Show coefficient sequences converge.

[ChemEng1]

Homework Statement

Consider P[0,1] the linear space of C[0,1] consisting of all polynomials. Show that the sequence {p_n} where p_n(t)=tⁿ has the property that its coefficient sequences converge but the sequence {p_n} does not converge in (P[0,1], ∞-norm).

Homework Equations

Observation: Graphing p_n(t)=tⁿ, as n→∞, p_n(t)→0 for t=[0,1). p_n(t)→1 for t=1.

The Attempt at a Solution

1. Does this approach work?
2. Should infinity norm be used to determine the coefficient convergence?
3. Should you always use the limit of the coefficient covergence to "test" the {p_n} convergence?
4. Is it enough to show that {p_n} doesn't converge to 0 in P[0,1] with ∞-norm?

Part 1: p_n(t) work.
For t=[0,1)
Observe t>t²>t³>t⁴>...>t^∞>0. As n→∞, tⁿ→0.

For t=1,
p_n(t)=1 for all n.

Part 2: {p_n} work.
Guess: norm[{p_n}]_∞→0.
To show: norm[{p_n}-0]_∞→0.

For a given t with 0≤t≤1, then norm[{p_n}-0]_∞=sup[|p_n(t)-0|]=sup{|t¹|, |t²|, ... , |tⁿ|}=1 (≠0). Therefore {p_n} does not converge to 0 in P[0,1] with ∞-norm.

 

[HallsofIvy]

Perhaps I am misunderstanding this but the "coefficient sequences" for t^n are (1, 0, 0, 0, ...), (0, 1, 0, 0, ...), (0, 0, 1, 0, ...)... and those do NOT converge.

 

[ChemEng1]

Wouldn't the coefficient sequences of tⁿ for a given t=[0,1] be {tⁿ, tⁿ⁺¹, tⁿ⁺², ... , t^∞}?

 

[ChemEng1]

Or am I that far off the mark?

 

[micromass]

Perhaps I am misunderstanding this but the "coefficient sequences" for t^n are (1, 0, 0, 0, ...), (0, 1, 0, 0, ...), (0, 0, 1, 0, ...)... and those do NOT converge.

Agreed, but it actually depends on how you define convergence of the coefficient sequences. If you just want separate convergence of the columns, then it does converge.

 

[ChemEng1]

Then I am officially lost.

 

[LCKurtz]

Then I am officially lost.

What is your definition of a coefficient sequence? Does your space consist of only polynomials with finitely many terms? Are your sequences of infinite length but finitely non-zero. Etc.

 

[ChemEng1]

What is your definition of a coefficient sequence? Does your space consist of only polynomials with finitely many terms? Are your sequences of infinite length but finitely non-zero. Etc.

1. The text refers to them as coordinate sequences, but doesn't formally define them. (The lecture referred to them as coefficient sequences.) It's the sequence of coefficients preceding similar basis elements generated across {x_n}.
2. The problem statement says all polynomials, so I think that includes polynomials with both finite and infinite terms.
3. p_n(t)<1, for 0≤t<1. p_n(t)=1, for t=1. I don't think any p_n(t)=0, although they converge to 0 as I tried to demonstrate.

Did I mess things up that badly?

 

[LCKurtz]

1. The text refers to them as coordinate sequences, but doesn't formally define them. (The lecture referred to them as coefficient sequences.) It's the sequence of coefficients preceding similar basis elements generated across {x_n}.
2. The problem statement says all polynomials, so I think that includes polynomials with both finite and infinite terms.
3. p_n(t)<1, for 0≤t<1. p_n(t)=1, for t=1. I don't think any p_n(t)=0, although they converge to 0 as I tried to demonstrate.

Did I mess things up that badly?

How do you expect to prove something about coordinate sequences if you don't know exactly what they are? I was expecting (guessing) you would answer my question that they are sequences that are finitely non-zero. And that the polynomial$$
a_0+a_1t+a_2t^2+...+a_nt^n$$ would map to the sequence$$
(a_0,a_1,a_2,...,a_n,0,0,0,...)$$ending in an infinite string of zeroes. That agrees with Hall's observation that$$
1\leftrightarrow e_0= (1,0,0,...)$$$$
t\leftrightarrow e_1=(0,1,0,0...)$$$$
t^2 \leftrightarrow e_2=(0,0,1,0,0...)$$and so on. So the coefficient sequences for your powers of t are $\{e_n\}$. You talk about this sequence converging. But you haven't defined what convergence in the sequence space means.

 

[ChemEng1]

I see that I have completely confused everything. Hopefully this post will help sort things out.

How do you expect to prove something about coordinate sequences if you don't know exactly what they are?

This proof from the book describes coordinate sequences as {\lambda^{n}_{k}}.

I was expecting (guessing) you would answer my question that they are sequences that are finitely non-zero.

I don't think the problem statement precludes polynomials with infinite amount of nonzero terms?

And that the polynomial a₀+a₁t+a₂t²+...+a_ntⁿwould map to the sequence (a₀,a₁,a₂,..., a_n,0,0,0,...) ending in an infinite string of zeroes.

The book's notation for coordinate sequences implies that they would have a cardinality of n, but not necessarily infinity.

You talk about this sequence converging. But you haven't defined what convergence in the sequence space means.

Hopefully the attached theorem shows what is meant by coordinate convergence better than I have. However, I don't think we can use that theorem as it applies to finite dimensional space, and this problem is infinitely dimensional.

 

[micromass]

I see that I have completely confused everything. Hopefully this post will help sort things out.


This proof from the book describes coordinate sequences as {\lambda^{n}_{k}}.
View attachment 43614

Aha, that actually makes sense. So you indeed need to prove that every column converges seperately.

I don't think the problem statement precludes polynomials with infinite amount of nonzero terms?

Let's settle this first. A polynomial by definition has a finite number of terms. Something like

t+2t^2+3t^3+...+nt^n+...

is not a polynomial. If you loook up your definition of polynomial in your book, you will see this.