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

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).
--
Robert Israel israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.

Relevant Pages

  • Re: Probably havent seen this one, but...
    ... Robert Israel wrote: ... take the critical point closest to zero (which is ... see to get a upper bound to this first extreme value. ... Don't you mean just when M is odd? ...
    (sci.math)
  • Re: Probably havent seen this one, but...
    ... that bad, but in the process, I keep wondering if this ratio has been ... and the closest I can come to pinning it on anything known is ... take the critical point closest to zero ... upper bound to this first extreme value. ...
    (sci.math)
  • Re: Probably havent seen this one, but...
    ... that bad, but in the process, I keep wondering if this ratio has been ... and the closest I can come to pinning it on anything known is ... take the critical point closest to zero ... upper bound to this first extreme value. ...
    (sci.math)
  • Re: Interesting math
    ... Multiple by an odd number. ... Multiplication by zero is both even ... even and odd to not violate our even and odd rule. ... making yourself a wild savage of imagination. ...
    (alt.usage.english)
  • Re: Interesting math
    ... And I wouldn't be too sure about "it" knowing that "zero is ... and odd numbers. ... rather than untrue. ... The same sort of issue arises across all sorts of mathematics. ...
    (alt.usage.english)