Trigonometric functions and Fourier series (part II)vipul/studenttalks/fourierseries... · 2007-02-13 · Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier

Trigonometricfunctions and

Fourier series (partII)

Vipul Naik

The Fouriertransform

Fourier series forfunctions to complexnumbers

The Fourier transformfor topological groups

The Fourier transformfor discrete groups

Computing theFourier coefficients

The square wave

The saw-tooth

Issues ofconvergence

Using kernels to provepointwise convergence

Fourier series on RReal-time streaming

Trigonometric functions and Fourier series(part II)

Vipul Naik

February 13, 2007



Vipul Naik






The square wave

The saw-tooth




Outline

The Fourier transformFourier series for functions to complex numbersThe Fourier transform for topological groupsThe Fourier transform for discrete groups

Computing the Fourier coefficientsThe square waveThe saw-tooth

Issues of convergenceUsing kernels to prove pointwise convergence




Vipul Naik






The square wave

The saw-tooth




Inner product for complex vector spaces

Let V be a C-vector space. An inner product is a bilinearform 〈 , 〉 : V × V → C satisfying:

I C-linearity in the first variable:

〈 a1 , b 〉+ 〈 a2 , b 〉 = 〈 a1 + a2 , b 〉〈 λa , b 〉 = λ 〈 a , b 〉

I Conjugate-linearity in the second variable:

〈 a , b1 〉+ 〈 a , b2 〉 = 〈 a , b1 + b2 〉〈 a , λb 〉 = λ 〈 a , b 〉



Vipul Naik






The square wave

The saw-tooth







〈 a1 , b 〉+ 〈 a2 , b 〉 = 〈 a1 + a2 , b 〉〈 λa , b 〉 = λ 〈 a , b 〉


〈 a , b1 〉+ 〈 a , b2 〉 = 〈 a , b1 + b2 〉〈 a , λb 〉 = λ 〈 a , b 〉



Vipul Naik






The square wave

The saw-tooth







〈 a1 , b 〉+ 〈 a2 , b 〉 = 〈 a1 + a2 , b 〉〈 λa , b 〉 = λ 〈 a , b 〉


〈 a , b1 〉+ 〈 a , b2 〉 = 〈 a , b1 + b2 〉〈 a , λb 〉 = λ 〈 a , b 〉



Vipul Naik






The square wave

The saw-tooth




Inner product for complex vector spaces (contd)

I Hermitian symmetry:

〈 a , b 〉 = 〈 b , a 〉

I Positive definiteness:

〈 a , a 〉 ∈ R+ for a 6= 0



Vipul Naik






The square wave

The saw-tooth




Inner product for complex vector spaces (contd)

I Hermitian symmetry:

〈 a , b 〉 = 〈 b , a 〉

I Positive definiteness:

〈 a , a 〉 ∈ R+ for a 6= 0



Vipul Naik






The square wave

The saw-tooth




Orthonormal set for functions to complexnumbers

For the space of functions from S1 to C, an orthonormal setis the set of functions x 7→ e inx where n ∈ Z.

Under pointwise multiplication, these functions form a groupisomorphic to the group Z of integers.The Fourier series for a function f is thus an expression:

∞∑n=−∞

aneinx

where the infinite summation converges to f .



Vipul Naik






The square wave

The saw-tooth





For the space of functions from S1 to C, an orthonormal setis the set of functions x 7→ e inx where n ∈ Z.Under pointwise multiplication, these functions form a groupisomorphic to the group Z of integers.

The Fourier series for a function f is thus an expression:

∞∑n=−∞

aneinx




Vipul Naik






The square wave

The saw-tooth





For the space of functions from S1 to C, an orthonormal setis the set of functions x 7→ e inx where n ∈ Z.Under pointwise multiplication, these functions form a groupisomorphic to the group Z of integers.The Fourier series for a function f is thus an expression:

∞∑n=−∞

aneinx




Vipul Naik






The square wave

The saw-tooth




Expression for the Fourier coefficients

Because the e inx are orthonormal, the value an is simply thesigned magnitude of the projection of f along e inx , namely:

an =1

π

∫ 2π

0f (x)e−inx dx

Now the map n 7→ an can be viewed as a function from Z toC which is uniquely determined by f . Thus, changingnotation, we denote by f (n) the value an corresponding to f .Then f is a map Z → C.



Vipul Naik






The square wave

The saw-tooth




Expression for the Fourier coefficients

Because the e inx are orthonormal, the value an is simply thesigned magnitude of the projection of f along e inx , namely:

an =1

π

∫ 2π

0f (x)e−inx dx

Now the map n 7→ an can be viewed as a function from Z toC which is uniquely determined by f . Thus, changingnotation, we denote by f (n) the value an corresponding to f .Then f is a map Z → C.



Vipul Naik






The square wave

The saw-tooth




The Fourier transform

The transform which takes as input a function f from S1 toC and gives as output the function f , is termed the Fouriertransform(defined).Thus the Fourier transform takes us from functions on S1 tofunctions on Z.



Vipul Naik






The square wave

The saw-tooth




Fourier transform in a little more generality

Let G be a topological Abelian group. Then consider the setof homomorphisms from G to C. This set gets equippedwith a natural group structure under pointwise addition. Callthis the dual group(defined) to G .Then, given a suitable inner product on G , we can get amap analogous to a Fourier transform: which takes acontinuous function f on G and outputs a function f on G .



Vipul Naik






The square wave

The saw-tooth




Double dual same as originalFor a locally compact topological Abelian group G , the dualof the dual group is again G . This suggests a kind of dualitybetween functions on G and functions on G .

The questions now, for the case of the Fourier transform, arethus:

I What are the functions on S1 for which the Fouriertransform gives a well-defined function on Z?The L2

functions

I What are the functions on Z that arise as Fouriertransforms of well-defined functions on S1?

I For which functions on S1 is it true that the sum of theFourier series given by the Fourier transform actuallyconverges?

I For which functions on Z is it true that the Fouriertransform of the limit of its Fourier series is the same asthe original function?



Vipul Naik






The square wave

The saw-tooth




Double dual same as originalFor a locally compact topological Abelian group G , the dualof the dual group is again G . This suggests a kind of dualitybetween functions on G and functions on G .The questions now, for the case of the Fourier transform, arethus:

I What are the functions on S1 for which the Fouriertransform gives a well-defined function on Z?

The L2

functions






Vipul Naik






The square wave

The saw-tooth






functions






Vipul Naik






The square wave

The saw-tooth






functions






Vipul Naik






The square wave

The saw-tooth






functions






Vipul Naik






The square wave

The saw-tooth




The additive group modulo p

At the very other end from groups like S1, are groups like(Fp,+), (additive group over a prime field).

For Fp, the dual group, called Fp, is simply the group ofone-dimensional characters of Fp under multiplication. Thisis isomorphic to Fp – however, there is no naturalisomorphism.The Fourier transform here thus takes functions on thegroup and returns functions on the character group.



Vipul Naik






The square wave

The saw-tooth





At the very other end from groups like S1, are groups like(Fp,+), (additive group over a prime field).For Fp, the dual group, called Fp, is simply the group ofone-dimensional characters of Fp under multiplication. Thisis isomorphic to Fp – however, there is no naturalisomorphism.

The Fourier transform here thus takes functions on thegroup and returns functions on the character group.



Vipul Naik






The square wave

The saw-tooth





At the very other end from groups like S1, are groups like(Fp,+), (additive group over a prime field).For Fp, the dual group, called Fp, is simply the group ofone-dimensional characters of Fp under multiplication. Thisis isomorphic to Fp – however, there is no naturalisomorphism.The Fourier transform here thus takes functions on thegroup and returns functions on the character group.



Vipul Naik






The square wave

The saw-tooth




Outline







Vipul Naik






The square wave

The saw-tooth




The [−π , π] convention

In this section, we shall shift from the [0 , 2π] situation tothe [−π , π] situation, viz where the function is periodicallyextended from its definition on the closed interval [−π , π].Some clear advantages are:

I The integral of any odd periodic function over thisinterval is zero.

I Thus, the coefficients of the sin functions in the Fourierseries expansion of an even function are 0

I Similarly the coefficients of the cos function in theFourier series expansion of an odd function are 0



Vipul Naik






The square wave

The saw-tooth











Vipul Naik






The square wave

The saw-tooth











Vipul Naik






The square wave

The saw-tooth











Vipul Naik






The square wave

The saw-tooth




The square waveThe square wave is defined as:

f (x) = −1 for − π ≤ x < 0

f (x) = 0 for x = 0

f (x) = 1 for 0 < x ≤ π

Clearly f is an odd function, hence its Fourier expansionmust only contain sin terms. The coefficient of sin nx is:

1

π

∫ π

−πf (x) sin nx dx

which simplifies to 0 if n is even and 4/(nπ) if n is odd.Thus the Fourier series for the square wave looks like:

4

π

∑n∈N

sin(2n − 1)t

2n − 1



Vipul Naik






The square wave

The saw-tooth





f (x) = −1 for − π ≤ x < 0

f (x) = 0 for x = 0

f (x) = 1 for 0 < x ≤ π


1

π

∫ π


which simplifies to 0 if n is even and 4/(nπ) if n is odd.

Thus the Fourier series for the square wave looks like:

4

π

∑n∈N

sin(2n − 1)t

2n − 1



Vipul Naik






The square wave

The saw-tooth





f (x) = −1 for − π ≤ x < 0

f (x) = 0 for x = 0

f (x) = 1 for 0 < x ≤ π


1

π

∫ π


which simplifies to 0 if n is even and 4/(nπ) if n is odd.Thus the Fourier series for the square wave looks like:

4

π

∑n∈N

sin(2n − 1)t

2n − 1



Vipul Naik






The square wave

The saw-tooth




Under linear changes

Suppose f is a 2π-periodic function. Then the functionx 7→ af (x) + b is also a 2π-periodic function.

The Fourier coefficients of this new function are related tothe old function as follows:

I The constant term a0 gets transformed to aa0 + b

I All the other terms get multiplied by a

We can use this to directly write the Fourier series for the0-1 square wave.



Vipul Naik






The square wave

The saw-tooth




Under linear changes

Suppose f is a 2π-periodic function. Then the functionx 7→ af (x) + b is also a 2π-periodic function.The Fourier coefficients of this new function are related tothe old function as follows:

I The constant term a0 gets transformed to aa0 + b

I All the other terms get multiplied by a

We can use this to directly write the Fourier series for the0-1 square wave.



Vipul Naik






The square wave

The saw-tooth




What this tells us

The square wave looks almost like something flat – the onlydifference being that there is a jump from −1 to 1. It isinteresting that so many sine functions are hiddenunderneath this apparently flat behaviour.

For the square wave, the coefficient of sin nx is only inverselinear in n, so by the typical standards of Fourier series, itdoes not taper very fast. This is to be expected since thesquare wave is about as democratic as one can get betweenfrequencies.



Vipul Naik






The square wave

The saw-tooth




What this tells us

The square wave looks almost like something flat – the onlydifference being that there is a jump from −1 to 1. It isinteresting that so many sine functions are hiddenunderneath this apparently flat behaviour.For the square wave, the coefficient of sin nx is only inverselinear in n, so by the typical standards of Fourier series, itdoes not taper very fast. This is to be expected since thesquare wave is about as democratic as one can get betweenfrequencies.



Vipul Naik






The square wave

The saw-tooth




The even saw-tooth

Define f (x) = |x | on the closed interval [−π , π] and extendf periodically to the whole real line.

Since f is even, all the sin coefficients are zero. This leavesus to compute the cosine coefficient and the constant term.By simple integration, the constant term is π/2 (this is theaverage height)Also:

an =2

π

∫ π

0x cos(nx) dx

This simplifies to −4πn2 for odd n and 0 for even n.

We thus get:

π

2− 4

π

∑n∈N

cos(2n − 1)t

(2n − 1)2



Vipul Naik






The square wave

The saw-tooth




The even saw-tooth

Define f (x) = |x | on the closed interval [−π , π] and extendf periodically to the whole real line.Since f is even, all the sin coefficients are zero. This leavesus to compute the cosine coefficient and the constant term.

By simple integration, the constant term is π/2 (this is theaverage height)Also:

an =2

π

∫ π

0x cos(nx) dx


We thus get:

π

2− 4

π

∑n∈N

cos(2n − 1)t

(2n − 1)2



Vipul Naik






The square wave

The saw-tooth




The even saw-tooth

Define f (x) = |x | on the closed interval [−π , π] and extendf periodically to the whole real line.Since f is even, all the sin coefficients are zero. This leavesus to compute the cosine coefficient and the constant term.By simple integration, the constant term is π/2 (this is theaverage height)

Also:

an =2

π

∫ π

0x cos(nx) dx


We thus get:

π

2− 4

π

∑n∈N

cos(2n − 1)t

(2n − 1)2



Vipul Naik






The square wave

The saw-tooth




The even saw-tooth

Define f (x) = |x | on the closed interval [−π , π] and extendf periodically to the whole real line.Since f is even, all the sin coefficients are zero. This leavesus to compute the cosine coefficient and the constant term.By simple integration, the constant term is π/2 (this is theaverage height)Also:

an =2

π

∫ π

0x cos(nx) dx


We thus get:

π

2− 4

π

∑n∈N

cos(2n − 1)t

(2n − 1)2



Vipul Naik






The square wave

The saw-tooth




Intuitive justification

Unlike the square wave function, which was very democraticamong all the functions, the saw-tooth clearly prefers lowerfrequencies – this is evidenced in the relatively faster rate atwhich the Fourier coefficients decline.



Vipul Naik






The square wave

The saw-tooth




Outline







Vipul Naik






The square wave

The saw-tooth




The convergence results we want

We ideally want to say that for functions of interest theFourier series converges at every point. This is equivalent torequiring that if we have a function which is orthogonal toall the sine and cosine functions, then it is identically zero.

The idea now is to focus at any particular point and showthat at that point the function takes the value zero. For thispurpose, we try to try to obtain a sequence of functions thatapproaches the Dirac delta for the point. In other words wetry to locate a sequence of functions fn such that〈 f , fn 〉 → f (p) for that point p.If each of these fn is a finite linear combination of the sineand cosine bunches, then 〈 f , fn 〉 is zero for all n andhence f (p) = 0.



Vipul Naik






The square wave

The saw-tooth





We ideally want to say that for functions of interest theFourier series converges at every point. This is equivalent torequiring that if we have a function which is orthogonal toall the sine and cosine functions, then it is identically zero.The idea now is to focus at any particular point and showthat at that point the function takes the value zero. For thispurpose, we try to try to obtain a sequence of functions thatapproaches the Dirac delta for the point. In other words wetry to locate a sequence of functions fn such that〈 f , fn 〉 → f (p) for that point p.

If each of these fn is a finite linear combination of the sineand cosine bunches, then 〈 f , fn 〉 is zero for all n andhence f (p) = 0.



Vipul Naik






The square wave

The saw-tooth





We ideally want to say that for functions of interest theFourier series converges at every point. This is equivalent torequiring that if we have a function which is orthogonal toall the sine and cosine functions, then it is identically zero.The idea now is to focus at any particular point and showthat at that point the function takes the value zero. For thispurpose, we try to try to obtain a sequence of functions thatapproaches the Dirac delta for the point. In other words wetry to locate a sequence of functions fn such that〈 f , fn 〉 → f (p) for that point p.If each of these fn is a finite linear combination of the sineand cosine bunches, then 〈 f , fn 〉 is zero for all n andhence f (p) = 0.



Vipul Naik






The square wave

The saw-tooth




The Dirichlet kernel

This is simply the average of the first n functions, and isgiven by:

t 7→ sin((n + 1/2)t)

sin(t/2)

with the suitable limiting value at t = 0, namely 2n + 1

This is the same as:

t 7→ 1 + 2n∑

k=1

cos kt



Vipul Naik






The square wave

The saw-tooth




The Dirichlet kernel

This is simply the average of the first n functions, and isgiven by:

t 7→ sin((n + 1/2)t)

sin(t/2)

with the suitable limiting value at t = 0, namely 2n + 1This is the same as:

t 7→ 1 + 2n∑

k=1

cos kt



Vipul Naik






The square wave

The saw-tooth




Properties of the Dirichlet kernel

The Dirichlet kernel has the following important properties:

I The integral is uniformly bounded. In fact, for any n,the integral of the corresponding Dirichlet kernel on anysub-interval is not more than 4π.

I As n →∞, the limiting value at 0 is ∞.

I The value of the Dirichlet kernel drops very rapidly justslightly away form 0.



Vipul Naik






The square wave

The saw-tooth




Riemann-Lebesgue property

To quantify the last statement (the value of the Dirichletkernel drops very rapidly just slightly after 0) the followingnotion was introduced.

A sequence fn of 2π-periodic functions is said to have theRiemann-Lebesgue property(defined) if for any r > 0:

limn→∞

∫ π

rf (t)fn(t) dt = 0

for f any L1 function.It turns out that the Dirichlet kernels do possess theRiemann-Lebesgue property. This, along with the fact thatthe value at 0 goes to infinity and the total area is bounded,suffices to show that limn→∞ f (t)Dn(t) dt is actually f (0).



Vipul Naik






The square wave

The saw-tooth





To quantify the last statement (the value of the Dirichletkernel drops very rapidly just slightly after 0) the followingnotion was introduced.A sequence fn of 2π-periodic functions is said to have theRiemann-Lebesgue property(defined) if for any r > 0:

limn→∞

∫ π

rf (t)fn(t) dt = 0

for f any L1 function.

It turns out that the Dirichlet kernels do possess theRiemann-Lebesgue property. This, along with the fact thatthe value at 0 goes to infinity and the total area is bounded,suffices to show that limn→∞ f (t)Dn(t) dt is actually f (0).



Vipul Naik






The square wave

The saw-tooth





To quantify the last statement (the value of the Dirichletkernel drops very rapidly just slightly after 0) the followingnotion was introduced.A sequence fn of 2π-periodic functions is said to have theRiemann-Lebesgue property(defined) if for any r > 0:

limn→∞

∫ π

rf (t)fn(t) dt = 0

for f any L1 function.It turns out that the Dirichlet kernels do possess theRiemann-Lebesgue property. This, along with the fact thatthe value at 0 goes to infinity and the total area is bounded,suffices to show that limn→∞ f (t)Dn(t) dt is actually f (0).



Vipul Naik






The square wave

The saw-tooth




Convolution product at a point

Here’s a rough sketch of the nature of result we want toestablish:

I Construct a sequence of functions fn, each a finite linearcombination of sines and cosines, with theRiemann-Lebesgue property. The sequence that weconstructed was that of Dirichlet kernels.

I Now, using the Riemann-Lebesgue property, argue thatfor any function f , limn→∞

∫f (t)fn(t) is f (0).

I More specifically argue that limn→infty

∫f (t + x)fn(t) is

f (x).

I Thus if f is orthogonal to all the sin and cos functions,f (x) must be equal to 0 for all x .

I In particular, if we take an arbitrary function, considerits Fourier series and call the difference f , then we canuse the above argument to show that the difference is 0.



Vipul Naik






The square wave

The saw-tooth








∫f (t)fn(t) is f (0).



f (x).





Vipul Naik






The square wave

The saw-tooth








∫f (t)fn(t) is f (0).



f (x).





Vipul Naik






The square wave

The saw-tooth








∫f (t)fn(t) is f (0).



f (x).





Vipul Naik






The square wave

The saw-tooth








∫f (t)fn(t) is f (0).



f (x).





Vipul Naik






The square wave

The saw-tooth




Riemann localization principle

The Riemann localization principle basically states that iftwo functions are locally similar, then the behaviour of theirFourier series in those neighbourhoods would be similar.More precisely:

Suppose f and g are 2π-periodic functions and further thatf = g almost everywhere on an open interval (t − r , t + r)with r > 0. Then the Fourier series both converge or bothdiverge at t. Moreover, if they converge, they take the samevalue on the whole interval.



Vipul Naik






The square wave

The saw-tooth




Riemann localization principle

The Riemann localization principle basically states that iftwo functions are locally similar, then the behaviour of theirFourier series in those neighbourhoods would be similar.More precisely:Suppose f and g are 2π-periodic functions and further thatf = g almost everywhere on an open interval (t − r , t + r)with r > 0. Then the Fourier series both converge or bothdiverge at t. Moreover, if they converge, they take the samevalue on the whole interval.



Vipul Naik






The square wave

The saw-tooth




The Fejer kernel

The Fejer kernel is the average of the values on the Dirichletkernel.



Vipul Naik






The square wave

The saw-tooth




Convergence result

We can show that for a periodic function f , if the left-handand right-hand limit exists at each point, and the limitingvalue of derivative with respect to this is also well-defined oneach side, the Fourier series converges pointwise to the meanof the left-hand and the right-hand limit.

Actually, we can prove a much stronger result: given any2π-periodic function with bounded variation, the Fourierseries converges to:

1

2

(f (t−) + f (t+)

)Finally, we can show that the Fourier series for any L2

function converges pointwise almost everywhere.



Vipul Naik






The square wave

The saw-tooth




Convergence result

We can show that for a periodic function f , if the left-handand right-hand limit exists at each point, and the limitingvalue of derivative with respect to this is also well-defined oneach side, the Fourier series converges pointwise to the meanof the left-hand and the right-hand limit.Actually, we can prove a much stronger result: given any2π-periodic function with bounded variation, the Fourierseries converges to:

1

2

(f (t−) + f (t+)

)

Finally, we can show that the Fourier series for any L2




Vipul Naik






The square wave

The saw-tooth




Convergence result

We can show that for a periodic function f , if the left-handand right-hand limit exists at each point, and the limitingvalue of derivative with respect to this is also well-defined oneach side, the Fourier series converges pointwise to the meanof the left-hand and the right-hand limit.Actually, we can prove a much stronger result: given any2π-periodic function with bounded variation, the Fourierseries converges to:

1

2

(f (t−) + f (t+)

)Finally, we can show that the Fourier series for any L2




Vipul Naik






The square wave

The saw-tooth




Outline







Vipul Naik






The square wave

The saw-tooth




Streaming musicSuppose a continuous stream of sound is coming in. Astream of sound can be viewed as a function f : R → Rwhere the domain R represents the time coordinate and therange R represents the instantaneous magnitude. Note thatthe study of such sound streams should be:

I Invariant under time-translation viz if the whole soundstream is displaced in time by a certain amount, theanalysis should not differ

I Invariant under spatial translation: That is, if aconstant sound stream (white noise) is added to thesound stream, the analysis should not differ (except totake out the white noise)

I Covariant under spatial/time dilation: If theinstantaneous magnitude at every point is multipled bya constant factor, or if the whole sound stream isstretched or compressed along the time axis, thereshould simply be some proportionate changes in theconstants rather than a qualitative change.



Vipul Naik






The square wave

The saw-tooth










Vipul Naik






The square wave

The saw-tooth










Vipul Naik






The square wave

The saw-tooth




As a combination of frequencies

Musical sounds arise as superpositions (pointwise sums) ofdifferent frequencies. There are the following complications:

I The frequencies may not even be related rationally.Thus, we may not be able to find any period for thesuperposition.For instance, the function

sin x + sin√

2x

has no period

I The sounds may start and end at different times. Thus,unlike the ideal sine wave which was always there andwill always be there, sounds in real life will start off andthen peter off.

I There are variable levels of noise



Vipul Naik






The square wave

The saw-tooth







sin x + sin√

2x

has no period





Vipul Naik






The square wave

The saw-tooth







sin x + sin√

2x

has no period





Vipul Naik






The square wave

The saw-tooth




Can we do a Fourier transform on R?

Here’s the idea. Given f : R → R, try writing f as an infinitelinear combination of functions of the form x 7→ e irx forevery real number r . In other words we need a map f 7→ fthat converts a function from R to R to another functionfrom R to R, but where the new function represents theamplitude corresponding to a given frequency.

Here’s a naive attempt at writing an expression for this:

f (r) =

∫ ∞

−∞f (x)e−irx dx

The problem of course is that even for the nicest continuousfunctions, the right-hand side may blow up to infinity. So weneed to restrict to continuous functions that become zerooutside some finite domain – the so-called continuousfunctions of compact support.



Vipul Naik






The square wave

The saw-tooth




Can we do a Fourier transform on R?

Here’s the idea. Given f : R → R, try writing f as an infinitelinear combination of functions of the form x 7→ e irx forevery real number r . In other words we need a map f 7→ fthat converts a function from R to R to another functionfrom R to R, but where the new function represents theamplitude corresponding to a given frequency.Here’s a naive attempt at writing an expression for this:

f (r) =

∫ ∞

−∞f (x)e−irx dx

The problem of course is that even for the nicest continuousfunctions, the right-hand side may blow up to infinity. So weneed to restrict to continuous functions that become zerooutside some finite domain – the so-called continuousfunctions of compact support.



Vipul Naik






The square wave

The saw-tooth




Schwarz functions and Schwarz spaces

The Schwarz space is the space of all functions f : R → Rsuch that for any polynomial function p, the map f (x)p(x)approaches ∞. If f is also infinitely differentiable, this isequivalent to demanding that all the derivatives approach 0as x →∞.

It turns out that the Fourier transform of any continuousfunction with compact support is a Schwarz function.In other words, if we take a sound stream that tapers off infinite time, the contribution of very high frequencies to thatsound stream goes down superpolynomially fast.



Vipul Naik






The square wave

The saw-tooth





The Schwarz space is the space of all functions f : R → Rsuch that for any polynomial function p, the map f (x)p(x)approaches ∞. If f is also infinitely differentiable, this isequivalent to demanding that all the derivatives approach 0as x →∞.It turns out that the Fourier transform of any continuousfunction with compact support is a Schwarz function.

In other words, if we take a sound stream that tapers off infinite time, the contribution of very high frequencies to thatsound stream goes down superpolynomially fast.



Vipul Naik






The square wave

The saw-tooth





The Schwarz space is the space of all functions f : R → Rsuch that for any polynomial function p, the map f (x)p(x)approaches ∞. If f is also infinitely differentiable, this isequivalent to demanding that all the derivatives approach 0as x →∞.It turns out that the Fourier transform of any continuousfunction with compact support is a Schwarz function.In other words, if we take a sound stream that tapers off infinite time, the contribution of very high frequencies to thatsound stream goes down superpolynomially fast.

Documents

Trigonometric functions and Fourier series (part II)vipul/studenttalks/fourierseries... · 2007-02-13 · Trigonometric functions and Fourier series (part II) Vipul Naik The Fourier