0
$\begingroup$

In linear algebra, the cosine of the angle between two vectors $a$ and $b$ is defined as $$\cos(a,b) = \frac{\langle a, b \rangle}{||a||\cdot||b||} .$$ The functional square root of the cosine has at various times been studied by mathematicians. It is the function $f(\cdot)$ such that $$f(f(x)) = \cos(x). $$ See for instance this MO question. I wonder whether it is possible to have a vector-based interpretation of this functional square root of the cosine, similar to the cosine itself as defined above.

I have also asked this question on MSE.

Improve this question
$\endgroup$
5
  • $\begingroup$ Could the person(s) who voted down and/or to close please explain his/her motivations? $\endgroup$
    – Max Lonysa Muller
    Commented Sep 12, 2020 at 13:11
  • $\begingroup$ @PiotrHajlasz I think this question is interesting because it may cast a renewed understanding of the functional square root of the cosine. It seems to me most effort to find this root has so far been directed at finding its Taylor series coefficients. While progress has been made in this direction, it has so far not enabled us to retrieve a closed-form analytic expression for the root. In contrast to many other functions, the cosine of two vectors has an elegant and simple representation in vectorial terms... (cont'd) $\endgroup$
    – Max Lonysa Muller
    Commented Sep 13, 2020 at 16:10
  • $\begingroup$ ... It therefore lends itself well to be interpreted from a geometric and linear-algebraic perspective. Such a perspective may be lacking in the current literature on functional equations $\endgroup$
    – Max Lonysa Muller
    Commented Sep 13, 2020 at 16:14
  • 1
    $\begingroup$ I am removing my vote to close. $\endgroup$
    – Piotr Hajlasz
    Commented Sep 15, 2020 at 22:27
  • $\begingroup$ Hmm, for me the idea of iteration of a function means, that the input and the output have the same structure. So they both may be a scalar, maybe a two-component vector or $2 \times 2$-matrix (for instance for $2$-D rotation matrices) , or infinite vectors of coefficients (for instance in the context of Carleman-matrices, which give a "vandermonde-vector" as input as well as output - here I use "vand. vector" meaning an infinite vector of an argument $x$ as $V(x) = [1,x,x^2,x^3,...]$). Could we define some structure first which is stable between input & output for your problem? $\endgroup$
    – Gottfried Helms
    Commented Oct 3, 2020 at 18:23

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .