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Final Answers
© 2000-2018   Gérard P. Michon, Ph.D.

(Involutive) Fourier Transform
&  Tempered Distributions

If you want to find the secrets of the Universe,
think in terms of energy, frequency and vibration.

Nikola Tesla  (1856-1943)

Related articles on this site:

Related Links (Outside this Site)

Fourier Transform  by  Eric Weisstein.
Self-Characteristic Distributions  by  Aria Nosratinia.

Wikipedia :   Tempered distributions and Fourier transform   |   Fourier transform   |   Deconvolution
Convergence of Fourier series

Mathematics for the Physical Sciences (1966)  by  Laurent Schwartz  (Dover)
Distributions et transformation de Fourier (1971)  François Roddier  (Ediscience)

VideosThe Fourier Transform and its Applications   by  Brad Osgood
(Stanford University, EE261)   1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30  [ Playlist ]
MIT 6.003 by Dennis Freeman (Fall 2011)   1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | [ Playlist ]
Introducing the Fourier Transform (20:46)  by  Grant Sanderson  (2018-01-26).


Fourier Transform  &  Tempered Distributions

(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.

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
 Vertical bar for alternatives
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.

The dual of the  Schwartz Space  is a set of distributions known as  tempered distributions.

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
Ff ) (s)   =   ò   e 2p i sx  f (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 function   f (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      QED
Vinculum Vinculum
2p i s p s

(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
  g (s)   =    ò   e 2p i sx  f (x)*  dx
 g   =   F ( f )
 f   =    F ( g )
  g (n)   =    ò   e -2p i nt  f (t)  dt

  f (t)   =    ò   e  2p i nt  g (n)  dn
 g   =   F ( f )*
 f   =    F ( g* )
  g (w)   =    ò   e - i wt  f (t)  dt

  f (t)   =     1    ò   e   i wt  g (w)  dw

 Come back later, we're
 still working on this one...

 Antoine Parseval 
 (1755-1836) (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).

 Come back later, we're
 still working on this one...

(2016-10-10)   Fourier transform of a delayed signal:
A time-delay corresponds to a  phase shift  in the frequency domain.

 Come back later, we're
 still working on this one...

(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 :
  f (x)   =    ò   e 2p i sx  g (s)*  ds
  g (s)   =    ò   e 2p i xs  f (x)*  dx
convolutionf1*f2 (x) g1(s)  g2(s)product
Dirac pulsed (x) 1uniform distribution
shifted pulsed (x+a) e 2p i a sphase shift
double spiked (x+a)  +   d (x-a) 2 cos 2pa scosine
d (x+a)  -   d (x-a) 2i sin 2pa ssine
signum functionsgn (x) i / ps 
Heaviside step H (x) = ½ [1+sgn(x)] ½ [d(s) + i/ps] 
square functionP(x) = H(x+½)-H(x-½) sinc pssampling function
triangle functionL (x)  =  P*P (x) sinc2 ps 
Gaussiane -p x2 e -p s2Gaussian
unit comb (shah) Shah function (x) Shah function (s) unit comb (shah)

Evaluation of Certain Fourier Transforms  by  Edmund Lam  (University of Hong-Kong, ELEC 8501, 2008).
Some Special Fourier Transform Pairs  by  James Vickers  (University of Southampton, 2004).
Specific Fourier Transforms  by  Brad Osgood  (Stanford, EE 261, 2008)  #7.
Fourier Transform of Tempered Distributions  by  Brad Osgood  (Stanford, EE 261, 2008)  #13.

(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 sx   e -p x2  dx     Differentiating both sides, we obtain:
  g' (s)   =    -i  ò   e 2p i sx    ( -2pe  -p x)  dx     which we integrate by parts:
  g' (s)   =    +i  ò   ( 2p i s  e 2p i sx )    e  -p x  dx     =     -2pg (s)

So,  g  satisfies the differential equation  dg = -2p s g ds  whose solution is:

g (s)   =   g (0)  e -p s2

Because of a well-known miracleg (0)  =  1.  So   g (s)  =  e -p s2      QED

(2019-10-10)   Central Limit Theorem   (CLT)
Sum of  Independent Identically Distributed  (IID) random variables

The distributions of a sum of two  independent  random variables is the convolution product of their respective distributions.

 Come back later, we're
 still working on this one...

Convolution and the Central Limit Theorem  by  Brad Osgood  (Stanford, EE 261, 2008)  #10.

(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.

The Huygens principle...

 Come back later, we're
 still working on this one...

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... 

 Come back later, we're
 still working on this one...

Fourier Transform and Diffraction  by  Brad Osgood  (Stanford, EE 261, 2008)  #15.

 Simeon Poisson 
 1781-1840 (2008-10-24)   Poisson summation formula.  Sampling formula.
The unit Dirac comb (shah function) is its own Fourier transform.

The unit comb  (Shah function)  is an infinite sum of  Dirac distributions :

Shah function (x)   =   d(x) + d(x-1) + d(x+1) + d(x-2) + d(x+2) + ... + d(x-n) + d(x+n) + ...

This correspond to a well-defined tempered distribution whose (Hermitian) pairing with a  Schwartz test function  is the sum of a convergent series.

 Come back later, we're
 still working on this one...

The cyrilic symbol  Shah function  (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.

Sampling and Interpolation Problem  by  Brad Osgood  (Stanford, EE 261, 2008)  #17.
Wikipedia :   Poisson summation formula   |   Nyquist-Shannon sampling theorem  (Shannon, 1949)

 Max von Laue 
 (1879-1960) (2016-10-24)   Crystallography
Crystals behave as diffraction patterns for X-rays.

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 still working on this one...

Crystal Gazing  by  Brad Osgood  (Stanford, EE 261, 2008)  #16.

(2016-10-30)   Spectrum and Support
The spectrum of a distribution is the support of its Fourier transform.

The  support  of a numerical function defined over some topological space is simply the  closure  of the set of points where that function isn't zero.

The support of a  distribution  ...

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 still working on this one...

Support of a distribution

(2016-10-30)   Quasicrystals
Discrete distributions with discrete spectra.

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 still working on this one...

If the Riemann Hypothesis is true,  then the nontrivial zeroes of the  Riemann Zeta function  form a quasicrystal in one dimension.  In 2008, Freeman Dyson  suggested that a classification of all one-dimensional quasicrystals is a worthy goal which might eventually yield a proof of the Riemann hypothesis (RH).

Aperiodic Penrose tilings (1977)   |   Quasicrystal
Quasicrystal and the RH by John Baez  (n-category café, 2013-06-14).
Quasicrystal and the RH by John Baez  (Mathoverflow, 2013-06+15).

(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

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 still working on this one...

(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.

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 still working on this one...

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