# Coco notes — Generalizations of Fourier evaluation

*by*Phil Tadros

Once I first discovered about Fourier sequence and integrals, I hated it

as a result of it appeared like a set of many ad-hoc definitions,

formally associated however very totally different.

To have a wider view of the topic, it helped me to appreciate that

Fourier sequence and integrals are a specific case of not one,

however many various constructions. Thus, they are often generalized in

broadly totally different instructions, resulting in in another way flavored views of

the unique idea.

$newcommand{R}{mathbf{R}}$

$newcommand{Z}{mathbf{Z}}$

$newcommand{Q}{mathbf{Q}}$

$newcommand{C}{mathbf{C}}$

$newcommand{U}{mathbf{U}}$

$newcommand{T}{mathbf{T}}$

$newcommand{ud}{mathrm{d}}$

- First you might have the
**4 traditional instances**that you could be study in

faculty: Fourier sequence of periodic capabilities, Fourier

transforms of integrable capabilities, the discrete-time Fourier

rework and the discrete Fourier rework.

These 4 traditional instances are elementary and you need to study

their definition and properties by coronary heart. - Then, you study
**sampling idea**and also you see that

a number of the traditional instances could also be obtained from the others.

For instance, the discrete Fourier rework might be

thought-about a specific case of Fourier sequence of a

periodic operate.

These relationships might be neatly organized within the

so-called Fourier-Poisson dice. - Later, you study
**distribution idea**, that

supplies a typical framework for indicators and their

samples utilizing Dirac combs. Thus every of the 4

traditional instances arises as a specific case of the

Fourier rework of tempered distributions on the

actual line. - A really totally different generalization is given by
**Pontryagin duality**. This begins by realizing that

the area of definition of every traditional case has

all the time the construction of a commutative group ($R$,

$Z$, $S^1$ or $Z/NZ$). Then, Pontryagin duality

supplies a basic development for Fourier evaluation

on commutative teams, and the 4 traditional instances

are explicit instances of it. - By enjoyable the situation of commutativity, you get
**non-commutative harmonic evaluation**. The case of a compact

non-commutative group is described utterly by the

Paley-Wiener theorem, and the final non-compact

non-commutative case is a big downside in illustration

idea, of which a lot is thought; particularly if the group has

some extra construction (semisimple, solvable). - The subsequent step is
**harmonic evaluation on homogeneous**. It seems that the group construction is just not

areas

important, and you are able to do virtually every thing simply by having a

group performing in your area, which needn’t be itself a bunch.

For instance, the sphere $S^2$ is just not a bunch, however there’s the

group of $3D$ rotations performing over it, and this results in

spherical harmonics. - Lastly there’s
**spectral geometry**, additionally referred to as the

spectral evaluation of the Laplace-Beltrami operator. In case your

area is only a potato (a compact Riemannian manifold), there

isn’t any group in anyway performing on it, however you continue to have a

Laplace-Beltrami operator, it has a discrete spectrum, and also you

can do the analogue of Fourier sequence on it. A big a part of

the traditional outcomes of Fourier sequence prolong to this case,

besides every thing associated to convolution—which is outlined

essentially utilizing the group construction.

Thus, what occurs whenever you ask a mathematician, *“what’s
Fourier evaluation?”* ?

If they’re a **actual analyst**, they’ll say that Fourier evaluation

are a set of examples within the research of tempered distributions.

If they’re an **algebraist**, they’ll say that Fourier evaluation

is a really explicit case of one-dimensional illustration idea.

If they’re a **geometer**, they’ll say that Fourier evaluation is

a specific case of spectral geometry for trivial flat manifolds.

Lastly, for those who ask a **complicated analyst**, they’ll say that

Fourier sequence are simply Taylor sequence evaluated on the unit circle.

And all of them shall be proper.

## 1. The 4 traditional instances

The **traditional instances** of Fourier evaluation are used to precise an

arbitrary operate $f(x)$ as a linear mixture of sinusoidal capabilities

of the shape $xto e^{ixi x}$. There are **4** instances, relying on

the area the place $x$ belongs.

### 1.1. Fourier sequence

Any periodic operate

$$

f:S^1toR

$$

might be expressed as a numerable linear mixture of sinusoidal waves. That is

referred to as the *Fourier sequence* of $f$

$$

f(theta) = sum_{ninZ} a_n e^{intheta}

$$

and the coefficients $a_n$ are computed as integrals of $f$

$$

a_n = frac{1}{2pi}int_{S^1} f(theta) e^{-intheta}

mathrm{d}theta

$$

### 1.2. Fourier rework

An arbitrary (integrable) operate

$$

f:RtoR

$$

might be expressed as a linear mixture of sinusoidal waves. The

coefficients of this linear mixture are referred to as the *Fourier
integral* of $f$, also referred to as Fourier rework or attribute

operate of $f$, relying on the context. Thus, $f$ is represented

as

$$

f(x) = int_R a(xi) e^{ixi x} mathrm{d} xi

$$

That is precisely analogous to the Fourier sequence above, however now the coefficients

$a$ of the linear mixture are listed by a steady index

$xiinmathbf{R}$ as an alternative of a discrete index $ninmathbf{Z}$. The values

of $a(xi)$ might be recovered by integrating once more the operate $f$:

$$

a(xi) = frac{1}{2pi}int_R f(x) e^{-ixi x} mathrm{d} x

$$

Discover that, even when their formulation look fairly related, the Fourier

sequence is just not a specific case of the Fourier rework. For

instance, a periodic operate isn’t integrable over the actual line

except it’s identically zero. Thus, you can not compute the Fourier

rework of a periodic operate.

### 1.3. Discrete Fourier rework

Within the finite case, you possibly can categorical any vector

start{equation}

(f_1, f_2, ldots f_N)

finish{equation}

as a linear mixture of “oscillating” vectors:

start{equation}

f_k = sum_l a_l e^{frac{2pi}{N}ikl}

finish{equation}

That is referred to as the *discrete Fourier rework*.

The coefficients $a_l$ might be recovered by inverting the matrix $M_{kl} =

e^{frac{2pi}{N}ikl}$, which is unitary. Thus

$$

a_l = frac{1}{N}sum_k f_k e^{-frac{2pi}{N}ikl}

$$

### 1.4. Discrete-time Fourier rework

Lastly, in case you have a doubly-infinite sequence:

start{equation}

ldots,f_{-2},f_{-1},f_0,f_1,f_2,ldots

finish{equation}

you possibly can categorical it as a linear mixture (integral) of sinusoidal capabilities

sampled on the integers, which is kind of a factor:

start{equation}

f_n = int_{S^1} a(theta) e^{intheta}

mathrm{d}theta

finish{equation}

The coefficients $a(theta)$ of this infinite linear mixture might be

recovered as a linear mixture of all of the values of $f$:

start{equation}

a(theta) = frac{1}{2pi}sum_n f_n e^{-intheta}

finish{equation}

Discover these two formulation are precisely the identical as Fourier sequence, however

reversing the roles of $a$ and $f$.

This is a crucial symmetry.

## 2. Pontryagin duality

Pontryagin duality extracts the essence of the definitions of Fourier

sequence, Fourier integrals and discrete Fourier transforms. The principle thought is

that we’ve got a *spatial area* $G$ and a *frequency area* $G^*$.

Then, any operate outlined on the spatial area

start{equation}

f:Gtomathbf{R}

finish{equation}

might be expressed as a linear mixture of sure capabilities $E$, listed by

the frequencies

start{equation}

f(x) = int_{G^*} a(xi) E(x,xi) mathrm{d} xi

finish{equation}

Right here the coefficients $a$ rely on the operate $f$ however the capabilities $E$

rely solely on the group $G$; they’re referred to as the *characters* of $G$.

The coefficients $a$ might be discovered by computing integrals over the spatial

area:

start{equation}

a(xi) = int_G f(x) overline{E(x,xi)} mathrm{d} xi

finish{equation}

the place the bar denotes complicated conjugation.

Discover that these formulation embody Fourier sequence, Fourier integrals, the

DFT and the DTFT as explicit instances, in response to the next desk

area | freq. | evaluation | synthesis | |

$G$ | $G^*$ | $displaystylewidehat{f}(xi)=int_G f(x) overline{E(xi,x)}ud x$ | $displaystyle f(x)=int_{G^*} widehat{f}(xi)E(xi,x)udxi$ | |

FS | $S^1$ | $Z$ |
$displaystyle f_n = frac{1}{2pi}int_0^{2pi} f(theta)e^{-intheta}udtheta$ |
$displaystyle f(theta)=sum_{ninmathbf{Z}} f_n |

FT | $R$ | $R$ | $displaystylewidehat{f}(xi)=frac{1}{sqrt{2pi}}int_R f(x)e^{-ixi x}ud x$ | $displaystyle f(x)=frac{1}{sqrt{2pi}}int_{mathbf{R}} widehat{f}(xi)e^{ixi x}ud xi$ |

DFT | $Z_N$ | $Z_N$ | $displaystylewidehat{f}_k=frac{1}{N}sum_{n=0}^{N-1}f_n,e^{-2pi ikn/N}$ | $displaystyle f_n=sum_{okay=0}^{N-1}widehat{f}_k,e^{2pi ikn/N}$ |

DTFT | $Z$ | $S^1$ | $displaystyle widehat{f}(theta)=sum_{ninmathbf{Z}} f_n e^{-intheta}$ | $displaystyle f_n = frac{1}{2pi}int_0^{2pi}widehat{f}(theta)e^{intheta}udtheta$ |

### 2.1. Domestically compact abelian teams

A topological group is a bunch along with a topology appropriate with the

group operation. A morphism between two topological teams is a mapping

which is on the identical time steady and a bunch morphism.

Right here we’re considering domestically compact abelian teams (LCAG). We

will denote the group operation by $x+y$, and the inverse of a bunch aspect

$x$ by $-x$.

The canonical instance of LCAG is $R^n$ with the same old topology and

the operation of sum of vectors. One other instance of LCAG is the

multiplicative group $U$ of complicated numbers of norm 1, which

topologically coincides with the unit circle $S^1$, and is isomorphic

to the additive group of actual numbers modulo $2pi$ referred to as the

one-dimensional torus $T=R/2piZ$. Different examples are any

finite abelian group with the discrete topology; or $Z$, the

additive group of integers with the discrete topology.

The group $U$ is essential within the following dialogue. It might

be denoted multiplicatively (by contemplating its parts as complicated numbers),

or additively (by contemplating its parts as angles). Each notations are

used henceforth, and they’re linked by the relation

[

e^{ialpha}e^{ibeta}

=

e^{i(alpha+beta)}

]

### 2.2. Characters and the twin group

Let $G$ be a LCAG. A personality of $G$ is a morphism from $G$ to

$U$. The set $G’$ of all characters of $G$ is a bunch (with the

operation of pointwise sum of mappings) and likewise a topological area (with the topology

of compact convergence). It seems that this group is domestically compact,

thus it’s a LCAG. It’s referred to as the twin group of $G$. There’s a canonical

morphism between $G$ and its bidual, and it may be seen simply that this

morphism is injective. The Pontryagin duality theorem states that $G$ is

isomorphic to its bidual. One other consequence states that $G$ is compact if an

provided that its twin is discrete.

For instance, the twin group of $R^n$ is itself. The integers

$Z$ and the unit circle $U$ are twin to one another.

The twin of any finite group is isomorphic (although non-canonically) to

itself.

The motion of a personality $xiin G’$ over a bunch aspect $xin G$ is

denoted by $E(xi,x)$ and even $e^{ixi x}$. Within the latter case, the complicated

conjugate of $e^{ixi x}$ is denoted by $e^{-ixi x}$. The exponential

notation is justified by the next properties, arising from the

definitions

- $E(xi,x)$ is a unit complicated quantity, thus it has the shape

$e^{itheta}$ for some actual quantity $theta$ - $E(xi,x+y) = E(xi,x)E(xi,y)$, by the definition of character
- $E(xi + eta,x) = E(xi,x)E(eta,x)$, by the definition of twin

group

### 2.3. Haar measures

Let $G$ be a LCAG. A non-vanishing measure over $G$ which is invariant by

translations is known as a Haar measure. Haar’s theorem states that there’s a

single Haar measure modulo multiplication by constructive constants. One other

consequence states that $G$ is compact if and provided that its complete Haar measure (any

one among them) is finite.

For instance, Lebesgue measure on $R^n$ is a Haar measure. The

counting measure of a discrete group is a Haar measure.

Given $G$, we repair a single Haar measure and we will speak concerning the areas

$L^p(G)$. The weather of this area are complex-valued capabilities such that

the $p$th energy of their norm has finite integral with respect to Haar’s

measure. Discover that the set $L^p(G)$ doesn’t rely on the precise selection

of normalization issue chosen for the definition of the Haar measure.

### 2.4. Fourier rework

Now we will outline a basic notion of Fourier transforms, for capabilities

belonging to the area $L^1(G)$. The Fourier

rework of a operate

start{equation}

f:Gtomathbf{C}

finish{equation}

is a operate

start{equation}

hat f:G’tomathbf{C}

finish{equation}

outlined by

start{equation}

hat f(xi) = int_G f(x) e^{-ixi x}mathrm{d} x

finish{equation}

Right here $e^{-ixi x}$ denotes the conjugate of the complicated quantity

$e^{ixi x}=E(xi,x)$. The inverse rework of a operate outlined on $G’$ is outlined equally, however

with out the conjugate:

start{equation}

examine f(x) = int_{G’} f(xi) e^{ixi x}mathrm{d} xi

finish{equation}

Observe that these definitions require deciding on Haar measures on $G$

and $G’$ (this quantities to fixing two arbitrary constants).

### 2.5. Harmonic evaluation on domestically compact abelian teams

To this point we’ve got simply given definitions: LCAG, characters, twin group, Haar

measure, and Fourier rework. Now it’s time to get better the principle outcomes

of harmonic evaluation.

The primary result’s the **Fourier inversion theorem** for $L^1(G)$, which

states that the inverse rework is definitely the inverse, for an acceptable

selection of scaling of the Haar measures on $G$ and $G’$. Such a pair of

measures are referred to as harmonized, or twin to one another. Within the following,

once we state a consequence involving integrals on $G$ and $G’$ we’ll all the time

assume that the Haar measures are harmonized.

The second result’s the **power conservation theorem** for $L^2(G)$,

which states that, when $f$ and $hat f$ are square-integrable, we’ve got

start{equation}

|f|_{L^2(G)}

=

|hat f|_{L^2(G’)}

finish{equation}

Specific instances of this theorem are the formulation of Parseval, Plancherel,

and many others.

The power conservation theorem is required to increase by continuity the

definition of Fourier transforms to $L^2(G)$

The third result’s the **convolution theorem**. First discover that the

group construction permits to outline the convolution of any two capabilities on

$L^1(G)$:

start{equation}

[f*g](x) = int_G f(y)g(x-y)mathrm{d} y

finish{equation}

Now, the convolution theorem says that the Fourier rework takes

convolution to point-wise multiplication

start{equation}

widehat{f*g} = hat f hat g

finish{equation}

There’s a lengthy checklist of outcomes, that may be discovered elsewhere. Allow us to point out

a final one. The twin group $G’$ is itself a LCAG, so it has a Fourier

rework in its personal proper. This mapping is the $L^2$ adjoint of the inverse

Fourier rework outlined from $G$.

Lastly, discover that within the case of finite teams all these outcomes are

trivial and so they quantity to elementary linear algebra. Within the steady case

they aren’t trivial, primarily as a result of we do not have an id aspect for

the convolution (e.g., the dirac delta operate), and to show the outcomes

one has to resort to successive approximations of the id.

The sequence of proofs sometimes begins by the convolution theorem,

which is used to show the conservation of power for capabilities that

belong to $L^1cap L^2$, then to increase by density the definition of

the Fourier rework to $L^2$ and at last to show the inversion

theorem. Besides the definition of the Haar measure and the

approximation of the id, that are explicit development, the

remainder of the proofs are similar to the corresponding proofs for the

case of Fourier transforms on the actual line. You simply need to examine

that each one the steps on the proof make sense in a bunch.

## 3. Sampling idea

Pontryagin duality provides an unified therapy of the 4 traditional

instances in Fourier evaluation: you’re all the time doing precisely the identical

factor, however in several teams. Nonetheless, it doesn’t say something

concerning the direct relationship between them. For instance, a Fourier

sequence the place all however a finite variety of the coefficients is zero can

be represented as a vector of size $N$. Does it have any

relationship with the discrete Fourier rework on $Z_N$? The

reply is *sure*, and it’s the most important results of sampling idea.

Allow us to begin with exactly this case. Suppose that we’ve got a

periodic operate $f(theta)$ whose Fourier sequence is finite (that is

referred to as a trigonometric polynomial). For

instance,

$$

f(theta)=sum_{n=0}^{N-1} f_ne ^{intheta}

$$

Now, we will do *three* various things with this object. **One**, we will categorical the coefficients $f_n$ as integrals of $f$:

$$

f_n = frac{1}{2pi}int_0^{2pi} f(theta)e^{-intheta}udtheta

$$

**Two**, we will take into account the vector of

coefficients $(f_0,ldots,f_{N-1})$ and compute its inverse DFT

$$

examine{f}_k = sum_{n=0}^{N-1} f_n,e^{2pi i nk/N}

$$

and **three**, only for enjoyable, we will consider the operate $f$

at $N$ factors evenly spaced alongside its interval

$$

fleft(frac{2picdot 0}{N}proper),

fleft(frac{2picdot 1}{N}proper),

fleft(frac{2picdot 2}{N}proper),

ldots

fleft(frac{2pi(N-1)}{N}proper)

$$

These three operations are, a-priori, unrelated. At the very least,

Pontryagin duality doesn’t say something about them, you’re

working with totally different teams $S^1$ and $Z_N$ that don’t have anything to

do with one another.

Nonetheless, various very humorous coincidences might be noticed:

- The $okay$-th pattern $fleft(frac{2pi okay}{N}proper)$ equals

$$

sum_{n=0}^{N-1}f_n,e^{2pi ikn/N}

$$

which is precisely $examine{f}_k$ - Thus, the vector of samples of the polynomial $f$ is the

IDFT of the vector of coefficients - Correspondingly, the vector of $N$ coefficients of the

polynomial $f$ is the DFT of the vector of $N$ uniform samples

of $f$ between $0$ and $2pi$. - In different phrases, the whole Fourier sequence of $f$ might be

obtained by evaluating the operate $f$ at $N$ factors. - In case you approximate the integral that evaluates $f_n$

from $f$ as a sum of $N$ step capabilities obtained by

sampling $f$, the computation is precise.

All these outcomes lie on the core of sampling idea.

They supply a fantastic, analog interpretation of the definition of

the discrete Fourier rework. In truth, whatever the

definition utilizing group characters, we might have outlined the discrete

fourier rework utilizing these outcomes! (property 3 above).

The **sampling theorem** takes many various kinds, but it surely all the time

quantities to a conservation of knowledge, or conservation of levels of

freedom. Thus, the properties above might be rephrased as

- Evaluating a trigonometric polynomial of $N$ coefficients

at $N$ factors is a linear map $C^NtoC^N$ - This linear map is invertible if and provided that the factors are totally different

(thus, the operate might be precisely recovered from $N$ of its samples) - If the factors are uniformly distributed, this linear map is

the discrete Fourier rework

The second assertion is usually referred to as the sampling theorem. The

situation that to get better a polynomial of $N$ coefficients

requires $N$ samples is known as the Nyquist situation. Since

it’s pure to think about trigonometric polynomials of the shape

$$

P(theta)=sum_{n=-N/2}^{N/2} p_n,e^{intheta}

$$

the Nyquist situation is usually said as *the sampling price should
be a minimum of the double of the maximal frequency*.

We have now thus associated Fourier sequence with the $N$-dimensional DFT, by way of

the operation of sampling at $N$ level. The reasoning is finite and

principally trivial. There are much more correspondences between the

4 traditional instances. For instance, Shannon-Whittaker interpolation

relates the Fourier rework with the discrete-time Fourier

rework: if the help of $hat f$ lies contained in the

interval $[-pi,pi]$, then $f$ might be recovered precisely by the

values $f(Z)$. A distinct development relates Fourier transforms

and Fourier sequence: if we’ve got a quickly lowering operate $f(x)$,

we will construct a $2pi$-periodic operate by folding it:

$$

tilde f(theta)=sum_{ninZ} f(theta+2pi n)

$$

and the Fourier sequence of $tilde f$ and the Fourier rework

of $f$ are intently associated.

All these relationships between the 4 traditional instances are neatly

encoded within the Fourier-Poisson dice, which is an superior commutative

diagram:

Allow us to outline the **folding** operation.

Suppose that $varphi$ is a quickly lowering easy operate.

We take a interval $P>0$ and outline the operate

$$

varphi_P(x):=sum_{kinZ}varphi(x+kP)

$$

Since $varphi$ is quickly lowering, this sequence converges

pointwise, and $varphi_P$ is a easy and $P$-periodic

operate. This folding operation transforms capabilities outlined on $R$ to

capabilities outlined on $R/2piZ$

Since $varphi$ is quickly lowering, we will compute it Fourier

rework

$$

widehatvarphi(y)=frac{1}{sqrt{2pi}}int_R f(x)e^{-ixy}ud x

$$

The **Poisson summation formulation** relates the Fourier rework

of $varphi$ with the Fourier sequence of $varphi_P$. Allow us to derive

it. Since $varphi_P$ is $P$-periodic, we will compute its Fourier

sequence

$$

varphi_P(x)=sum_{ninZ}c_nexpfrac{2pi inx}{P}

$$

which converges pointwise for any $x$ since $varphi_P$ is easy.

The Fourier coefficients $c_n$ are

$$

c_n = frac{1}{P}int_0^Pvarphi_P(x)expfrac{-2pi inx}{P}ud x

$$

by increasing the definition of $varphi_P$:

$$

c_n = frac{1}{P}int_0^Pleft(sum_{kinZ}varphi(x+kP)proper)expfrac{-2pi inx}{P}ud x

$$

and now, since $varphi$ is quickly lowering, we will interchange

the sum and the integral, (for instance, if $varphi$ is compactly

supported, the sum is finite):

$$

c_n = frac{1}{P}sum_{kinZ}int_0^Pvarphi(x+kP)expfrac{-2pi inx}{P}ud x

$$

now, by the change of variable $y=x+kP$,

$$

c_n =

frac{1}{P}sum_{kinZ}int_{kP}^{(okay+1)P}varphi(y)expfrac{-2pi

iny}{P}ud y

$$

the place we’ve got used the truth that $exp$ is $2pi i$-periodic to

simplify $e^{2pi ink}=1$. This sum of integrals over contiguous

intervals is just an integral over $R$, thus

$$

c_n =

frac{1}{P}int_Rvarphi(y)e^{-iyleft(frac{2pi n}{P}proper)}ud y

$$

the place we acknowledge the Fourier rework of $varphi$, thus

$$

c_n =frac{sqrt{2pi}}{P}widehatvarphileft(

frac{2pi n}{P}

proper).

$$

Changing the coefficients $c_n$ into the Fourier sequence

of $varphi_P$ we get hold of the id

$$

sum_{kinZ}varphi(x+kP)

=

frac{sqrt{2pi}}{P}

sum_{ninZ}

widehatvarphileft(

frac{2pi n}{P}

proper)

expfrac{2pi inx}{P}

$$

which might be evaluated at $x=0$ to present the Poisson summation formulation

$$

sum_{kinZ}varphi(kP)

=

frac{sqrt{2pi}}{P}

sum_{ninZ}

widehatvarphileft(

frac{2pi n}{P}

proper).

$$

The actual case $P=sqrt{2pi}$ is gorgeous:

$$

sum_{kinZ}varphileft(ksqrt{2pi}proper)

=

sum_{ninZ}widehatvarphileft(nsqrt{2pi}proper)

$$

As we’ll see under, the language of distribution idea permits to

categorical this formulation

as $$Xi_{sqrt{2pi}}=widehatXi_{sqrt{2pi}}$$ (the Fourier rework of a

Dirac comb is a Dirac comb).

## 4. Distributions

Discover that almost all of sampling idea might be accomplished with out recourse to

distributions. Certainly, Shannon, Nyquist, Whittaker, Borel, all

said and proved their outcomes method earlier than the invention of

distributions. These days, distribution idea supplies a satisfying

framework to state all of the traditional sampling leads to a unified

kind. It’s tough to evaluate which methodology is easier, as a result of the

traditional sampling outcomes all have elementary proofs, whereas the

detailed definition of tempered distributions is a bit concerned.

It’s higher to be conversant in each prospects.

In **classical sampling idea**, you pattern a steady

operate $f:RtoC$ by evaluating it at a discrete set of factors,

for instance $Z$, thus acquiring a sequence of

values $ldots,f(-2),f(-1),f(0),f(1),f(2),ldots$, which might be

interpreted as a operate $tilde f:ZtoC$. Thus, the sampling

operation is a mapping between very totally different areas: from the

steady capabilities outlined over $R$ into the capabilities outlined

over $Z$.

If you carry out **sampling utilizing distributions**, you pattern a

easy operate $f$ by multiplying it by a Dirac comb. Thus,

the sampling operation is linear a mapping between subspaces of the identical

area: tempered distributions.

### 4.1. Distributions: overview

Distributions are an extension of capabilities identical to the actual

numbers $R$ are an extension of the rationals $Q$. A lot of the

operations that may be accomplished with $Q$ might be accomplished with $R$, after which

some extra. Nonetheless, there’s a value to pay: there are some

operations that solely make sense on the smaller set. For instance,

whereas the “denominator” operate on $Q$ can’t be prolonged

meaningfully to $R$, the weather of $R$ can’t be enumerated like

these of $Q$, and many others. Nonetheless, if you wish to work with limits, the

area $Q$ is generally ineffective and also you want $R$.

There are a couple of areas of distributions. The three most well-known are

- $mathcal{D}’$ the area of all distributions
- $mathcal{S}’$ the area of tempered distributions
- $mathcal{E}’$ the area of compactly supported distributions

Every of those areas is a large generalization of an already very massive

area of capabilities:

- $mathcal{D}’$ incorporates all capabilities of $L^1_{loc}$
- $mathcal{S}’$ incorporates all capabilities of $L^1_{loc}$ that

are slowly rising (bounded, or going to infinity at a polynomial

price) - $mathcal{E}’$ incorporates all compactly supported integrable

capabilities

Right here $L^1_{loc}$ denotes the set of domestically integrable capabilities,

that’s, complex-valued capabilities such that $int_K|f| <+infty$ for

any compact $Ok$.

These are the properties that we earn with respect to the unique

areas:

- Most operations on capabilities prolong naturally to

distributions: sums, product by scalars, product by a operate,

affine adjustments of variable - Any distribution is infinitely derivable, and the by-product

belongs to the identical area - Any distribution is domestically integrable
- The Fourier rework is an isometry within the area of

tempered distributions - There’s a very simple to make use of definition of restrict of

distributions

And these are the costs to pay for the daring:

- You can’t consider a distribution at a degree
- You can’t multiply two distributions
- There isn’t any technique to outline a norm within the vector area of

distributions

### 4.2. Distributions: definition

There are a number of, quite totally different, definitions of distribution.

Essentially the most sensible definition right this moment appears to be because the topological

duals of areas of check capabilities:

- $mathcal{D}$ the area of all $mathcal{C}^infty$

capabilities of compact help - $mathcal{S}$ the area of all quickly

lowering $mathcal{C}^infty$ capabilities - $mathcal{E}$ the area of all $mathcal{C}^infty$

capabilities

Discover that $mathcal{D}$ and $mathcal{E}$ make sense for capabilities

outlined over an arbitrary open set, however $mathcal{S}$ solely makes

sense on the entire actual line.

The one downside with that is that the topologies on these areas of

check capabilities aren’t trivial to assemble. For instance, there’s

no pure technique to outline helpful norms on these areas. Thus,

topologies should be constructed utilizing households of seminorms, or by

different means (within the case of $mathcal{D}$). That is out of the scope

of this doc, but it surely’s an ordinary development that may be simply

discovered elsewhere (e.g., Gasqued-Witomski).

The essential topological property that we want is the definition of

**restrict of a sequence of distributions**. We are saying that {that a}

sequence $T_n$ of distributions converges to a distribution $T$ when

$$

T_n(varphi)to T(varphi)qquadtextrm{for any check operate } varphi

$$

Thus, the restrict of distributions is lowered to the restrict of scalars.

A sequences of distributions is convergent if and solely whether it is

“pointwise” convergent. That is way more easy than the case of

capabilities, the place there are a number of totally different and incompatible notions

of convergence.

A distribution is, by definition, a linear map on the area of check

capabilities. The next notations are frequent for the results of

making use of a distribution $T$ to a check operate $varphi$:

$$

T(varphi)

quad=quad

left<T,varphiright>

quad=quad

int Tvarphi

quad=quad

int T(x)varphi(x)ud x

$$

The final notation is especially insidious, as a result of for a generic

distribution, $T(x)$ doesn’t make sense. Nonetheless, it’s an abuse of

notation because of the following lemma:

**Lemma.** Let $f$ be a domestically integrable operate (slowly

rising, or compactly supported). Then the linear map

$$

T_f : varphimapstoint f(x)varphi(x)ud x

$$

is well-defined and steady on $mathcal{D}$ (or $mathcal{S}$,

or $mathcal{E}$). Thus it’s a distribution.

The lemma says that any operate might be interpreted as a

distribution.

This is essential, as a result of **all** the following

definitions on the area of distributions are crafted in order that, when

utilized to a operate they’ve the anticipated impact.

For instance, the **by-product of a distribution** $T$ is outlined by

$$

left<T’,varphiright>

:=

left<T,-varphi’proper>

$$

Two observations: (1) this definition is smart, as a result of $varphi$

is all the time a $mathcal{C}^infty$ operate, and so is $-varphi’$.

And (2) this definition extends the notion of by-product when $T$

corresponds to a derivable operate $f$. We write

$$

T_{f’}= {T_f}’

$$

to point that the proposed definition is appropriate with the

corresponding development for capabilities.

The same trick is used to increase the shift $tau_a$, scale $zeta_a$ and

symmetry $sigma$ of capabilities (the place $a>0$):

start{eqnarray*}

tau_a f(x) &:= f(x-a)

zeta_a f(x) &:= f(x/a)

sigma f(x) &:= f(-x)

finish{eqnarray*}

to the case of distributions:

start{eqnarray*}

left<tau_a T,varphiright> &:= left<T,tau_{-a}varphiright>

left<zeta_a T,varphiright> &:= left<T,a^{-1}zeta_{a^{-1}}varphiright>

left<sigma T,varphiright> &:= left<T,sigmavarphiright>

finish{eqnarray*}

and the compatibility might be checked by simple change of

variable.

After common capabilities, crucial instance of distribution

is the **Dirac delta**, outlined by $delta(varphi):=varphi(0)$.

Within the recurring notation we write

$$

intdelta(x)varphi(x)ud x = varphi(0)

$$

as a result of this kind could be very amenable to adjustments of variable.

An equal definition is $delta(x)=H'(x)$ the place $H$ is the

indicator operate of constructive numbers. This is smart as a result of $H$

is domestically integrable, and its by-product is well-defined within the

sense of distributions. The Dirac delta belongs to all three

areas $mathcal{D}’$, $mathcal{S}’$ and $mathcal{E}’$.

Utilizing Diracs, we will outline many different distributions, by making use of

shifts, derivatives, and vector area operations. For instance, the

**Dirac comb** is outlined as

$$

Xi(x)=sum_{ninZ}delta(x-n)

$$

the place the infinite sequence is to be interpreted as a restrict. That is

well-defined in $mathcal{D}’$ (the place the sum is finite because of the

compact help of the check operate)

and $mathcal{S}’$ (the place the sequence is trivially convergent due the

fast lower of the check operate) however not on $mathcal{E}’$ (the place

the sequence is just not essentially convergent for arbitrary check

capabilities, for instance $varphi=1inmathcal{E}$).

We will do different loopy issues, like $sum_{nge 0}delta^{(n)}(x-n)$,

which can also be properly outlined when utilized to a check operate. However we

can’t do every thing. For instance $sum_{nge 0}delta^{(n)}(x)$ is

not properly outlined, as a result of there’s not a assure that the sum of

all derivatives of a check operate on the identical level converges.

### 4.3. Fourier rework of distributions

How you can outline the Fourier rework of a distribution?

We have to discover a definition that extends the definition that we

have already got for capabilities, thus $widehat{T_f}=T_{widehat{f}}$.

It’s simple to examine that the definition

$$

left<widehat{T},varphiright>

:=

left<T,widehat{varphi}proper>

$$

does the trick, as a result of it corresponds to Plancherel Theorem when $T$

is a domestically integrable operate.

Nonetheless, discover that this definition doesn’t make sense

in $mathcal{D}’$: if $varphiinmathcal{D}$, then it has compact

help, so its Fourier rework doesn’t,

thus $widehat{varphi}notinmathcal{D}$.

The area $mathcal{S}$, referred to as the Schwartz area, has the attractive

property of being invariant by Fourier transforms. Certainly, the Fourier

rework, with acceptable normalization constants, is an $L^2$

isometry on $mathcal{S}$. Thus, tempered distributions are the

pure area the place to carry out Fourier transforms.

Now, we will compute the Fourier rework, within the sense of

distributions, of many capabilities! For instance, what’s the Fourier

rework of the operate $f(x)=1$? This operate is a temperate

distribution, so it will need to have a Fourier rework, would not it?

Certainly it does, and it may be simply discovered from the definitions:

$$

left<widehat{1},varphiright>

=

left<1,widehat{varphi}proper>

=

intwidehat{varphi}(x)ud x

=

frac{1}{sqrt{2pi}}varphi(0)

$$

So, the Fourier rework of a continuing is a Dirac!

By combining this consequence with the derivatives we will compute the

Fourier rework of polynomials. For instance $f(x)=x^2$ has the

property that $f”$ is fixed, thus $widehat{f”}$ is a Dirac, and

then $widehat{f}$ is the second by-product of a Dirac.

### 4.4. Sampling with Diracs

Can we compute the Fourier rework of $f(x)=e^x$ ? No, as a result of it

is just not a slowly rising operate, and it doesn’t correspond to any

tempered distribution.

Nonetheless, the operate $f(x)=e^{ix}$ is definitely slowly rising (it’s

bounded), so it has a Fourier rework as a tempered Distribution

that’s $widehat{f}(xi)=delta(xi-1)$. Utilizing trigonometric

identities, we discover the Fourier transforms of $sin$ and $cos$,

that are additionally sums of Diracs:

start{eqnarray*}

widehat{cos}(xi) &=frac{delta(x-1)+delta(x+1)}{2}

widehat{sin}(xi) &=frac{delta(x-1)-delta(x+1)}{2i}

finish{eqnarray*}

And, as we’d say in Catalan, *the mom of the
eggs*footnote{“La mare dels ous”, or in french “où il gît le

lièvre”. I have no idea a equally colourful expression in english}: the

Fourier rework of a Dirac comb is one other Dirac comb.

Now I do not see

know the way to show this by combining the identities above, but it surely has

a easy proof by expressing the Dirac comb because the by-product of a

sawtooth operate and making use of it to a check operate, as accomplished on the

earlier part.

## 5. Spectral geometry

Spectral idea supplies a brutal generalization of a big a part of

Fourier evaluation. We get rid of the group construction (and thus with

the likelihood to have convolutions, that are based mostly on the motion

of the group). In trade, we have to work inside a compact area,

endowed by a Riemannian metric. For instance, a compact sub-manifold

of Euclidean area. The canonical instance is $S^1$, that in

the classical case results in Fourier sequence. Right here, we get better all

the outcomes of Fourier sequence (besides these associated to periodic

convolution) for capabilities outlined on our manifold.

Let $M$ be a compact Riemannian manifold (with or with out boundary), and

let $Delta$ be its Laplace-Beltrami operator, outlined

as $Delta=*d*d$, the place $d$ is the outside by-product (which is impartial

of the metric) and $*$ is the Hodge duality between $p$-forms

and $d-p$-forms (which is outlined utilizing the metric).

The next are normal leads to differential geometry (see e.g.

Warner’s ebook chapter

6 https://link.springer.com/content/pdf/10.1007%2F978-1-4757-1799-0_6.pdf)

- (1) There’s a sequence of $mathcal{C}^infty(M)$

capabilities $varphi_n$ and constructive

numbers $lambda_ntoinfty$ such that

$$Deltavarphi_n=-lambda_nvarphi_n$$ - (2) The capabilities $varphi_n$, suitably normalized, are an

orthonormal foundation of $L^2(M)$.

These outcomes generalize Fourier sequence to an arbitrary easy manifold $M$.

Any square-integrable operate $f:MtoR$ is written uniquely as

$$f(x)=sum_nf_nvarphi_n(x)$$ and the coefficients $f_n$ are computed by

$$f_n=int_Mfvarphi_n.$$ Some explicit instances are the recurring Fourier and

sine bases (however not the cosine foundation), bessel capabilities for the disk, and

spherical harmonics for the floor of a sphere.

$M$ | $varphi_n$ | $-lambda_n$ | |

interval |
$[0,2pi]$ | $sinleft(frac{nx}{2}proper)$ | $n^2/4$ |

circle | $S^1$ | $sin(ntheta),cos(ntheta)$ | $n^2$ |

sq. | $[0,2pi]^2$ | $sinleft(frac{nx}{2}proper)sinleft(frac{mtheta}{2}proper)$ | $frac{n^2+m^2}{4}$ |

torus | $(S^1)^2$ | $sin(nx)sin(my),ldots$ | $n^2+m^2$ |

disk | $|r|le1$ | $sin,cos(ntheta)J_n(rho_{m,n}r)$ | $rho_{m,n}$ roots of $J_n$ |

sphere | $S^2$ | $Y^m_l(theta,varphi)$ | $l^2+l$ |

The eigenfunctions $varphi_n$ are referred to as the vibration modes of $M$, and the

eigenvalues $lambda_n$ are referred to as the (squared) elementary frequencies of $M$.

A number of geometric properties of $M$ might be interpreted when it comes to the

Laplace-Beltrami spectrum. For instance, if $M$ has $okay$ related parts,

the primary $okay$ eigenfuntions shall be supported successively on every related

part. On a related manifold $M$, the primary vibration mode might be

taken to be constructive $varphi_1ge0$, thus all the opposite modes have

non-constant indicators (as a result of they’re orthogonal to $varphi_1$). In

explicit, the signal of $varphi_2$ cuts $M$ in two components in an optimum method,

it’s the Cheeger minimize of $M$, maximizing the perimeter/space ratio of the minimize.

The zeros of $varphi_n$ are referred to as the nodal curves (or nodal units) of $M$,

or additionally the Chladni patterns. If $M$ is a subdomain of the aircraft, these

patterns might be discovered by slicing an object within the form of $M$, pouring a

layer of sand over it, and letting it vibrate by high-volume sound waves at

totally different frequencies. For many frequencies, the sand won’t kind any

explicit sample, however when the frequency coincides with

a $sqrt{lambda_n}$, the sand will accumulate over the set $[varphi_n=0]$,

which is the set of factors of the floor that don’t transfer when the floor

vibrates at this frequency. Within the typical case, the variety of related

parts of $[varphi_n>0]$ grows linearly with $n$, thus the

capabilities $varphi_n$ turn into extra oscillating (much less common) as $n$ grows.

Usually, symmetries of $M$ come up as multiplicities of eigenvalues.

The Laplace-Beltrami spectrum ${lambda_1,lambda_2,lambda_3,ldots}$ is

intently associated, however not similar, to the geodesic size spectrum, that

measures the sequence of lengths of all closed geodesics of $M$. The grand

outdated man of this idea is Yves Colin de Verdière, pupil of Marcel Berger.

Geometry is just not normally a spectral invariant, however non-isometric manifolds

with the identical spectrum are tough to return by. The primary pair of distinct

however isospectral manifolds was wound in 1964 by John Milnor, in dimension 16.

The primary instance in dimension 2 was present in 1992 by Gordon, Webb and

Wolperd, and it answered negatively the well-known query of Marc Kac “Are you able to

hear the form of a drum?’.

In 2018, we’ve got some ways to assemble discrete and steady households of

isospectral manifolds in dimensions two and above.