Say that we have a continuous, differentiable function f(x) and we have found the best approximation (in the sense of the infinity norm) of f from some set of functions forming a finite dimensional vector space (say, polynomials of degree less than n or trigonometric polynomials of degree less...
Thank you for your reply!
Yes, f(x) is continuous. And indeed f(x) + g(x,y) is monotone.
What I meant to ask was if there is a way to explicitly find that value for x at which f+g=0 other than the "brute force" way of inverting the expression? Or perhaps, more generally - does an equation of...
Say we have two functions with the following properties:
f(x) is negative and monotonically approaches zero as x increases.
g(x,y) is a linear function in x and is, for any given y, tangent to f(x) at some point x_0(y) that depends on the choice of y in a known way.
Additionally, for any...
Hi!
Consider the function
\frac{d^n}{dx^n} \sum_{k=1}^m \sin{kx}, \quad 0 \leq x \leq \pi/2 .
If n is odd this function attains its largest value, \sum_{k=1}^m k^n at x=0 . But what about if n is even? Where does the maximum occur and what value does it take?
Any help is much...
Thank you mfb for your reply!
Yes, that was my original idea as well. If g is the approximation in the RHS of (1) , then I reasoned that the optimal result should be when (f-g) \perp f . However, (f-g, f) is a linear function in the coefficients a_n so there are no extrema (I am...
Hi!
Say that we wish to approximate a function f(x), \, x\in [0, 2\pi] by a trigonometric polynomial such that
f(x) \approx \sum_{|n|\leq N} a_n e^{inx} \qquad (1)
The best approximation theorem says that in a function space equipped with the inner product
(f,g) = \frac{1}{2...
Hi!
I've encountered the series below:
\sum_{l=0}^{k-1} (r+l)^j (r+l-k)^i
where r, k, i, j are positive integers and i \leq j .
I am interested in expressing this series as a polynomial in k - or rather - finding the coefficients of that polynomial as i,j changes. I have reasons to...
Hi!
It just dawned on me that any such matrices (I suppose there are only 4 places A could go ^^, ) are related by simple permutations. Since any permutation matrix has determinant + or - 1 then what you say must be true.
Thank you for the enlightenment! =)
The Cauchy expansion says that
\text{det} \begin{bmatrix}
A & x \\[0.3em]
y^T & a
\end{bmatrix}
= a \text{det}(A) - y^T \text{adj}(A) x ,
where A is an n-1 by n-1 matrix, y and x are vectors with n-1 elements, and a is a scalar.
There is a proof in Matrix Analysis by Horn and...
Hi all!
I have no application in mind for the following question but it find it curious to think about:
Say that we have a square matrix where the sum of the elements in each row and each column is zero. Clearly such a matrix is singular. Suppose that no row or column of the matrix is the...
That's a fair point.
I was playing around with a different matrix - Hermitian and also symmetric about the anti-diagonal. Turns out that the eigenvectors are closely related to the eigenvectors of the matrix R above so I was curious about their structure. It seems reasonable that the upper and...
Hi folks!
I've encountered the matrix below and I'm curious about its properties;
R=
\begin{pmatrix}
0 & N-S\\
N+S & 0
\end{pmatrix}
where R, N and S are real matrices, R is 2n by 2n, N is n by n symmetric and S is n by n skew-symmetric.
Clearly R is symmetric so the...