(2016-10-25) Direct Product of Two One-Variable Functions
Always well-defined as a two-variable function.
This is actually a special case of a tensor product, so defined:
( f . g ) (x,y) = f (x) g (y)
(2008-10-21) Convolution Product
(or convolution, for short)
Motivational introduction to convolution products and distributions.
Whenever it makes sense, the following integral
(from -¥ to +¥)
is known as the value at point x of the convolution product
of f and g.
f * g (x)
=
ò
f (u) g (x-u) du
So defined, the convolution operator is commutative
(HINT: Change the variable of integration
from u to
w = x-u ). It's also associative :
f * g
* h (x)
=
òò
f (u) g (v) h (x-u-v) du dv
More generally, a convolution product of several functions
at a certain value
x is the integral of their ordinary (pointwise) product over the (oriented)
hyperplane where the sum of their arguments is equal to the constant x.
This viewpoint makes the commutativity and associativity of convolution obvious.
Loosely speaking, a key feature of convolution is that a
convolution product
of two functions is at least as nice a function
as either of its factors.
This happens trivially when that "other factor" is Dirac's
d distribution (a unit spike at point zero)
which is, by definition,
the neutral element for the convolution operation:
d * f
=
f * d
=
f
One night in 1944, my late teacher
Laurent Schwartz
(1915-2002) had the idea that extended functions
(now called distributions ) could be
fully specified by their convolution products over a set
of suitable very well-behaved test functions, up to irrelevant differences
over a set of measure zero. (Not all functions correspond to a distribution,
but locally summable ones do.)
Schwartz was awarded a Fields Medal (in 1950)
for the successful development of that idea.
Test functions are always chosen to be smooth
and so that the derivative of a test function is always a test function.
Because a convolution can be differentiated by differentiating
either factor, we may define the derivative of
any distribution f via its convolution into
any test function j :
f ' * j
=
f * j'
There's no such thing as a non-differentiable distribution !
Although a convolution product is well-defined when one factor
(a function or a distrivution) is nice enough,
it's not at all defined in some cases,
utlimately because the right-hand side of the following defining
equation can be problematic:
ò
(f * g) (u)
j (u) du =
òò
f (x) g (y)
j (x+y) dx dy
This is so because j (x+y)
is never a suitable two-dimensional test function on the plane
(note that it's constant over any anti-diagonal).
However, for example, when the support of either of the two
distributions f or g
is bounded, the right-hand side makes perfect sense for any single-dimensional
test function j
and, therefore, the convolution product
f*g
is well-defined.
For distributions which do not have that kind of restrictions,
the above right-hand side is not guaranted to make sense and the convoltion product
is not necessarily well-defined.
A distribution is uniquely specified when its convolution into any test function
is known, but much than that is required.
Indeed, if we assume that the translation of a test-function is a test function,
all we need to specify a distribution is the value at zero of its convolution product
into any test function.
This means that a distribution is fully specified by a functional
over the set of test functions (by definition, a numerical functional
maps a funtion to a number. Not all such finctionals are acceptable, though.
A distribution is actually a continuous linear funtional
over test-functions.
The above goes to show that all test functions must be infinitely smooth
and sufficiently concentrated about a point to make the relevant integrals
converge. Also making the set of test functions stable by translation
is an easy way to ensure that all parts of a distribution are relevant.
(2008-10-23) Duality between (some) functions and functionals
An Hermitian product defined over dual spaces ("bras" and "kets").
Let's consider a pair f and g of
complex functions of a real variable.
In the same spirit as the above convolution product,
we may define an inner product (endowed with
Hermitian symmetry)
for some such pairs of functions
via the following definite integral (from -¥ to
+¥) whenever it makes sense.
Here, f (u)*
denotes the complex conjugate of f (u).
< f | g >
=
ò
f (u)*g (u) du
This notation
(Dirac's notation)
is firmly linked with
Hermitian symmetry in the minds of all physicists
familiar with Dirac's bra-ket notation (pun intended).
Here, kets are well-behaved
test functions and bras,
which we shall define as the duals of kets,
are the new mathematical animals called distributions,
presented in the next article.
Other introductions to the Theory of distributions usually
forgo the complex conjugation and use a mere pairing
rather than a full-fledged Hermitian product.
They use a comma rather than a vertical bar as a separator:
< f , g >
=
ò
f (u) g (u) du
(2008-10-23) Theory of Distributions
The set of distributions is the
dual of the set of test functions.
A linear form is a linear function which associates
a scalar to a vector.
For a vector space of finite dimension,
the linear forms constitute another vector space of the same dimension
dubbed the dual of the original one.
On the other hand, the dual of a space of infinitely many dimensions need
not be isomorphic to it.
Actually, something stange happens among spaces of infinitely
many dimensions: The smaller the space, the larger its dual...
Thus, loosely speaking, the dual of a very restricted
space of test functions is a very large space
of new mathematical objects called
distributions.
The name "distribution" comes from the fact that those objects provide
a rigorous basis for classical 3-dimensional distributions of
electric charges which need not be ordinary volume densities
but may also be "concentrated" on surfaces, lines or points.
The alternate term of "generalized functions" is best shunned,
because distributions do generalize measure densities rather than
pointwise functions.
(Two functions which differ in finitely many points
correspond to the same distribution, because they are
"physically" undistinguishable.)
The most restricted space of test functions
conceived by Laurent Schwartz
(in 1944) is that of the
smooth functions of compact support.
It is thus the space which yields, by duality, the most general type of
distributions.
Such distributions turn out to be
too general in the context of Fourier analysis
because the Fourier transform of a function of compact support
is never itself a function of compact support.
So, Schwartz introduced a larger space of test functions,
stable under Fourier transform, whose duals are called "tempered distributions"
for which the Fourier transform is well-defined by duality, as explained below.
The support of a function is the
closure of the set of all points for which it's nonzero.
Compactness is a very general topological concept
(a subset of a topological space is compact when every
open cover contains a finite subcover).
In Euclidean spaces of finitely many dimensions,
a set is compact if and only if it's both closed and bounded
(that's the Heine-Borel Theorem ).
Thus, the support of a function of a real variable is compact
when that function is zero outside of a finite
interval.
Examples of smooth functions (i.e.,
infinitely differentiable functions) of compact support are
not immediately obvious. Here is one:
x (x) =
exp (
1
)
if x is between -1 and 1
1-x2
x (x) = 0
elsewhere
(2008-10-22) Schwartz Functions
The smooth functions whose derivatives are all rapidly decreasing.
A function of a vector x is said to be
rapidly decreasing when its product into any
power of ||x||
tends to zero as ||x|| tends to infinity.
Schwartz functions are smooth functions whose
partial derivatives of any order are all rapidly decreasing
in the above sense.
The set of all Schwartz functions is called the
Schwartz Space.
(2008-10-22) Tempered Distributions The natural domain of definition of the Fourier transform.
Not all distributions have a Fourier transform
but tempered ones do.
The Fourier transform of a tempered distribution is a tempered distribution.
Two functions which differ in only finitely many points
(or, more generally, over any set of vanishing
Lebesgue measure)
represent the same tempered distribution.
However, the explicit pointwise formulas giving the
inverse transform of the Fourier transform of a function,
if they yield a function at all, can only yield a function
verifying the following relation:
f (x) = ½
[ f - (x) + f + (x) ]
If it's not continuous,
such a function only has discrete jump discontinuities
where its value is equal to the average of its lower and upper limits.
When a distribution can be represented by a function, it's wise
to equate it to the representative function which which has the above property,
because it's the only one which can be retrieved pointwise
from its Fourier transform without using dubious ad hoc methods.
(2008-10-24) The Involutive Definition of the Fourier Transform
The basic definition for functions extends to tempered distributions.
I am introducing a viewpoint
(the involutive convention) which defines the Fourier transform so it's equal to its own
inverse (i.e., it's an involution ).
This eliminates all the reciprocal coefficients and sign changes which
have hampered the style of generations of scholars.
(The complex conjugation which is part of our definition was
shunned in the many competing historical definitions of the Fourier transform.)
It is consistent with the Hermitian symmetry
imposed above on the "pairing"
of distributions and test functions
to blur the unnecessary distinction between that "pairing" and
a clean Hermitian product.
The Involutive Fourier Transform F
F ( f ) (s)
=
ò
e 2p i sxf (x)* dx
As usual,
the integral is understood to be a definite integral
from -¥ to
+¥.
f (x)* is the
complex conjugate of f (x).
Example: Square function ( P )
and sine cardinal (sinc)
The square function P (x)
= ½ [ sgn(½+x) + sgn(½-x) ]
and the sampling functionf (s) = sinc(ps)
are Fourier transforms of each other.
Proof:
As the square function
P vanishes outside the
interval
[-½, ½] and is equal to 1 on the interior
of that interval, the Fourier transform f
of P is given by:
f (s)
=
ò
½
e 2pisx dx
=
e pis
-
e - pis
=
sin ps
=
sinc ps
-½
2pi s
ps
(2008-11-02) Competing Definitions of the Fourier Transform
Several definitions of the Fourier transform have been used.
Only the above definition makes the Fourier transform
its own inverse. (Well, technically, you could replace
i by -i in that definition
and still obtain an involution, but this
amounts to switching right and left in the orientation of the
complex plane.)
A few competing definitions are tabulated below as pairs of transforms
which are inverses of each other
The first of each pair is usually called the direct
Fourier transform and the other one is the matching inverse
Fourier transform, but the opposite convention can also be used.
The last column gives expressions in terms of the
involutive Fourier transform F
introduced above (and listed first).
Competing Definitions of
the Fourier Transform and its Inverse
1
g (s) =
ò
e 2p i sxf (x)* dx
g = F ( f )
f = F ( g )
2
g (n) =
ò
e -2p i ntf (t) dt
f (t) =
ò
e 2p i ntg (n) dn
g = F ( f )*
f = F ( g* )
3
g (w) =
ò
e - i wtf (t) dt
f (t) =
1
ò
e i wtg (w)
dw
2p
(2008-10-23) Parseval's Theorem (1799)
In modern terms: The Fourier transform is unitary.
The Swiss mathematician Michel
Plancherel (1885-1967) is credited with the modern formulation of the theorem.
The core idea occurs in a statement about series published in 1799
by Antoine Parseval
(1755-1836).
(2016-10-10) Fourier transform of a delayed signal:
A time-delay corresponds to a phase shift in the frequency domain.
(2008-10-24) Important distributions and their Fourier transforms Our definition makes the Fourier transform equal to its own inverse.
The following table can be read both ways:
The right entry is the Fourier transform of the left one and vice-versa.
Pairs of tempered distributions which are Fourier transforms of each other :
(2016-10-24) Product of Two Tempered Distributions (if defined)
It's the Fourier transform of the convolution of their Fourier transforms.
If we weren't using the involutive definition of the
Fourier transform, we would have to replace one of the occurences of
"Fourier transform" in the above definition by "inverse Fourier transform".
Because the convolution of two tempered distributions isn't always defined,
neither is their product in the above sense.
However, the above is consistent with the ordinary (pointwise)
product of continuous functions and does extend well beyond that.
(2008-10-24) The Fourier transform of a Gaussian curve is Gaussian.
The unit Gaussian distribution is a fixed-point of the Fourier involution.
f (x) =
e -p x2
is its own Fourier transform.
Proof :
Let g be the Fourier transform of f.
We have:
g (s) =
ò
e 2p i sxe -p x2
dx
Differentiating both sides, we obtain:
(2008-10-24) Two-dimensional Fourier transform.
Under coherent monochromatic light, a translucent film produces a distant light
whose intensity is the Fourier transform of the film's opacity.
One practical way to observe the "distant" monochromatic image of a translucent plane
is to put it at the focal point of a lens.
The Fourier image is observed at any chosen distance past the lens.
From that Image, an identical lens can be used to reconstruct the original light
in its own focal plane.
Interestingly, that type of setup provides an easy way to observe the convolution
of two images...
Just take a photographic picture of the Fourier transform of the
first image by putting it in the focal plane of your camera and shining a broad
laser beam through it. Make a transparent slide from that picture.
This slide may be then be used as a sort of spatial filter...
This correspond to a well-defined tempered distribution whose (Hermitian) pairing
with a Schwartz test function is the sum of a convergent series.
The cyrilic symbol
(shah) has been used to denote the unit Dirac comb for at
least 45 years (and probably a lot more).
François Roddier was using that notatiom in 1971 as if
it was already well-established.
(2009-07-12) Radon transform (1917)
The relation between a tomographic scan and two-dimensional density.
This transform and its inverse were introduced in 1917 by the Austrian mathematician
Johann Radon (1887-1956).
In the plane, a ray
(a nonoriented straight line)
is uniquely specified by:
The inclination
j of the [upward] normal unit vector
n
(0≤j<p).
The value
r = n.OM
(the same for any point M on the line).
The cartesian equation of such a ray depends on
the parameters j and
r :
x cos j +
y sin j =
r
In the approximation of geometrical optics,
the optical density of a bounded transmission
medium along a given "ray of light" is the logarithm
of the ratio of the incoming light intensity to the outgoing intensity.
That's to say that density = log (opacity) since the opacity of
a substrate is defined as the aforementioned ratio.
In that context, logarithms are usually understood to be decimal
(base 10) logarithms.
Opacities are multiplicative whereas densities are simply additive
along a path. We use the latter exclusively.
This traditional vocabulary forces us to state that
the light-blocking capability of a substrate around a given point is its
optical density per unit of length which we denote by the symbol
m.
It varies with location:
m =
m(x,y) ≥ 0
The (total) density along a straight ray specified by the parameters
r and
j defined above can be expressed by a two-dimensional
integral using
a one-dimensional d distribution:
Rm (r,j) =
òò
m(x,y)
d(r -
x cos j - y sin j)
dx dy
(2016-10-25) Continuity of Functionals
The great simplicity of defining distributions as continuous
functionals on some suitable set of well-behaved test functions is entirely due
the fact that continuity can indeed be well-defined in the realm of
functional analysis.
That notion is the most difficult part of our subject.
Fortunately, the beauty and the simplicity of the theory of distributions
can be enjoyed by taking this deep part of its foundations for granted.
The following discussion is therefore entirely optional for most readers.
) is its own Fourier transform.
