Chebyshev polynomials Puns

Initially, I just wanted to show how Chebyshev approximations can be calculated with the Fourier transform and then decided to go crazy and add Chebyshev type II and Legendre and Taylor series and compare how their errors differ. I could extend this to generalized Jacobi polynomials if anybody is interested, but those 3 are sort of a greatest hits compilation of Jacobi polynomials.

https://www.desmos.com/calculator/nktfc1hdqv

I have The Chebyshev Polynomial C_{n}.

I have to argue that C_{n+1} can be written as:

C_{n+1}(t) = 2^n (t − θ_{0,0})(t − θ_{0,1})…(t − θ_{0,n})

Where the parentheses can be written as w^c_{n}(t)

and where θ_{0,k}, k={0,1,…,n},

k = 0,1,…,n are the n+1 roots of C_{n+1}.

I have to show that max_{t∈[−1,1]} for |w^n_{c}(t)| = 1/(2^n).

Have tried many things in vain. Know I have to find some kind of properties of the Chebyshev Polynomial. But I'm lost. Don't know which to use. Would be very grateful for any help.

How does chebyshev polynomial minimize the problem of Runges Phenomenon ?

I have been learning this notion for a bit now in my trig course, yet I can't seem to be able to prove the following statement about chebyshev polynomials:

If n is odd, Pn(x) contains only odd powers of x. Similarly, if n is even, Pn(x) contains

only even powers of x.

Can somebody give me a hint?

I'm trying to understand the approximation error for the interpolation polynomial using Chebyshev nodes. I understand the formula for the interval [-1,1], but when we have an arbitrary interval [a,b], I don't know how to deduce the formula.

I want to know ho to go from the previous-to-last formula to the last formula of this wikipedia article: https://en.wikipedia.org/wiki/Chebyshev_nodes#Approximation

The program ERF calculates the error function:

erf(x) = 2 / √π * ∫ e^-(t^2) dt from t = 0 to t = x

The calculation uses the series:

erf(x) = 2 * e^(-(x^2)) / √π * Σ( 2^n * x^(2n+1) / (1 * 3 * ... * (2n+1)) from n=0 to ∞

The program BERN calculates Bernoulli numbers for positive integers n > 2. The calculator is set to Radian mode.

The formula used:

B_n = (-cos(n * π/2))^((n+2)/2) * 2 * n! * Σ(2 * j * π)^(-n) from j=1 to ∞

Chebyshev Polynomials

The program CHEBY calculates the value of a first-order Chebyshev polynomial using the recurring formula where:

T_n(x) = 2 * x * T_n(x) - T_(n-1)(x)

where T_0(x) = 1, T_1(x) = x

Link: http://edspi31415.blogspot.com/2019/10/hp-42sdm42-error-function-bernoulli.html

Hello,

Wiki has the proof that Chebyshev polynomials have the minimal ∞-norm in [-1,1], among the polynomials of degree n with leading coefficient 1:

https://en.wikipedia.org/wiki/Chebyshev_polynomials#Minimal_.E2.88.9E-norm

What I'm trying to find though, is the proof that Chebyshev polynomials are unique among such polynomials - that there is no other different polynomial (with the above attributes) that has the same ∞-norm in [-1,1] as the Chebyshev polynomial of degree n.

Edit: from what I understand, it shouldn't be something very complicated. It's supposed to be very similar to the above linked proof. I just cant understand how to get there...

Thank you.

Will Chebyshev polynomials introduce any intermodulation distortion? I was under the impression that this would not occur. If I am correct, what are ways to introduce intermodulation distortion? Is there a way similar to Chebyshev polynomials?

Thanks in advance.

Can someone explain the scientific and/or engineering significance of the Chebyshev polynomials? They seem to be of a particular and esoteric significance.

I'm not a mathematician. I initially got interested in my title question after I read G.H. Hardy's boast that his math was "useless," except that his specialty, number theory, turned out to be valuable in cryptography. The only other example I know of is some of Ramanujan's math, which apparently has applications in current work on quantum gravity. Please tell me about some other examples.

I have been messing around a bit with chebyshev polynomials with team playing around with plugins, and we are unsure how to properly use them, and how their properties work. I also was think of sending the signals through a soft clipping function, since we have problems with hard clipping, but yeah I have unsure of what's a good idea, and I over-rely on experimentation.

Using my interpolation function on f(x)=cos(x) and the interval [a,b] = [-pi/2, pi/2 using second-order interpolation and both, Chebyshev economisation and equidistant data points, I am trying to create a plot that contains 2 plots for the data.

I am unsure if I am performing my interpolation correctly, I am not allowed to use the built-in Lagrange function in SymPy to solve the problem, the question I've been asked is to write a function that takes a function f, order n, array (of any length) z and the array x with the data points for the polynomial interpolation, and returns the values pn(z) of the polynomial interpolation does my interpolation function answer this question?

Also, I don't think I've set up my plots correctly, when trying to plot just using the Chebyshev nodes I get the error: * x and y must have the same first dimension, but have shapes (10,) and (1,)*. How would I fix this error or is there a better way to plot my data instead of what I am currently attempting?

I am new to python so any help is very much appreciated

```

import numpy as np

import matplotlib.pyplot as plt

def chebs(a, b, n):

i = np.array(range(n))

x = np.cos((2*i+1)*np.pi/(2*(n)))

return 0.5*(b-a)*x+0.5*(b+a)

x_vals = chebs(-np.pi/2, np.pi/2, 10)

from sympy import Symbol

t = Symbol('t')

def f(x):

return np.cos(x)

def interpolation(x, z, t):

d = len(x)

if len(x) != len(z):

print("Error: the length of x and z is different")

else:

L = 0

for i in range (d):

p = 1

for j in range (d):

if j != i:

p *= (t-x[j]/(x[i] - x[j]))

L += z[i]*p

return L

x = np.linspace(-np.pi/2, np.pi/2, 10)

plt.plot(x, interpolation(x, f(x), 2), label="Lagrange")

plt.plot(x_vals, interpolation(x_vals, f(x), 2), label= "Chebyshev")

plt.legend(loc='upper left')

plt.show()

Do your worst!

For context I'm a Refuse Driver (Garbage man) & today I was on food waste. After I'd tipped I was checking the wagon for any defects when I spotted a lone pea balanced on the lifts.

I said "hey look, an escaPEA"

No one near me but it didn't half make me laugh for a good hour or so!

Pilot on me!!

Dad jokes are supposed to be jokes you can tell a kid and they will understand it and find it funny.

This sub is mostly just NSFW puns now.

If it needs a NSFW tag it's not a dad joke. There should just be a NSFW puns subreddit for that.

Edit* I'm not replying any longer and turning off notifications but to all those that say "no one cares", there sure are a lot of you arguing about it. Maybe I'm wrong but you people don't need to be rude about it. If you really don't care, don't comment.

What did 0 say to 8 ?

" Nice Belt "

So What did 3 say to 8 ?

" Hey, you two stop making out "

I won't be doing that today!

You take away their little brooms

There hasn't been a post all year!

This morning, my 4 year old daughter.

Daughter: I'm hungry

Me: nerves building, smile widening

Me: Hi hungry, I'm dad.

She had no idea what was going on but I finally did it.

Thank you all for listening.

Initially, I just wanted to show how Chebyshev approximations can be calculated with the Fourier transform and then decided to go crazy and add Chebyshev type II and Legendre and Taylor series and compare how their errors differ. I could extend this to generalized Jacobi polynomials if anybody is interested, but those 3 are sort of a greatest hits compilation of Jacobi polynomials.

https://www.desmos.com/calculator/nktfc1hdqv

My h/w question asks me to "write a function that takes the limits of an interval and the order n of the polynomial interpolation pn(x), and returns the data points xi that can be used for Chebyshev interpolation."

I have written the following function that produces a list of evenly distributed points xi on a given interval [a,b] to be used however I don't think I have incorporated the degree of my polynomial into this function, here n is just the number of points I decide to have, does this still answer the question?

```

import numpy as np

def chebs(a, b, n):

i = np.array(range(n))

x = np.cos((2*i+1)*np.pi/(2*(n)))

return 0.5*(b-a)*x+0.5*(b+a)

```