Frejanne Ruoff
Smooth functions, orthogonal polynomials and
rapidly decreasing sequences
Bachelor thesis, June 8, 2011
Supervisor: dr. M.F.E. de Jeu
Mathematisch Instituut, Universiteit Leiden
Table of contents
1 Introduction
1
2 Formulation of the main theorem
2
3 Smooth functions
3
4 Orthogonal polynomials
6
5 Rapidly decreasing sequences
9
6 Proof of the main theorem
11
7 Concluding remarks
12
8 References
12
A Banach space C l ([a, b])
13
ii
1
Introduction
Let w : [a, b] → R be a real-valued L1 -function on a bounded interval with the following
property:
∃ δ > 0, ∃ C > 0 and ∃ ν > 0 such that for all xt ∈ [a, b] :
w(x) ≥ C |x − xt |ν
(1.1)
for almost all x ∈ [a, b] with |x − xt | < δ.
An example of a function w with this property is
w(x) ≥ C
for all x ∈ [a, b], where C > 0. Using w as a weight function, we define the weighted
L2 space L2 ([a, b], w(x)dx). Let the inner product on L2 ([a, b], w(x)dx) for all f, g ∈
L2 ([a, b], w(x)dx) be as follows
Z b
f (x)g(x)w(x)dx.
(f, g) =
a
1, x, x2 , . . .
We note that
are linearly independent elements of L2 ([a, b], w(x)dx). If we
apply the Gram-Schmidt process, the process will give us a sequence of polynomials
{pn }∞
n=0 such that the degree of pn is exactly n. The sequence of polynomials is an orthonormal system in L2 ([a, b], w(x)dx). On general grounds this is even an orthonormal
basis which results in the fact that we can write every arbitrary f ∈ L2 ([a, b], w(x)dx)
as
f=
∞
X
(f, pn ) pn .
n=0
Let us define the map F : L2 ([a, b], w(x)dx) → `2 that assigns to every element the sequence of coefficients with respect to the orthonormal basis {pn }∞
n=0 . Now the question
rises what will happen with the image of a ‘decent’ function. An informal meta-principle
tells us that smoothness will result in good convergence. This gives us the idea that
we are able to say something about F(f ) when f is an infinitely differentiable function.
It turns out to be even more beautiful than one could hope for: infinitely differentiable functions are mapped bijectively via F to the rapidly decreasing sequences. A
rapidly decreasing sequence {an }∞
n=0 has the property that it goes to 0 faster than every negative power of n. The main goal of this thesis is to give a proof of this statement.
This, however, will not be the first thesis or paper on this subject. In the late sixties
Zerner stated a theorem [3] very similar to the theorem we will prove in this thesis. The
main differences are the following. Firstly, his theorem covers a more restricted case,
namely the case where the weight function w has the property w(x) ≥ C > 0 for all
x ∈ [a, b]. On the other hand, his theorem treated several dimensions instead of one,
generalising the situation more than we will. He also gave a sketch of a proof of the
theorem. Shortly after his French publication in Comptes Rendus, Pavec gave a full
proof [2] of the theorem given the sketch by Zerner and taking w(x) ≡ 1 for all x ∈ [a, b].
These two underlying papers have been used to construct the proof given here and we
will reflect on their results and use at the end of this thesis.
1
2
Formulation of the main theorem
We will have a closer and more formal look at the statement we want to prove. Let
a, b ∈ R and a < b throughout this whole thesis. We start by defining the following
polynomial space.
Definition 2.1. The space Pk ([a, b]) is the space of real-valued polynomials in one
variable on the interval [a, b] with a, b ∈ R and with degree exactly k ∈ N.
The polynomials in this thesis are all real-valued, because certain theorems are only
applicable to real-valued polynomials. The ‘decent’ functions we mentioned in the introduction make up the following space.
Definition 2.2. Define the space C ∞ ([a, b]) ⊂ L2 ([a, b], w(x)dx) as the space of infinitely differentiable real-valued functions on the interval [a, b].
We will define the rapidly decreasing sequences formally now.
Definition 2.3. A sequence {an }∞
n=0 is called a rapidly decreasing sequence if for all
k ∈ N:
lim nk an = 0.
n→∞
The space consisting of these rapidly decreasing sequences is key in our proof.
Definition 2.4. Define the space (s) as the space of rapidly decreasing sequences
{an }∞
n=0 .
Now we are ready to state the theorem that will be proven in this thesis.
Theorem 2.5. The restriction of F to the subset C ∞ ([a, b]) is a bijection between the
vector spaces C ∞ ([a, b]) and (s).
First, we will prove that F(C ∞ ([a, b])) ⊂ (s). Then we will turn to orthogonal polynomials and give a bound for
(j)
kpk k∞
in terms of kpk k2 for all pk ∈ Pk ([a, b]) and j = 0, 1, 2, . . .. Afterwards we will use that
bound to show that (s) ⊂ F(C ∞ ([a, b])). At that point we only have to conclude that
F is indeed a bijection between vector spaces.
This result gives rise to the statement that says that there is an isomorphism between
C ∞ ([a, b]) and (s) as topological vector spaces. We will not prove that statement.
2
3
Smooth functions
In this section we will prove that F maps C ∞ ([a, b]) injectively into (s).
Let f ∈ C ∞ ([a, b]) be arbitrary. We define the following distances:
ck = d(f, Pk ([a, b]))∞ =
αk = d(f, Pk ([a, b]))2 =
inf
kf − pk k∞
pk ∈Pk ([a,b])
inf
pk ∈Pk ([a,b])
kf − pk k2 .
We now use a result from [1] that will, after some elaboration, help us to bound the
∞
sequence {αk }∞
k=0 . Let g ∈ C ([−1, 1]). First define
∆n (x) =
(
1
max
if n = 0,
√
1−x2 1
, n2
n
if n = 1, 2, . . . .
This function has the modulus of continuity ω(g, h) defined as follows:
ω(g, h) = max |g(x + t) − g(x)| .
x,t,|t|≤h
Now we are ready to state a consequence of the theorem, cf. [1, p. 66].
Theorem 3.1. For every g ∈ C ∞ ([−1, 1]) and for every q = 1, 2, . . . there is a sequence
of polynomials pn (x) for which
|g(x) − pn (x)| ≤ Mq ∆n (x)q ω(g (q) , ∆n (x)),
−1 ≤ x ≤ 1,
(3.1)
n = q, q + 1, . . . ;
the constant Mq depends only upon q.
We take x ∈ [−1, 1] arbitrarily and apply this theorem. We then have that for n ∈ N>0
1
n
∆n (x) ≤
and for n = 0
∆n (x) = 1
by definition of ∆n . Clearly
ω(g (q) , ∆n (x)) =
max
x,t,|t|≤∆n (x)
(q)
g (x + t) − g (q) (x)
≤ 2 · kg (q) k∞ .
When we substitute these results into (3.1), we get
|g(x) − pn (x)| ≤ 2Mq kg (q) k∞
3
1
.
nq
Since this holds for all x ∈ [−1, 1], we have
kg − pn k∞,[−1,1] ≤ 2Mq kg (q) k∞
1
,
nq
n ≥ q.
(3.2)
We can scale our f ∈ C ∞ ([a, b]), turning it into a function g in C ∞ ([−1, 1]). Then,
turning to the sequence {ck }∞
k=0 , (3.2) implies
ck = inf kf − pk k∞ ≤ Cq kf (q) k∞
pk ∈Pk
1
,
kq
k ≥ q;
(3.3)
where Cq depends only on q.
Corollary 3.2. From (3.3) we conclude that for every f ∈ C ∞ ([a, b])
lim k l ck = 0
for all l ∈ N
k→∞
and thus {ck }∞
k=0 ∈ (s).
The most important lemma of this section will now be stated and proved.
Lemma 3.3. Let F : L2 ([a, b], w(x)dx) → `2 be the map assigning to every element f
the sequence of coefficients with respect to the orthonormal basis of L2 ([a, b], w(x)dx).
Then F(f ) ∈ (s) for all f ∈ C ∞ ([a, b]).
Proof. We will denote F(f ) by {uk }∞
k=0 . We will have a closer look at the sequence
∞
{αk }k=0 :
αk = inf kf − pk k2
pk ∈Pk
Z
b
|f (x) − pk (x)| w(x)dx
= inf
pk ∈Pk
a
Z
≤ inf
pk ∈Pk
21
2
b
kf −
pk k2∞ w(x)dx
12
a
Z
= inf kf − pk k∞
pk ∈Pk
21
b
w(x)dx
a
and therefore we have that for all k ∈ N:
21
Z b
αk ≤
w(x)dx
· ck .
a
Combining this with corollary 3.2 yields that {αk }∞
k=0 ∈ (s).
Since Pk = span (p0 , p1 , . . . , pk ), we have
αk2 =
X
r≥k+1
4
|ur |2
and so
|uk+1 | ≤ |αk | .
We have showed already that {αk }∞
k=0 ∈ (s). Therefore we may now conclude that
∞
{uk }k=0 ∈ (s). Since we defined {uk }∞
k=0 as being the image of an arbitrary f in
C ∞ ([a, b]) under F, this gives the sought for result.
5
4
Orthogonal polynomials
We will now have a look at polynomials of degree k ∈ N and bound the supremum norm
of their j-th derivative by their 2-norm. We will use the following, cf. [1, p. 40].
Theorem 4.1 (Markov Inequality). If qk ∈ Pk ([−1, 1]), then
kqk0 (x)k∞ ≤ k 2 kqk k∞ ,
−1 ≤ x ≤ 1.
Let pk ∈ Pk ([a, b]). If we want the theorem to be applicable to pk , we have to transform
the interval used in this theorem. Therefore we define the polynomial pk ∈ Pk ([−1, 1])
as follows:
b−a
a+b
pk (x) = pk
.
x+
2
2
Then
p0k (x)
=
p0k
b−a
a+b
x+
2
2
·
b−a
2
and therefore we get for all x ∈ [a, b]
0
2
a+b
2 p (x) = p0
x
−
·
k
k b−a
b−a
b − a
2
kpk k∞ .
≤ k2
b−a
Since this holds for all x ∈ [a, b], we get the following corollary.
Corollary 4.2. If pk ∈ Pk ([a, b]), then
kp0k k∞ ≤ k 2
2
kpk k∞ .
b−a
We will now prove the following key statement.
Theorem 4.3. Let δ > 0, C > 0, ν > 0 so that property (1.1) of the weight function w
given in the introduction hold and let j = 0, 1, 2, . . .. For all polynomials pk ∈ Pk ([a, b])
with k ∈ N such that
b−a
≤ δ,
2k 2
we have
(j)
kpk k∞ ≤
k!
(k − j)!
2 2
b−a
j
s
k ν+1
6
2ν (ν + 1)(ν + 2)(ν + 3)
kpk k2 .
C(b − a)ν+1
Proof. Let pk ∈ Pk be arbitrary. Furthermore, let xt ∈ [a, b] such that
|pk (xt )| = kpk k∞
and we may assume without loss of generality that pk (xt ) ≥ 0. This implies that
pk (xt ) = kpk k∞ .
Take δ > 0, C > 0, ν > 0 so that property (1.1) of the weight function w given in the
introduction hold. Then for all x ∈ [a, b] with pk (xt ) − kp0k k∞ |xt − x| ≥ 0
pk (x) ≥ pk (xt ) − kp0k k∞ |xt − x|
kp0k k∞
|xt − x|
= kpk k∞ 1 −
kpk k∞
kp0k k∞
|xt − x| .
≥ kpk k∞ 1 −
kpk k∞
From corollary 4.2 we know that
kp0k k∞
2k 2
≤
kpk k∞
b−a
and thus
2k 2
pk (x) ≥ kpk k∞ 1 −
|xt − x|
b−a
for all x ∈ [a, b] with |xt − x| ≤
b−a
.
2k2
We will look at the 2-norm of pk . Note that at least one of the two intervals
b−a
b−a
xt , xt +
and
x
−
,
x
t
t
2k 2
2k 2
is contained in the interval [a, b]. Since both intervals will give the same result, we use
the first interval mentioned for the following computation. Let k ∈ N>0 be big enough
for b−a
≤ δ to hold.
2k2
kpk k22
Z
b
=
|pk (x)|2 w(x)dx
a
Z
≥
xt + b−a
2
2k
|pk (x)|2 w(x)dx
xt
xt + b−a
2
2
2k 2
≥
1−
(x − xt ) w(x)dx
b−a
xt
2
Z xt + b−a
2k2
2k 2
≥
kpk k2∞ 1 −
(x − xt ) C (x − xt )ν dx
b−a
xt
Z
=
2k
kpk k2∞
(b − a)ν+1
2
Ckpk k∞ ν 2(ν+1)
2 k
(ν + 1)(ν +
7
2)(ν + 3)
.
From this we know now
s
kpk k∞ ≤ k ν+1
2ν (ν + 1)(ν + 2)(ν + 3)
· kpk k2 .
C (b − a)ν+1
This result can be combined with corollary 4.2. That will give us
s
2ν (ν + 1)(ν + 2)(ν + 3)
ν+1
0
2 2
· kpk k2 .
k
kpk k∞ ≤ k
b−a
C (b − a)ν+1
When we apply corollary 4.2 repeatedly, say j times, we obtain the desired result for
the j-th derivative of pk , namely
s
2 j
k!
2
2ν (ν + 1)(ν + 2)(ν + 3)
(j)
kpk k∞ ≤
k ν+1
kpk k2 .
(k − j)!
b−a
C(b − a)ν+1
The theorem we have just proven gives rise to a remarkable corollary, that we state
below.
2
Corollary 4.4. If {pk }∞
k=0 is defined to be an orthonormal basis for L ([a, b], w(x)dx),
then for every j = 0, 1, 2, . . . there is a constant Cj such that
(j)
kpk k∞ ≤ Cj k 2j+ν+1 ,
8
k = 0, 1, 2, . . . .
5
Rapidly decreasing sequences
The results we found in section 4 will help us to prove here that (s) ⊂ F(C ∞ ([a, b])).
Lemma 5.1. Let F : L2 ([a, b], w(x)dx) → `2 be the map assigning to every element f
the sequence of coefficients with respect to the orthonormal basis of L2 ([a, b], w(x)dx).
Then (s) ⊂ F(C ∞ ([a, b])).
Proof. Take an arbitrary sequence {αn }∞
n=0 ∈ (s). In appendix A we prove that, for
l = 0, 1, 2, . . ., C l ([a, b]) is a Banach space with the norm defined by
kf kl =
l
X
kf (i) k∞ .
i=0
Recall that {pn }∞
n=0 was defined as being an orthonormal system of polynomials of
degree n ∈ N that formed a basis for L2 ([a, b], w(x)dx). Furthermore, for all n ∈ N, we
know from corollary 4.4
kpn kl =
l
X
kp(j)
n k∞
j=0
≤
l
X
Cj n2j+ν+1
j=0
≤
l
X
Cj n2j+ν+1
j=0
≤ C̃l n2l+ν+1
for some C̃l ≥ 0. We have
∞
X
|αn | kpn kl ≤
n=0
∞
X
|αn | C̃l n2l+ν+1 .
n=0
m
Since {αn }∞
n=0 ∈ (s), we know that for all m ∈ N holds that limn→∞ αn n = 0 for all
2l+ν+1
m ∈ N. From a certain n0 ∈ N on, the product |αn | C̃l n
will be less or equal to
1
and
we
know
that
n2
∞
X
1
< ∞.
n2
n=0
We can now conclude that
∞
X
|αn | kpn kl < ∞.
n=0
9
P∞
is absolutely convergent in C l ([a, b]) for all l ∈ N, which implies that
l
n=0 αn pn is convergent in C ([a, b]). Now set
Hence
P∞
n=0 αn pn
sm :=
m
X
αn p n .
n=0
For l = 0, 1, 2, . . . let
in C l ([a, b]).
gl = lim sm
m→∞
Since the inclusion C l+1 ([a, b]) ⊂ C l ([a, b]) is continuous for l = 0, 1, 2, . . ., we see that
g0 = g1 = g2 = . . . = f
for some f ∈ C ∞ ([a, b]). Certainly
sm → f
in C([a, b])
implies that
sm → f
in L2 ([a, b], w(x)dx),
so that
F(sm ) → F(f )
in `2 .
Since F(sm ) = (α0 , α1 , α2 , . . . , αn , 0, 0, . . .), we see that F(f ) = (α0 , α1 , α2 , . . .), as
required. Therefore (s) ⊂ F(C ∞ ([a, b])).
10
6
Proof of the main theorem
We will prove the main theorem using different lemmas that have been proven in previous chapters. Firstly, define [a, b] as a finite closed interval with a, b ∈ R. Let w be
a real-valued weight function in L1 ([a, b], dx) for L2 ([a, b], w(x)dx) with the following
property: ∃ δ > 0, ∃ C > 0 and ∃ ν > 0 such that for all xt ∈ [a, b]
w(x) ≥ C |x − xt |ν
for almost all x ∈ [a, b] with |x − xt | < δ. The sequence of polynomials {pn }∞
n=0 is the
corresponding orthonormal basis of L2 ([a, b], w(x)dx). The map
F : L2 ([a, b], w(x)dx) → `2
(6.1)
assigns to every element the sequence of coefficients with respect to that orthonormal
basis.
Lemma 3.3 and lemma 5.1, both of which we have given a proof, have told us that F
is injective and surjective respectively. They therefore give the proof of the following
theorem.
Theorem 6.1. The restriction of the map F to the subset C ∞ ([a, b]) assigning to
every element f the sequence of coefficients with respect to the orthonormal basis of
L2 ([a, b], w(x)dx), is a bijection between the vector spaces C ∞ ([a, b]) and (s).
This result gives rise to the statement that says that there is an isomorphism between
C ∞ ([a, b]) and (s) as topological vector spaces.
11
7
Concluding remarks
The proof that we have seen in this thesis has been constructed with the help of the
proof by Pavec [2]. That paper covers a case which is more restrictive in the sense that
it is assumed that w(x) ≥ C > 0. At the same time it is more general because it treats
bounded domains (with sufficiently smooth boundary) in an arbitrary dimension. The
proof given in [2] that F(f ) ∈ (s) is a bit brief, consisting basically of a reference to
[1]. However, we have not been able to find a multi-variable result as needed in [2],
and the proof of the underlying result in [1] (i.e., of our theorem 3.1) does not seem
to generalize easily to several variables. The proof of the other key step, namely that
(s) ⊂ F(C ∞ ([a, b])), as given in [2], is in fact given for w ≡ 1 but this is easily adapted
for w(x) ≥ C > 0. Our conclusion at this moment is that the result as claimed in [2] and
[3] may well be true but that the argumentation seems not to be entirely complete yet.
For one dimension it is certainly sound, and as we have shown, it can be generalized to
more general weight functions than those bounded away from zero.
8
References
[1] G.G. Lorentz, Approximation of Functions, Holt, Rinehart and Winston, New York,
1966.
[2] M.R. Pavec, Isomorphisme entre D(Ω) et (s), d’après une note de M. Zerner, Publications des Séminaires de Mathématiques (Fasc. 1: Séminaires d’Analyse fonctionnelle), Rennes 6 (1969).
[3] M. Zerner, Développement en séries de polynômes orthonormaux des fonctions
indéfiniment différentiables, C. R. Acad. Sc. Paris Sér. A-B 268 (1969), A218-A220.
12
A
Banach space C l ([a, b])
In this appendix we will prove that the space C l ([a, b]), l = 0, 1, 2, . . ., with a certain
norm is a Banach space.
Lemma A.1. The space C l ([a, b]) with the norm defined by
kf kl =
l
X
kf (i) k∞
i=0
is a Banach space.
l
Proof. Take an arbitrary Cauchy sequence {fn }∞
n=0 in C ([a, b]), which is equivalent
with
n o∞
0 ∞
{fn }∞
,
f
,
.
.
.
,
fn(l)
n n=0
n=0
n=0
all being Cauchy sequences in C([a, b]) with the supremum norm. We will prove the
lemma using induction on l ∈ N.
0 ∞
Suppose l = 1. We already know that {fn }∞
n=0 and {fn }n=0 are both Cauchy in C([a, b])
with the norm k.k∞ . The space C([a, b]) with the norm k.k∞ is a Banach space. That
means that there are g, h ∈ C([a, b]) such that
lim fn = g
n→∞
lim f 0
n→∞ n
=h
in C([a, b]) with k.k∞
in C([a, b]) with k.k∞ .
Let x ∈ [a, b] be arbitrary. From the fundamental theorem of calculus we know that
Z x
fn (x) :=
fn0 (t)dt
a
is a differentiable function, so that for n → ∞, since fn0 → h uniformly,
Z x
g(x) :=
h(t)dt.
a
This implies that g is a differentiable function and g 0 = h, so that
lim f 0
n→∞ n
= g0
in C([a, b]) with k.k∞ .
Therefore g ∈ C 1 ([a, b]). Now we may conclude that the Cauchy sequence {fn }∞
n=0 in
C 1 ([a, b]) converges to an element of C 1 ([a, b]) and thus that C 1 ([a, b]) is a Banach space
with the norm k.k1 .
13
Let us assume that C l ([a, b]) is a Banach space for all l ≤ L. Now take l = L + 1
(L+1) ([a, b]). Then there exist g ∈ C L ([a, b]) and
and let {fn }∞
n=0 be a C-sequence in C
h ∈ C([a, b]) such that
lim fn = g
n→∞
lim f 0
n→∞ n
= g0
in C([a, b]) with k.k∞
in C([a, b]) with k.k∞
..
.
lim f (L) =
n→∞ n
lim f (L+1)
n→∞ n
g (L)
in C([a, b]) with k.k∞
=h
in C([a, b]) with k.k∞ .
Again, we will use the fundamental theorem of calculus. We know that
Z x
(L)
fn(L+1) (t)dt
fn (x) :=
a
(L+1)
is a differentiable function, so that for n → ∞, since fn
Z x
(L)
g (x) :=
h(t)dt.
→ h uniformly,
a
This results in g (L) being differentiable and therefore g (L+1) = h. Then
lim f (L+1)
n→∞ n
= g (L+1)
in C([a, b]) with k.k∞ .
We now know that g (L) ∈ C 1 ([a, b]) and thus that the Cauchy sequence {fn }∞
n=0 in
l
l
l
C ([a, b]) converges to an element of C ([a, b]). We conclude that C ([a, b]) is a Banach
space with the norm k.kl .
14