Download Trigonometric functions and Fourier series (Part 2)

Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
Trigonometric functions and Fourier series
(part II)
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
Vipul Naik
The square wave
The saw-tooth
Issues of
convergence
February 13, 2007
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Outline
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier transform
Fourier series for functions to complex numbers
The Fourier transform for topological groups
The Fourier transform for discrete groups
Computing the Fourier coefficients
The square wave
The saw-tooth
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Issues of convergence
Using kernels to prove pointwise convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Fourier series on R
Real-time streaming
Inner product for complex vector spaces
Let V be a C-vector space. An inner product is a bilinear
form h , i : V × V → C satisfying:
I
C-linearity in the first variable:
h a1 , b i + h a2 , b i = h a1 + a2 , b i
h λa , b i = λ h a , b i
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Inner product for complex vector spaces
Let V be a C-vector space. An inner product is a bilinear
form h , i : V × V → C satisfying:
I
C-linearity in the first variable:
h a1 , b i + h a2 , b i = h a1 + a2 , b i
h λa , b i = λ h a , b i
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
I
Conjugate-linearity in the second variable:
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
h a , b1 i + h a , b2 i = h a , b1 + b2 i
h a , λb i = λ h a , b i
Inner product for complex vector spaces
Let V be a C-vector space. An inner product is a bilinear
form h , i : V × V → C satisfying:
I
C-linearity in the first variable:
h a1 , b i + h a2 , b i = h a1 + a2 , b i
h λa , b i = λ h a , b i
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
I
Conjugate-linearity in the second variable:
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
h a , b1 i + h a , b2 i = h a , b1 + b2 i
h a , λb i = λ h a , b i
Inner product for complex vector spaces (contd)
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
I
Hermitian symmetry:
ha, b i=hb , ai
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Inner product for complex vector spaces (contd)
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
I
Hermitian symmetry:
ha, b i=hb , ai
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
I
Positive definiteness:
h a , a i ∈ R+ for a 6= 0
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Orthonormal set for functions to complex
numbers
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
For the space of functions from S 1 to C, an orthonormal set
is the set of functions x 7→ e inx where n ∈ Z.
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Orthonormal set for functions to complex
numbers
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
For the space of functions from S 1 to C, an orthonormal set
is the set of functions x 7→ e inx where n ∈ Z.
Under pointwise multiplication, these functions form a group
isomorphic to the group Z of integers.
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Orthonormal set for functions to complex
numbers
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
For the space of functions from S 1 to C, an orthonormal set
is the set of functions x 7→ e inx where n ∈ Z.
Under pointwise multiplication, these functions form a group
isomorphic to the group Z of integers.
The Fourier series for a function f is thus an expression:
∞
X
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
an e inx
n=−∞
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
where the infinite summation converges to f .
Expression for the Fourier coefficients
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
e inx
Because the
are orthonormal, the value an is simply the
signed magnitude of the projection of f along e inx , namely:
1
an =
π
Z
0
2π
f (x)e −inx dx
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Expression for the Fourier coefficients
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
e inx
Because the
are orthonormal, the value an is simply the
signed magnitude of the projection of f along e inx , namely:
1
an =
π
Z
2π
f (x)e −inx dx
0
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Now the map n 7→ an can be viewed as a function from Z to
C which is uniquely determined by f . Thus, changing
notation, we denote by fˆ(n) the value an corresponding to f .
Then fˆ is a map Z → C.
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
The Fourier transform
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
The transform which takes as input a function f from S 1 to
C and gives as output the function fˆ, is termed the Fourier
transform(defined).
Thus the Fourier transform takes us from functions on S 1 to
functions on Z.
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Fourier transform in a little more generality
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
Let G be a topological Abelian group. Then consider the set
of homomorphisms from G to C. This set gets equipped
with a natural group structure under pointwise addition. Call
this the dual group(defined) to G .
Then, given a suitable inner product on G , we can get a
map analogous to a Fourier transform: which takes a
continuous function f on G and outputs a function fˆ on Ĝ .
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Double dual same as original
For a locally compact topological Abelian group G , the dual
of the dual group is again G . This suggests a kind of duality
between functions on G and functions on Ĝ .
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Double dual same as original
For a locally compact topological Abelian group G , the dual
of the dual group is again G . This suggests a kind of duality
between functions on G and functions on Ĝ .
The questions now, for the case of the Fourier transform, are
thus:
I
What are the functions on S 1 for which the Fourier
transform gives a well-defined function on Z?
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Double dual same as original
For a locally compact topological Abelian group G , the dual
of the dual group is again G . This suggests a kind of duality
between functions on G and functions on Ĝ .
The questions now, for the case of the Fourier transform, are
thus:
I
I
What are the functions on S 1 for which the Fourier
transform gives a well-defined function on Z?The L2
functions
What are the functions on Z that arise as Fourier
transforms of well-defined functions on S 1 ?
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Double dual same as original
For a locally compact topological Abelian group G , the dual
of the dual group is again G . This suggests a kind of duality
between functions on G and functions on Ĝ .
The questions now, for the case of the Fourier transform, are
thus:
I
What are the functions on S 1 for which the Fourier
transform gives a well-defined function on Z?The L2
functions
I
What are the functions on Z that arise as Fourier
transforms of well-defined functions on S 1 ?
I
For which functions on S 1 is it true that the sum of the
Fourier series given by the Fourier transform actually
converges?
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Double dual same as original
For a locally compact topological Abelian group G , the dual
of the dual group is again G . This suggests a kind of duality
between functions on G and functions on Ĝ .
The questions now, for the case of the Fourier transform, are
thus:
I
What are the functions on S 1 for which the Fourier
transform gives a well-defined function on Z?The L2
functions
I
What are the functions on Z that arise as Fourier
transforms of well-defined functions on S 1 ?
I
For which functions on S 1 is it true that the sum of the
Fourier series given by the Fourier transform actually
converges?
I
For which functions on Z is it true that the Fourier
transform of the limit of its Fourier series is the same as
the original function?
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
The additive group modulo p
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
At the very other end from groups like S 1 , are groups like
(Fp , +), (additive group over a prime field).
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
The additive group modulo p
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
At the very other end from groups like S 1 , are groups like
(Fp , +), (additive group over a prime field).
For Fp , the dual group, called Fˆp , is simply the group of
one-dimensional characters of Fp under multiplication. This
is isomorphic to Fp – however, there is no natural
isomorphism.
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
The additive group modulo p
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
At the very other end from groups like S 1 , are groups like
(Fp , +), (additive group over a prime field).
For Fp , the dual group, called Fˆp , is simply the group of
one-dimensional characters of Fp under multiplication. This
is isomorphic to Fp – however, there is no natural
isomorphism.
The Fourier transform here thus takes functions on the
group and returns functions on the character group.
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Outline
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier transform
Fourier series for functions to complex numbers
The Fourier transform for topological groups
The Fourier transform for discrete groups
Computing the Fourier coefficients
The square wave
The saw-tooth
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Issues of convergence
Using kernels to prove pointwise convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Fourier series on R
Real-time streaming
The [−π , π] convention
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
In this section, we shall shift from the [0 , 2π] situation to
the [−π , π] situation, viz where the function is periodically
extended from its definition on the closed interval [−π , π].
Some clear advantages are:
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
The [−π , π] convention
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
In this section, we shall shift from the [0 , 2π] situation to
the [−π , π] situation, viz where the function is periodically
extended from its definition on the closed interval [−π , π].
Some clear advantages are:
I
The integral of any odd periodic function over this
interval is zero.
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
The [−π , π] convention
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
In this section, we shall shift from the [0 , 2π] situation to
the [−π , π] situation, viz where the function is periodically
extended from its definition on the closed interval [−π , π].
Some clear advantages are:
I
I
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
The integral of any odd periodic function over this
interval is zero.
Computing the
Fourier coefficients
Thus, the coefficients of the sin functions in the Fourier
series expansion of an even function are 0
Issues of
convergence
The square wave
The saw-tooth
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
The [−π , π] convention
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
In this section, we shall shift from the [0 , 2π] situation to
the [−π , π] situation, viz where the function is periodically
extended from its definition on the closed interval [−π , π].
Some clear advantages are:
I
I
I
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
The integral of any odd periodic function over this
interval is zero.
Computing the
Fourier coefficients
Thus, the coefficients of the sin functions in the Fourier
series expansion of an even function are 0
Issues of
convergence
Similarly the coefficients of the cos function in the
Fourier series expansion of an odd function are 0
Fourier series on R
The square wave
The saw-tooth
Using kernels to prove
pointwise convergence
Real-time streaming
The square wave
The square wave is defined as:
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
f (x) = −1 for − π ≤ x < 0
f (x) = 0 for x = 0
f (x) = 1 for 0 < x ≤ π
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
The square wave
The square wave is defined as:
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
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 expansion
must only contain sin terms. The coefficient of sin nx is:
Z
1 π
f (x) sin nx dx
π −π
which simplifies to 0 if n is even and 4/(nπ) if n is odd.
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
The square wave
The square wave is defined as:
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
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 expansion
must only contain sin terms. The coefficient of sin nx is:
Z
1 π
f (x) sin nx dx
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 X sin(2n − 1)t
π
2n − 1
n∈N
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Under linear changes
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
Suppose f is a 2π-periodic function. Then the function
x 7→ af (x) + b is also a 2π-periodic function.
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Under linear changes
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
Suppose f is a 2π-periodic function. Then the function
x 7→ af (x) + b is also a 2π-periodic function.
The Fourier coefficients of this new function are related to
the 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 the
0-1 square wave.
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
What this tells us
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
The square wave looks almost like something flat – the only
difference being that there is a jump from −1 to 1. It is
interesting that so many sine functions are hidden
underneath this apparently flat behaviour.
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
What this tells us
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
The square wave looks almost like something flat – the only
difference being that there is a jump from −1 to 1. It is
interesting that so many sine functions are hidden
underneath this apparently flat behaviour.
For the square wave, the coefficient of sin nx is only inverse
linear in n, so by the typical standards of Fourier series, it
does not taper very fast. This is to be expected since the
square wave is about as democratic as one can get between
frequencies.
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
The even saw-tooth
Define f (x) = |x| on the closed interval [−π , π] and extend
f periodically to the whole real line.
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
The even saw-tooth
Define f (x) = |x| on the closed interval [−π , π] and extend
f periodically to the whole real line.
Since f is even, all the sin coefficients are zero. This leaves
us to compute the cosine coefficient and the constant term.
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
The even saw-tooth
Define f (x) = |x| on the closed interval [−π , π] and extend
f periodically to the whole real line.
Since f is even, all the sin coefficients are zero. This leaves
us to compute the cosine coefficient and the constant term.
By simple integration, the constant term is π/2 (this is the
average height)
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Trigonometric
functions and
Fourier series (part
II)
The even saw-tooth
Define f (x) = |x| on the closed interval [−π , π] and extend
f periodically to the whole real line.
Since f is even, all the sin coefficients are zero. This leaves
us to compute the cosine coefficient and the constant term.
By simple integration, the constant term is π/2 (this is the
average height)
Also:
Z
2 π
an =
x cos(nx) dx
π 0
This simplifies to
We thus get:
−4
πn2
for odd n and 0 for even n.
Vipul Naik
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
π
4 X cos(2n − 1)t
−
2 π
(2n − 1)2
n∈N
Intuitive justification
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
Unlike the square wave function, which was very democratic
among all the functions, the saw-tooth clearly prefers lower
frequencies – this is evidenced in the relatively faster rate at
which the Fourier coefficients decline.
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Outline
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier transform
Fourier series for functions to complex numbers
The Fourier transform for topological groups
The Fourier transform for discrete groups
Computing the Fourier coefficients
The square wave
The saw-tooth
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Issues of convergence
Using kernels to prove pointwise convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Fourier series on R
Real-time streaming
The convergence results we want
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
We ideally want to say that for functions of interest the
Fourier series converges at every point. This is equivalent to
requiring that if we have a function which is orthogonal to
all the sine and cosine functions, then it is identically zero.
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
The convergence results we want
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
We ideally want to say that for functions of interest the
Fourier series converges at every point. This is equivalent to
requiring that if we have a function which is orthogonal to
all the sine and cosine functions, then it is identically zero.
The idea now is to focus at any particular point and show
that at that point the function takes the value zero. For this
purpose, we try to try to obtain a sequence of functions that
approaches the Dirac delta for the point. In other words we
try to locate a sequence of functions fn such that
h f , fn i → f (p) for that point p.
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
The convergence results we want
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
We ideally want to say that for functions of interest the
Fourier series converges at every point. This is equivalent to
requiring that if we have a function which is orthogonal to
all the sine and cosine functions, then it is identically zero.
The idea now is to focus at any particular point and show
that at that point the function takes the value zero. For this
purpose, we try to try to obtain a sequence of functions that
approaches the Dirac delta for the point. In other words we
try to locate a sequence of functions fn such that
h f , fn i → f (p) for that point p.
If each of these fn is a finite linear combination of the sine
and cosine bunches, then h f , fn i is zero for all n and
hence f (p) = 0.
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Trigonometric
functions and
Fourier series (part
II)
The Dirichlet kernel
Vipul Naik
This is simply the average of the first n functions, and is
given by:
t 7→
sin((n + 1/2)t)
sin(t/2)
with the suitable limiting value at t = 0, namely 2n + 1
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Trigonometric
functions and
Fourier series (part
II)
The Dirichlet kernel
Vipul Naik
This is simply the average of the first n functions, and is
given 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 + 2
n
X
k=1
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
cos kt
Fourier series on R
Real-time streaming
Properties of the Dirichlet kernel
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
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 any
sub-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 just
slightly away form 0.
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Riemann-Lebesgue property
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
To quantify the last statement (the value of the Dirichlet
kernel drops very rapidly just slightly after 0) the following
notion was introduced.
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Riemann-Lebesgue property
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
To quantify the last statement (the value of the Dirichlet
kernel drops very rapidly just slightly after 0) the following
notion was introduced.
A sequence fn of 2π-periodic functions is said to have the
Riemann-Lebesgue property(defined) if for any r > 0:
Z π
lim
f (t)fn (t) dt = 0
n→∞ r
for f any
L1
function.
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Riemann-Lebesgue property
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
To quantify the last statement (the value of the Dirichlet
kernel drops very rapidly just slightly after 0) the following
notion was introduced.
A sequence fn of 2π-periodic functions is said to have the
Riemann-Lebesgue property(defined) if for any r > 0:
Z π
lim
f (t)fn (t) dt = 0
n→∞ r
L1
for f any
function.
It turns out that the Dirichlet kernels do possess the
Riemann-Lebesgue property. This, along with the fact that
the 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).
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Convolution product at a point
Here’s a rough sketch of the nature of result we want to
establish:
I
Construct a sequence of functions fn , each a finite linear
combination of sines and cosines, with the
Riemann-Lebesgue property. The sequence that we
constructed was that of Dirichlet kernels.
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Convolution product at a point
Here’s a rough sketch of the nature of result we want to
establish:
I
I
Construct a sequence of functions fn , each a finite linear
combination of sines and cosines, with the
Riemann-Lebesgue property. The sequence that we
constructed was that of Dirichlet kernels.
Now, using the Riemann-Lebesgue
property, argue that
R
for any function f , limn→∞ f (t)fn (t) is f (0).
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Convolution product at a point
Here’s a rough sketch of the nature of result we want to
establish:
I
I
I
Construct a sequence of functions fn , each a finite linear
combination of sines and cosines, with the
Riemann-Lebesgue property. The sequence that we
constructed was that of Dirichlet kernels.
Now, using the Riemann-Lebesgue
property, argue that
R
for any function f , limn→∞ f (t)fn (t) is f (0).
R
More specifically argue that limn→infty f (t + x)fn (t) is
f (x).
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Convolution product at a point
Here’s a rough sketch of the nature of result we want to
establish:
I
I
I
I
Construct a sequence of functions fn , each a finite linear
combination of sines and cosines, with the
Riemann-Lebesgue property. The sequence that we
constructed was that of Dirichlet kernels.
Now, using the Riemann-Lebesgue
property, argue that
R
for any function f , limn→∞ f (t)fn (t) is f (0).
R
More specifically argue that limn→infty f (t + x)fn (t) is
f (x).
Thus if f is orthogonal to all the sin and cos functions,
f (x) must be equal to 0 for all x.
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Convolution product at a point
Here’s a rough sketch of the nature of result we want to
establish:
I
I
I
I
I
Construct a sequence of functions fn , each a finite linear
combination of sines and cosines, with the
Riemann-Lebesgue property. The sequence that we
constructed was that of Dirichlet kernels.
Now, using the Riemann-Lebesgue
property, argue that
R
for any function f , limn→∞ f (t)fn (t) is f (0).
R
More specifically argue that limn→infty f (t + x)fn (t) is
f (x).
Thus if f is orthogonal to all the sin and cos functions,
f (x) must be equal to 0 for all x.
In particular, if we take an arbitrary function, consider
its Fourier series and call the difference f , then we can
use the above argument to show that the difference is 0.
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Riemann localization principle
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
The Riemann localization principle basically states that if
two functions are locally similar, then the behaviour of their
Fourier series in those neighbourhoods would be similar.
More precisely:
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Riemann localization principle
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
The Riemann localization principle basically states that if
two functions are locally similar, then the behaviour of their
Fourier series in those neighbourhoods would be similar.
More precisely:
Suppose f and g are 2π-periodic functions and further that
f = g almost everywhere on an open interval (t − r , t + r )
with r > 0. Then the Fourier series both converge or both
diverge at t. Moreover, if they converge, they take the same
value on the whole interval.
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
The Fejer kernel
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
The Fejer kernel is the average of the values on the Dirichlet
kernel.
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Convergence result
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
We can show that for a periodic function f , if the left-hand
and right-hand limit exists at each point, and the limiting
value of derivative with respect to this is also well-defined on
each side, the Fourier series converges pointwise to the mean
of the left-hand and the right-hand limit.
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Convergence result
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
We can show that for a periodic function f , if the left-hand
and right-hand limit exists at each point, and the limiting
value of derivative with respect to this is also well-defined on
each side, the Fourier series converges pointwise to the mean
of the left-hand and the right-hand limit.
Actually, we can prove a much stronger result: given any
2π-periodic function with bounded variation, the Fourier
series converges to:
1
f (t − ) + f (t + )
2
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Convergence result
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
We can show that for a periodic function f , if the left-hand
and right-hand limit exists at each point, and the limiting
value of derivative with respect to this is also well-defined on
each side, the Fourier series converges pointwise to the mean
of the left-hand and the right-hand limit.
Actually, we can prove a much stronger result: given any
2π-periodic function with bounded variation, the Fourier
series converges to:
1
f (t − ) + f (t + )
2
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Finally, we can show that the Fourier series for any L2
function converges pointwise almost everywhere.
Outline
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier transform
Fourier series for functions to complex numbers
The Fourier transform for topological groups
The Fourier transform for discrete groups
Computing the Fourier coefficients
The square wave
The saw-tooth
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Issues of convergence
Using kernels to prove pointwise convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Fourier series on R
Real-time streaming
Streaming music
Suppose a continuous stream of sound is coming in. A
stream of sound can be viewed as a function f : R → R
where the domain R represents the time coordinate and the
range R represents the instantaneous magnitude. Note that
the study of such sound streams should be:
I Invariant under time-translation viz if the whole sound
stream is displaced in time by a certain amount, the
analysis should not differ
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Streaming music
Suppose a continuous stream of sound is coming in. A
stream of sound can be viewed as a function f : R → R
where the domain R represents the time coordinate and the
range R represents the instantaneous magnitude. Note that
the study of such sound streams should be:
I Invariant under time-translation viz if the whole sound
stream is displaced in time by a certain amount, the
analysis should not differ
I Invariant under spatial translation: That is, if a
constant sound stream (white noise) is added to the
sound stream, the analysis should not differ (except to
take out the white noise)
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Streaming music
Suppose a continuous stream of sound is coming in. A
stream of sound can be viewed as a function f : R → R
where the domain R represents the time coordinate and the
range R represents the instantaneous magnitude. Note that
the study of such sound streams should be:
I Invariant under time-translation viz if the whole sound
stream is displaced in time by a certain amount, the
analysis should not differ
I Invariant under spatial translation: That is, if a
constant sound stream (white noise) is added to the
sound stream, the analysis should not differ (except to
take out the white noise)
I Covariant under spatial/time dilation: If the
instantaneous magnitude at every point is multipled by
a constant factor, or if the whole sound stream is
stretched or compressed along the time axis, there
should simply be some proportionate changes in the
constants rather than a qualitative change.
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
As a combination of frequencies
Musical sounds arise as superpositions (pointwise sums) of
different 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 the
superposition.
For instance, the function
√
sin x + sin 2x
has no period
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
As a combination of frequencies
Musical sounds arise as superpositions (pointwise sums) of
different 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 the
superposition.
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 and
will always be there, sounds in real life will start off and
then peter off.
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
As a combination of frequencies
Musical sounds arise as superpositions (pointwise sums) of
different 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 the
superposition.
For instance, the function
√
sin x + sin 2x
has no period
I
I
The sounds may start and end at different times. Thus,
unlike the ideal sine wave which was always there and
will always be there, sounds in real life will start off and
then peter off.
There are variable levels of noise
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Can we do a Fourier transform on R?
Here’s the idea. Given f : R → R, try writing f as an infinite
linear combination of functions of the form x 7→ e irx for
every real number r . In other words we need a map f 7→ fˆ
that converts a function from R to R to another function
from R to R, but where the new function represents the
amplitude corresponding to a given frequency.
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Can we do a Fourier transform on R?
Here’s the idea. Given f : R → R, try writing f as an infinite
linear combination of functions of the form x 7→ e irx for
every real number r . In other words we need a map f 7→ fˆ
that converts a function from R to R to another function
from R to R, but where the new function represents the
amplitude corresponding to a given frequency.
Here’s a naive attempt at writing an expression for this:
Z ∞
fˆ(r ) =
f (x)e −irx dx
−∞
The problem of course is that even for the nicest continuous
functions, the right-hand side may blow up to infinity. So we
need to restrict to continuous functions that become zero
outside some finite domain – the so-called continuous
functions of compact support.
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Schwarz functions and Schwarz spaces
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Schwarz space is the space of all functions f : R → R
such that for any polynomial function p, the map f (x)p(x)
approaches ∞. If f is also infinitely differentiable, this is
equivalent to demanding that all the derivatives approach 0
as x → ∞.
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Schwarz functions and Schwarz spaces
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Schwarz space is the space of all functions f : R → R
such that for any polynomial function p, the map f (x)p(x)
approaches ∞. If f is also infinitely differentiable, this is
equivalent to demanding that all the derivatives approach 0
as x → ∞.
It turns out that the Fourier transform of any continuous
function with compact support is a Schwarz function.
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming
Schwarz functions and Schwarz spaces
Trigonometric
functions and
Fourier series (part
II)
Vipul Naik
The Schwarz space is the space of all functions f : R → R
such that for any polynomial function p, the map f (x)p(x)
approaches ∞. If f is also infinitely differentiable, this is
equivalent to demanding that all the derivatives approach 0
as x → ∞.
It turns out that the Fourier transform of any continuous
function with compact support is a Schwarz function.
In other words, if we take a sound stream that tapers off in
finite time, the contribution of very high frequencies to that
sound stream goes down superpolynomially fast.
The Fourier
transform
Fourier series for
functions to complex
numbers
The Fourier transform
for topological groups
The Fourier transform
for discrete groups
Computing the
Fourier coefficients
The square wave
The saw-tooth
Issues of
convergence
Using kernels to prove
pointwise convergence
Fourier series on R
Real-time streaming