π * I ⟪v, ⬝⟫ f v
.
If ‖v‖^n * ‖f v‖
is integrable, then the Fourier transform of f
is C^n
.
If ‖v‖^n * ‖f v‖
is integrable, then the n
-th derivative of the Fourier transform of f
is
the Fourier transform of fun v ↦ (-2 * π * I) ^ n ⟪v, ⬝⟫^n f v
.