Mittag-Leffler star expansion coefficients again & Runge's theorem
mike3
Regarding the Mittag-Leffler expansion mentioned here:
http://eom.springer.de/s/s087230.htm
I got to see the rest of the proof mentioned in reference 2.
Apparently the coefficients are those of a sequence of polynomials
P_n(w) = c_0^(n) + c_1^(n) w + ... + c_{k_n}^(n) w^{k_n}
that converge to the function f(z) = 1/(1 - z) in the "domain G
bounded by u >= 1, v = 0 (the part of the real axis gong from 1 to
oo)" (so I suppose that's the complex plane minus z = u + iv where u
>= 1, v = 0, i.e. minus the real interval [1, oo). This domain is the
same as the Mittag-Leffler star of 1/(1 - z), no?), which is
guaranteed to exist by "Runge's theorem". I looked that up, Runge's
theorem doesn't seem to tell _how to get those polynomials_, only that
they exist. Is there some way to construct such a sequence of
polynomials?
mike3
> Hi.
>
> Regarding theMittag-Lefflerexpansion mentioned here:
I.e. is there a more "constructive" theorem that could be useful here?
mike3
*bump* Does anyone have any answers to these questions or not?
FredJeffries
Don't know if this is what you are looking for, but G H Hardy in
"Divergent Series" in note 8.10 on page 197 gives three "elegant
representations" of the function 1/(1-z) in its Mittag-Leffler star:
http://books.google.com/books?id=jPccoUKsLdQC&pg=PA197
Section 4.11 of that book is devoted to "Lindelof's and Mittag-
Leffler's Methods" of summation:
http://books.google.com/books?id=jPccoUKsLdQC&pg=PA77
mike3
But they don't seem to be a representation as a sequence of
polynomials, at least not directly. Note they involve that limit on
the delta parameter. That's what's got me hung up.
What's needed is a sequence of polynomials which converges uniformly
to 1/(1-z) in it's ML star.
mike3
Ehh.. there's a qualifier I noticed.
"convergence is uniform inside G [ML star of 1/(1-z)], _in particular
on the compact set E(F, L) subset G_". The set E(F, L) is
the set of all points w = (z - z0)/(zeta - z0) with z being in F
and zeta being in L. z0 appears to be the "center" of the star. The
set F however looks to be various: it is constructed from some
arbitrary
compact subset of the star of the function whose Mittag Leffler
expansion
we are seeking that contains the star's center, by unioning every line
segment from the center to every point in that subset. And L is "a"
(sounds arbitrary!) "closed rectifiable Jordan curve" such that
L is in the star fof the original function and also "I(L)" (??? don't
know what that is) is a superset of F.
The arbitrariness of the choice of the set F is bothersome, yet
supposedly the coefficients for these polynomials can be found "once
and
for all" and it's done. Also I'm not sure what to make of the "in
particular"
wording. Does that mean it's only uniform on that compact set?
This leads me to question the usefulness of any of this for
constructing
coefficients... Why is it so darned awfully difficult to track this
puppy
down? Why is it so horribly obscure?