If and
are two absolutely integrable functions on a Euclidean space
, then the convolution
of the two functions is defined by the formula
A simple application of the Fubini-Tonelli theorem shows that the convolution is well-defined almost everywhere, and yields another absolutely integrable function. In the case that
,
are indicator functions, the convolution simplifies to
where denotes Lebesgue measure. One can also define convolution on more general locally compact groups than
, but we will restrict attention to the Euclidean case in this post.
The convolution can also be defined by duality by observing the identity
for any bounded measurable function . Motivated by this observation, we may define the convolution
of two finite Borel measures on
by the formula
for any bounded (Borel) measurable function , or equivalently that
for all Borel measurable . (In another equivalent formulation:
is the pushforward of the product measure
with respect to the addition map
.) This can easily be verified to again be a finite Borel measure.
If and
are probability measures, then the convolution
also has a simple probabilistic interpretation: it is the law (i.e. probability distribution) of a random varible of the form
, where
are independent random variables taking values in
with law
respectively. Among other things, this interpretation makes it obvious that the support of
is the sumset of the supports of
(when the supports are compact; the situation is more subtle otherwise) and
, and that
will also be a probability measure.
While the above discussion gives a perfectly rigorous definition of the convolution of two measures, it does not always give helpful guidance as to how to compute the convolution of two explicit measures (e.g. the convolution of two surface measures on explicit examples of surfaces, such as the sphere). In simple cases, one can work from first principles directly from the definition (2), (3), perhaps after some application of tools from several variable calculus, such as the change of variables formula. Another technique proceeds by regularisation, approximating the measures involved as the weak limit (or vague limit) of absolutely integrable functions
(where we identify an absolutely integrable function with the associated absolutely continuous measure
) which then implies (assuming that the sequences
are tight) that
is the weak limit of the
. The latter convolutions
, being convolutions of functions rather than measures, can be computed (or at least estimated) by traditional integration techniques, at which point the only difficulty is to ensure that one has enough uniformity in
to maintain control of the limit as
.
A third method proceeds using the Fourier transform
of (and of
). We have
and so one can (in principle, at least) compute by taking Fourier transforms, multiplying them together, and applying the (distributional) inverse Fourier transform. Heuristically, this formula implies that the Fourier transform of
should be concentrated in the intersection of the frequency region where the Fourier transform of
is supported, and the frequency region where the Fourier transform of
is supported. As the regularity of a measure is related to decay of its Fourier transform, this also suggests that the convolution
of two measures will typically be more regular than each of the two original measures, particularly if the Fourier transforms of
and
are concentrated in different regions of frequency space (which should happen if the measures
are suitably “transverse”). In particular, it can happen that
is an absolutely continuous measure, even if
and
are both singular measures.
Using intuition from microlocal analysis, we can combine our understanding of the spatial and frequency behaviour of convolution to the following heuristic: a convolution should be supported in regions of phase space
of the form
, where
lies in the region of phase space where
is concentrated, and
lies in the region of phase space where
is concentrated. It is a challenge to make this intuition perfectly rigorous, as one has to somehow deal with the obstruction presented by the Heisenberg uncertainty principle, but it can be made rigorous in various asymptotic regimes, for instance using the machinery of wave front sets (which describes the high frequency limit of the phase space distribution).
Let us illustrate these three methods and the final heuristic with a simple example. Let be a singular measure on the horizontal unit interval
, given by weighting Lebesgue measure on that interval by some test function
supported on
:
Similarly, let be a singular measure on the vertical unit interval
given by weighting Lebesgue measure on that interval by another test function
supported on
:
We can compute the convolution using (2), which in this case becomes
and we thus conclude that is an absolutely continuous measure on
with density function
:
In particular, is supported on the unit square
, which is of course the sumset of the two intervals
and
.
We can arrive at the same conclusion from the regularisation method; the computations become lengthier, but more geometric in nature, and emphasises the role of transversality between the two segments supporting and
. One can view
as the weak limit of the functions
as (where we continue to identify absolutely integrable functions with absolutely continuous measures, and of course we keep
positive). We can similarly view
as the weak limit of
Let us first look at the model case when , so that
are renormalised indicator functions of thin rectangles:
By (1), the convolution is then given by
where is the intersection of two rectangles:
When lies in the square
, one readily sees (especially if one draws a picture) that
consists of an
square and thus has measure
; conversely, if
lies outside
,
is empty and thus has measure zero. In the intermediate region,
will have some measure between
and
. From this we see that
converges pointwise almost everywhere to
while also being dominated by an absolutely integrable function, and so converges weakly to
, giving a special case of the formula (4).
Exercise 1 Use a similar method to verify (4) in the case that
are continuous functions on
. (The argument also works for absolutely integrable
, but one needs to invoke the Lebesgue differentiation theorem to make it run smoothly.)
Now we compute with the Fourier-analytic method. The Fourier transform of
is given by
where we abuse notation slightly by using to refer to the one-dimensional Fourier transform of
. In particular,
decays in the
direction (by the Riemann-Lebesgue lemma) but has no decay in the
direction, which reflects the horizontally grained structure of
. Similarly we have
so that decays in the
direction. The convolution
then has decay in both the
and
directions,
and by inverting the Fourier transform we obtain (4).
Exercise 2 Let
and
be two non-parallel line segments in the plane
. If
is the uniform probability measure on
and
is the uniform probability measure on
, show that
is the uniform probability measure on the parallelogram
with vertices
. What happens in the degenerate case when
and
are parallel?
Finally, we compare the above answers with what one gets from the microlocal analysis heuristic. The measure is supported on the horizontal interval
, and the cotangent bundle at any point on this interval points in the vertical direction. Thus, the wave front set of
should be supported on those points
in phase space with
,
and
. Similarly, the wave front set of
should be supported at those points
with
,
, and
. The convolution
should then have wave front set supported on those points
with
,
,
,
,
, and
, i.e. it should be spatially supported on the unit square and have zero (rescaled) frequency, so the heuristic predicts a smooth function on the unit square, which is indeed what happens. (The situation is slightly more complicated in the non-smooth case
, because
and
then acquire some additional singularities at the endpoints; namely, the wave front set of
now also contains those points
with
,
, and
arbitrary, and
similarly contains those points
with
,
, and
arbitrary. I’ll leave it as an exercise to the reader to compute what this predicts for the wave front set of
, and how this compares with the actual wave front set.)
Exercise 3 Let
be the uniform measure on the unit sphere
in
for some
. Use as many of the above methods as possible to establish multiple proofs of the following fact: the convolution
is an absolutely continuous multiple
of Lebesgue measure, with
supported on the ball
of radius
and obeying the bounds
for
and
for
, where the implied constants are allowed to depend on the dimension
. (Hint: try the
case first, which is particularly simple due to the fact that the addition map
is mostly a local diffeomorphism. The Fourier-based approach is instructive, but requires either asymptotics of Bessel functions or the principle of stationary phase.)
23 comments
Comments feed for this article
26 July, 2013 at 4:13 pm
Fred Lunnon
Missing sentence perhaps, after
This can easily be verified to again be a finite Borel measure. As
[Sorry, that sentence fragment has now been deleted. -T.]
26 July, 2013 at 10:26 pm
Alok Tiwari
Is there a typo in equation 1?
[Corrected, thanks – T.]
27 July, 2013 at 1:57 am
mircea
If you utilize exterior product then you can extend formula (3) also to currents; the other ideas also nicely extend to this, plus you get a nice geometric operation. This is a way of producing new submanifolds, e.g. the Windows symbol (as an oriented submanifold) is the “convolution” of the two curves dividing it in 4. Also, e.g. for an oriented circle C in R² the convolution of C with itself is zero.
27 July, 2013 at 3:34 am
omar aboura
In “Another technique proceeds by reularisation, …” reularisation should be regularisation.
In “…given by weighting Lebesgue masure…” masure should be measure.
In the equation after “We can compute the convolution {\mu*\nu} using (2), which in this case becomes”, \int_{{\bf R}^n} should be \int_{{\bf R}^2}.
In “…the actual waave front set” waave should be wave.
[Corrected, thanks – T.]
27 July, 2013 at 5:31 am
Anonymous
In the paragraph beginning with
“A third method proceeds using the Fourier transform…”
should the integration variable be x in the definition of Fourier transform of measure mu?
27 July, 2013 at 5:33 am
John Mangual
* The object you’ve described in Exercise 2 is the Minkowski Sum of two line segments in the plane. I know what that is. You offer
three4 different ways to work out the problem.If I remember correctly, you can approximate measures as linear combinations of step functions (… dusting off analysis notes …) So you’re defining a measure version of Minkowski addition? I need to draw pictures.
* Using the identity:
we see that a point measure is infinitely spread out in space. That’s definitely one way of expressing wave-particle duality. Wavepackets are a compromise, kind-of localized in space and kind-of localized in time.
In Ex 2, you seem to be describing a “quantum line segment” localized along a line, or even a “quantum parallelogram”. I wonder now what these shapes look like in momentum space?
* Naively, I picture the wave-front as a bunch of circle emanating from the point with increasing radius. I am having trouble reconciling Hormander’s definition
with the mental image given from optics
27 July, 2013 at 6:00 am
Edgar's Creative
Reblogged this on Edgar's Creative and commented:
Another brilliant take by Terry Tao on an important Mathematical subject. Breathtaking!
27 July, 2013 at 9:22 pm
Kamran Alam Khan
Reblogged this on Observer.
30 July, 2013 at 6:48 am
Kamran Alam Khan
Reblogged this on Observer.
5 August, 2013 at 4:10 am
Marcelo de Almeida
Reblogged this on Being simple and commented:
About Bessel functions and other special functions, I suggest the well-known online book DLMF: http://dlmf.nist.gov/10
5 August, 2013 at 6:23 am
Eytan Paldi
This online updated version of the original A&S is very useful indeed! (for example, Auluck’s formula – used in the upper bound for
– appears as formula 10.22.5 in DLMF , but not in the original A&S !)
5 August, 2013 at 7:54 am
Marcelo de Almeida
This online book is/was very important to me.
5 January, 2016 at 6:29 pm
Anonymous
What is the “surface measure” in this note? I read throughout your math 245abc notes but I don’t see this concept.
[Surface measure on a k-dimensional manifold is the restriction of k-dimensional Hausdorff measure to that manifold. It can also be defined locally using a local parameterisation of the manifold as the pushforward of k-dimensional Lebesgue measure, multiplied by an appropriate Jacobian factor; see Exercise 12 of https://terrytao.wordpress.com/2009/05/19/245c-notes-5-hausdorff-dimension-optional/ . A third definition is to take Lebesgue measure restricted to an
-neighbourhood of the manifold, normalise by
, and take weak limits. -T.]
28 January, 2017 at 7:17 pm
Abraham
Can anyone point me to a reference/textbook that shows that the support of the convolution of two probability measures, f \ast g, defined on uncountable sample spaces is the sumset of the support of the individual measures? I can only find results of the form supp{ f\ast g} \subseteq supp{f} \oplus supp{g}. I can see how it follows when the sample space is countable.
30 January, 2017 at 9:10 pm
Terence Tao
Use the inequality
for any two balls
, and the fact that
whenever
is in the support of
.
16 June, 2017 at 12:57 pm
Anonymous
Would you elaborate what you mean by “The convolution
can also be defined by duality…” ? What is the duality here?
22 June, 2017 at 5:29 pm
Terence Tao
A locally integrable function
(or more generally, a measure or a distribution) can be defined (up to almost everywhere equivalence) by specifying its inner products
against a suitable class of test functions
in a predual space. (In some cases, a representation theorem, such as the Riesz representation theorem, may need to be invoked in order to guarantee that
actually exists.)
30 April, 2018 at 8:03 am
Anonymous
In Folland’s Real Analysis, it is said that if
is the “closure” of
, then
. But in Wolff’s harmonic analysis lecture notes, the author states that
where the “closure” is not used. Which version is correct?
30 April, 2018 at 8:25 am
Terence Tao
The answer depends on the definition of support (set-theoretic support, topological support, or essential support). But for compactly supported functions both inclusions hold for all notions of support.
1 May, 2018 at 4:31 am
Anonymous
Presumably, there is a default version for an analyst when the definition is not given explicitly (e.g. I don’t see which definition is used in this post)? Folland does states explicitly that the support of
is defined topologically as the closure of the subset of
where
is non-zero. Wolff did not give the definition but restricted the discussion mostly in the Euclidean space in his notes and when he wrote

functions do not need to have compact support.
1 May, 2018 at 10:33 am
Terence Tao
If you keep reading Wolff’s notes until the point where the concept of support is actually used, you should be able to determine from that context what the intended definition of support is, and whether the closure operation was implicitly omitted. (See also my general advice on how to read mathematical texts without “crashing” due to “compilation errors”.)
16 November, 2018 at 8:18 am
Joaquin
Insignificant typo in the Fourier transform of mu. Integration variable must be x.
[Corrected, thanks – T.]
2 July, 2019 at 2:44 pm
Alex Iosevich
The bound in Exercise 3 (for dimensions three and higher) appears to depend only on non-vanishing curvature. At least for the small |x| bound, smoothness does not appear to play any role (beyond what is needed to define curvature).