Re: Probably haven't seen this one, but...

Robert Israel <israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote:
David W. Cantrell <DWCantrell@xxxxxxxxxxx> writes:

[This is to supersede my previous (incorrect) posting.]

Robert Israel <israel@xxxxxxxxxxx> wrote:
On May 4, 8:18 am, junoexpress <MTBrenne...@xxxxxxxxx> wrote:
On May 3, 1:12 pm, Robert Israel
<isr...@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote:
junoexpress <MTBrenne...@xxxxxxxxx> writes:
Hi,

I'm doing some work with a function that is the ratios of
(normalized) sinc functions, having the form:
f:x-> Sinc(Mx)/Sinc(x) for M a natural number and |x| < 1

I'm having to work out some properties of this function, which
are not that bad, but in the process, I keep wondering if this
ratio has been analyzed before (in other words, I hate to
present a lot of detailed derivations only to have someone else
say, "Oh yeah, that's just the Gluckenheimer function" and if
it has been looked at before, maybe I could get some deeper
insight into the solution also.)

So I come to the gurus. Is this a function which anyone has
seen analyzed before? To my knowledge, it is not in
Abrahamowitz and Stegan, and the closest I can come to pinning
it on anything known is to say that it's the ratio of 2
spherical Bessel functions (which doesn't seem like a function
that's probably been analyzed).

Since sinc(x) = sin(x)/x (for x <> 0), your function is just
f(x) = sin(Mx)/(M sin(x)). This can also be written as
U_{M-1}(cos(x))/M where U_k is the k'th Chebyshev polynomial of
the second kind.

This is a very nice observation. There is one problem I am having
in applying them to my problem. I need to know (or have a decent
bounds) on the absolute value of the first extreme value of U_m(x)
after x=0. So in the case of U_2(x), you could solve for the
critical points, take the critical point closest to zero (which is
not equal to zero) x*, and then compute |U2(x*)|. But of course for
M>4, this strategy will not work, and there is no obvious way I can
see to get a (close) upper bound to this first extreme value.
(Could always do this computationally, but if there was a proven
bound, that of course would be better).

|sin(Mx)/sin(x)| <= |1/sin(x)|

The peaks will be quite close to points where sin(Mx) = (+/-) 1,

Don't you mean just when M is odd?

No, both even and odd. The point is simply that sin(Mx) varies
rapidly while 1/sin(x) varies slowly (away from the points where
sin(x)=0).

Ah. You were talking about the peaks of |sin(Mx)/sin(x)|, and so of course,
as you say, parity of M doesn't matter.

But Matt had mentioned "the first extreme value of U_m(x) after x=0" and
so I had mistakenly thought that you too were talking about the extrema
of U_M(x).

David
.

Relevant Pages

  • Re: Probably havent seen this one, but...
    ... and the closest I can come to pinning it on anything known ... take the critical point closest to zero ... upper bound to this first extreme value. ... Don't you mean just when M is odd? ...
    (sci.math)
  • Re: Power series for Cos x converge for all x?
    ... israel@math.ubc.ca (Robert Israel) wrote: ... >>close to zero, it may require an awful lot of terms to get a good ... > sum to the x^term, so the error will always be less than the ... > absolute value of the last term. ...
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  • Re: Can we define f(z)=sqrt(z*sin(z)) as analytic function?
    ... Robert Israel wrote: ... Juryu wrote: ... but is not zero at z=0? ... fis an odd function. ...
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  • Re: Why sci.math?
    ... > israel@xxxxxxxxxxx (Robert Israel) writes: ... >>>In the UK at a certain level in junior school (I don't recall the age ... > Even to me, it looks odd. ...
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