Find a monic polynomial orthogonal to all polynomials of lower degrees.

In summary, we discussed the space of continuous functions and how to find a monic polynomial that is orthogonal to all polynomials of lower degrees. We also looked at specific examples for degrees 0, 1, 2, and 3 and explored the process of finding such a polynomial.
  • #1
Dustinsfl
2,281
5
Space of continuous functions.

Inner product [tex]<f,g>=\int_{-1}^{1}f(x)g(x)dx[/tex].

Find a monic polynomial orthogonal to all polynomials of lower degrees.

Taking a polynomial of degree 3.

[tex]x^3+ax^2+bx+c[/tex]

Need to check [tex]\gamma, x+\alpha, x^2+\beta x+ \lambda[/tex]

[tex]\int_{-1}^{1}(\gamma x^3+\gamma a x^2 +\gamma bx + \gamma c)dx[/tex]
[tex]=\frac{\gamma x^4}{4}+\frac{\gamma a x^3}{3}+\frac{\gamma b x^2}{2}+\gamma c x|_{-1}^{1}[/tex]
[tex]=\frac{2\gamma a}{3}+2\gamma c=0\Rightarrow c=-\frac{a\gamma}{3}[/tex]

[tex]\int_{-1}^{1}(x^3+ax^2+bx+c)(x+\beta)dx[/tex]
[tex]\int_{-1}^{1}\left(x^4+ax^3+bx^2-\frac{a\alpha x}{3}+\beta x^3 +\alpha\beta x^2+b\beta x-\frac{a\alpha\beta}{3}\right)dx=6+10b+10a\beta-10a\alpha\beta=0[/tex]

What do I do with that?
 
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  • #2
well, are you given the degree you polynomial is supposed to be, or are you suppose to find a formula for any n (degree)?

for n = 0, we can choose p0(x) = 1 (we don't have any polynomials of lesser degree, so any constant will do. i like 1, don't you?).

for n = 1, the only requirement is that <p1(x),c> = 0 for any constant polynomial k(x) = c, that is:

[tex]\int_{-1}^1(ax+b)c\ dx = 0[/tex]

or: 2b = 0, so b = 0, thus p1(x) = ax. again there is no reason not to choose a = 1.

for n = 2, we need <p2,c> = 0, and <p2,ax+b> = 0

if p2(x) = rx2+sx+u, this means r = -3u, from the first inner product, and s = 0 from the second.

so p2(x) = u(3x2 - 1). again, any non-zero choice will do, although one might be inclined to choose u such that <p2(x),p2(x)> = 1.

now, for n = 3:

you may as well assume that γ ≠ 0, since it is arbitrary, which gives:

c = -a/3, not c = -aγ/3 (just divide by γ).

in your second inner product, you start with x+β, instead of x+α, and somehow wind up with something with α's and β's. huh? pick a variable for the constant term of your generic linear polynomial, and stick with it.
 

Related to Find a monic polynomial orthogonal to all polynomials of lower degrees.

1. What is an inner product?

An inner product is a mathematical operation that takes two vectors and produces a single scalar value. It is often used to measure the angle between two vectors or to determine the length of a vector. In other words, it is a way to quantify the relationship between two vectors in a vector space.

2. What is the difference between inner product and dot product?

The dot product is a specific type of inner product that is defined for Euclidean vector spaces. It is calculated by multiplying the corresponding components of two vectors and then summing the results. However, an inner product is a more general concept that can be defined for any vector space. It may involve different mathematical operations depending on the properties of the vector space.

3. How is inner product integration used in physics?

In physics, inner product integration is used to calculate physical quantities such as energy, work, and momentum. For example, in quantum mechanics, the inner product of two wavefunctions is used to calculate the probability amplitude for a particle to be in a certain state. In general, inner product integration is used to describe the relationship between different physical quantities in a mathematical way.

4. What are some applications of inner product integration?

Inner product integration has many applications in mathematics and physics. In addition to its use in calculating physical quantities, it is also used in signal processing, image processing, and data analysis. It is also a fundamental concept in functional analysis, which is a branch of mathematics that studies vector spaces of functions.

5. How does inner product integration relate to orthogonality?

Inner product integration is closely related to orthogonality. In fact, two vectors are considered orthogonal if their inner product is equal to zero. This means that they are perpendicular to each other in a geometric sense. Orthogonality is an important concept in linear algebra and is used in many applications, such as finding the best fit line for a set of data points.

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