How to convince myself that I can take n=1 here?

In summary, the conversation discusses the use of the inner product formula for two functions, as well as the question of whether or not the value of n should be taken as 1. It is argued that the degree of f does not necessarily determine the value of n, so it cannot be assumed that n is equal to 1.
  • #1
Hall
351
87
Homework Statement
In the linear space ##P_n## of all real polynomials of degree ##\leq## ##n##, define
##
\langle f,g \rangle = \sum_{k=0}^{n} f \left(\frac{k}{n}\right) ~g\left( \frac{k}{n} \right)
##
If ##f(t)=t##, find all real polynomials ##g## orthogonal to ##f##.
Relevant Equations
##\langle f(t)=t, g \rangle = \sum_{k=0}^{n} \frac{k}{n} ~g\left( \frac{k}{n} \right)##
The Homework Statement reads the question.

We have
$$
\langle f,g \rangle = \sum_{k=0}^{n} f\left(\frac{k}{n}\right) ~g\left( \frac{k}{n} \right)
$$
If ##f(t) = t##, we have degree of ##f## is ##1##, so, should I take ##n = 1## in the above inner product formula and proceed as follows
$$
\langle f(t)=t, g \rangle = \sum_{k=0}^{1} \frac{k}{1} ~g\left( \frac{k}{1} \right)$$
$$\sum_{k=0}{1} \frac{k}{1} ~g\left( \frac{k}{1} \right) = 0 $$
$$f(0) ~g(0) + f(1) ~g(1) = 0$$
$$g(1) = 0 ~~~\implies ~~ g(t) = a(1-t) ~~~~~~~~~~~~\text{for any real a}
?
$$
I'm finding it hard to convince myself that I can take ##n=1## just because the degree of ##f## is 1, the ##n## given to us is the fixed degree upto which polynomials are allowed in our linear space ##P_n##.

If I don't take ##n=1## the thing would get a little messier:
$$\sum_{k=0}^{n} \frac{k}{n} g\left( \frac{k}{n} \right) = 0 $$
$$\frac{1}{n} \sum_{k=1}^{n} k ~g\left(\frac{k}{n} \right) = 0 $$
$$\sum_{k=1}^{n} k ~g\left(\frac{k}{n} \right) = 0 $$
$$\text{One possible solution for g is} $$
$$g (t) = a~(t - 1/n) (t- 2/n) \cdots (t - n/n)
$$

Do I have to take ##n=1## or not?
 
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  • #2
Hall said:
I'm finding it hard to convince myself that I can take n=1 just because the degree of f is 1, the n given to us is the fixed degree upto which polynomials are allowed in our linear space Pn.
I agree. It could be that n = 3, for instance, but the function f(t) = t would still belong to that space. I don't think you can assume that n = 1.
 

Related to How to convince myself that I can take n=1 here?

1. How can I convince myself that I am capable of taking on n=1 in my research?

One way to convince yourself is to focus on your past achievements and successes. Remind yourself of times when you have successfully completed a similar task or project. This will help boost your confidence and belief in your abilities.

2. What steps can I take to build my confidence in tackling n=1 in my research?

Start by breaking down the task into smaller, manageable steps. This will make it less overwhelming and more achievable. Also, seek support and guidance from your peers or mentors. Surrounding yourself with positive and encouraging individuals can help boost your confidence.

3. Is it normal to doubt my abilities when taking on n=1 in my research?

Yes, it is completely normal to doubt yourself when taking on a new and challenging task. It is important to acknowledge these doubts and fears, but also to remind yourself of your skills and strengths. Remember that it is natural to feel unsure, but it is important to not let these doubts hold you back.

4. How can I handle setbacks or failures when working on n=1 in my research?

Setbacks and failures are a normal part of any research process. It is important to not let them discourage you or make you doubt your abilities. Instead, use them as learning opportunities and reflect on what went wrong and how you can improve for the future. Remember that failure is not a reflection of your worth or capabilities.

5. What are some positive affirmations I can use to convince myself that I can take on n=1 in my research?

Some positive affirmations you can use include: "I am capable and competent in my research abilities," "I have successfully overcome challenges in the past and I can do it again," and "I am confident in my skills and knowledge." Repeat these affirmations to yourself regularly to boost your self-belief and confidence.

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