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December 10

analytic functions

Suppose I specify the values of a function at a set (maybe infinite) of discrete points (like x = 0, x = 1, and so on). Can I always find an analytic function defined on the whole real line that matches the value of the other function at those points? 74.15.136.30 (talk) 21:36, 10 December 2011 (UTC)[reply]

If it's a finite number of points, then you can do even better and find a polynomial that goes through them all. For an infinite number of points, you would need a full power series (ie. an analytic function) but I think that would be enough. I don't know a proof off the top of my head, though. --Tango (talk) 23:04, 10 December 2011 (UTC)[reply]

You would need to insist that your points had distinct x-coordinates. For example, if you had the points (0,0) and (0,1) then you wouldn't be able to find such a function. Assuming all of the x-coordinates are different, then you have a very interesting question. As Tango said, if you have a finite number of points then you can find a polynomial that passes through all of them. If you have n points then generically (e.g. not colinear) there will be a unique order (n–1) polynomial. If you consider only the first two points then you'll get a linear polynomial, if you consider the first three points then (generically) you'll get a quadratic, etc. Considering more and more points will give a sequence of higher and higher order polynomials. The question is: does that sequence of polynomials have a limit that is analytic on the whole real line? (Also, you need to ask yourself if the limit depends on the order in which you add points. — Fly by Night (talk) 23:57, 10 December 2011 (UTC)[reply]

Since the OP spoke in terms of a function on a set, rather than a graph, I understood the intention to be a single-valued function (since that's what the word usually means). If you do work in terms of a graph, then you are right that you need to add an additional caveat that no points share an x-coordinate. --Tango (talk) 00:04, 11 December 2011 (UTC)[reply]

The answer is yes, so long as the set of values of x at which you wish to specify f(x) does not have an accumulation point. Equivalently, this set must have only finitely many elements in any bounded interval. This is a special case of Exercise 1 of Section 5.2.3 of "Complex Analysis" by Ahlfors. (If your set does have an accumulation point, this clearly can't work, because if, say, x_n → a, there's nothing to prevent you from requiring the values f(x_n) to tend to infinity, and then there will be no way to choose f(a) so that f is continuous there.) 96.46.201.210 (talk) 00:43, 11 December 2011 (UTC)[reply]

Our article on finite difference deals with the subject. Bo Jacoby (talk) 07:51, 11 December 2011 (UTC).[reply]

Can you explain the relation of that article to the question? 96.46.201.210 (talk) 12:22, 12 December 2011 (UTC)[reply]

If the newton series is convergent it defines an analytical function f satisfying f(x_i) = y_i for i=1,2,3,... Bo Jacoby (talk) 01:31, 13 December 2011 (UTC).[reply]

Isn't the Newton series defined in terms of a function that is already given? I must be missing something.

No, the newton series depend only on the points (x_i,y_i). Bo Jacoby (talk) 17:01, 13 December 2011 (UTC).[reply]

Based on the Wikipedia article, this appears to be defined only when you know, for some fixed a, all the values of f(a+n), where n varies through all natural numbers. Is that correct? 96.46.201.210 (talk) 21:06, 14 December 2011 (UTC)[reply]

No. The article Newton_polynomial contains the general formula. Bo Jacoby (talk) 08:20, 15 December 2011 (UTC).[reply]

All right, I see. Are you saying that this notion leads to another proof of the statement by Ahlfors? 96.46.201.210 (talk) 00:26, 16 December 2011 (UTC)[reply]

I don't think so. The condition for convergence of the newton series, a_n → ∞, is sufficient but not necessary. Bo Jacoby (talk) 13:18, 16 December 2011 (UTC).[reply]

The OP specified that the points were discrete, so we're ok there. For those of us that don't own Complex Analysis by Ahlfors, could you sketch the proof? --Tango (talk) 00:01, 13 December 2011 (UTC)[reply]

Let a_n be a sequence of (distinct) complex numbers with a_n → ∞, and let A_n be any complex numbers. Then there is an entire function f(z) with f(a_n) = A_n for all n. To define f, Ahlfors suggests the following. Let g be an entire function with a simple zero at each a_n. Such a g exists by the Weierstrass factorization theorem. Now set

f(z)=\sum _{n=1}^{+\infty }g(z){\frac {e^{\gamma _{n}(z-a_{n})}}{z-a_{n}}}\cdot {\frac {A_{n}}{g'(a_{n})}},

and show that the sum converges for some choice of numbers γ_n.

In fact, I presume we need to show the convergence is uniform on compact subsets in order to prove that f is analytic. When z=a_n it looks at first as if the nth term is undefined, but in this case g(z)/(z - a_n) should be interpreted as g′(a_n); the resulting function is still analytic. It is also clear that if such γ_n's exist, then we'll have f(a_n) = A_n as required. Once the γ_n's have been chosen, to prove uniform convergence on compact sets, it's enough to restrict attention to a disk |z| ≤ R, and in that case, we can focus attention on what happens once n is large enough that |a_n| ≥ 2R, say (and ignore the finitely many terms at the beginning where this is not the case). Restricting attention to the matter of uniform absolute convergence, now we can ignore the g(z) factor since it is bounded in the disk, and ignore the 1/(z - a_n) factor, since it's bounded by 1/R. If we choose γ_n = λ_na̅_n, where λ_n is an undetermined nonnegative real number, then we'll have Re(γ_n(z - a_n)) ≤ - λ_n|a_n|²/2. Then we just need to pick λ_n large enough so that the series

\sum e^{-\lambda _{n}|a_{n}|^{2}/2}{\frac {|A_{n}|}{|g'(a_{n})|}}

converges. This can be achieved for example by taking λ_n = n + max[0, log(|A_n|/|g′(a_n)|)]. This is because eventually |a_n|² ≥ 2, and from then on the whole term is bounded by e^-n.96.46.201.210 (talk) 09:38, 13 December 2011 (UTC)[reply]

I've just noticed that this problem is generalized by Theorem 15.13 in Rudin's "Real and Complex Analysis", which says that given an open set Ω in the plane and a subset A of Ω with no accumulation points in Ω, it is possible to find an analytic function defined on Ω with, at each point of A, finitely many prescribed derivatives (including the value of the function itself, which is the zeroth derivative). 96.46.201.210 (talk) 09:56, 13 December 2011 (UTC)[reply]