When analytic continuation from discrete set is unique?

Question

Under which additional conditions an analytic continuation of a function defined on an infinite discrete set of points (i.e. no limit points) can be made unique? Any theorem on the subject please?

Regards,

Answer 1

One theorem in this vein I know of is Carlson's theorem, which basically says that if we start with a function $f$ that maps from the natural numbers, $\mathbb{N}$, to the complex plane, and which grows at most exponentially, at i.e.

$$|f(n)| \in O(a^n)$$

for some base $a > 1$, then it admits no more than one analytic extension $F$ to domain $\mathbb{C}$ such that, along the imaginary axis, the growth bound

$$|F(iy)| \in O(e^{\pi|y|})$$

holds.

Although, it's more often stated as that if both bounds above hold for a given complex function $F$ that is zero at each integer, then $F$ is identically zero (i.e. $F = (z \mapsto 0)$).

One reference of use is, which also gives a slightly more general form for subsets of natural numbers:

http://www.ams.org/journals/tran/1956-083-02/S0002-9947-1956-0081944-8/S0002-9947-1956-0081944-8.pdf

Answer 2

"Under which additional conditions..." is a little broad for a definitive answer. One situation where this happens is when we place appropriate growth conditions on the functions. For example:

Let $D=D(0,1)$. Suppose $(z_n)\subset D$, $|x_n|\to 1$ but $\sum(1-|z_n|)=\infty$. Suppose $f,g\in H(D)$ are bounded. If $f(z_n)=g(z_n)$ for all $n$ then $f=g$.

Proof: Look for "Blaschke condition" on wikipedia or in Rudin _Real and Complex Analysis. (If you don't find a suitable article on wikipedia say so and one will appear...)

Another example: Say $f\in PW$ (for "Paley-Wiener") if $f$ is entire, $\int_{-\infty}^\infty|f(t)|^2\,dt<\infty$ and $f(z)\le ce^{\pi|z|}$.

If $f,g\in PW$ and $f(n)=g(n)$ for all $n\in\Bbb Z$ then $f=g$.