Carlson's theorem

Carlson's theorem gives conditions under which an entire function is uniquely determined by its values at the nonnegative integers. It combines a bound on exponential type with growth restrictions in a half-plane.

Connect

Entire function: A complex-valued function holomorphic on the entire complex plane. The theorem applies to entire functions and constrains their growth.
Holomorphic function: A complex function that is complex differentiable in a neighborhood of every point in its domain. Entire functions are holomorphic on the whole complex plane.
Fritz Carlson: A Swedish mathematician known for work in complex analysis and the theorem bearing his name. Carlson established the uniqueness result now associated with his name.
Mellin transform: An integral transform that converts multiplicative scaling into shifts of a complex parameter. Uniqueness arguments can identify analytic transforms from their values at integer parameters.
Identity theorem: A theorem stating that holomorphic functions agreeing on a set with an accumulation point in their domain agree everywhere on a connected domain. Integer samples have no finite accumulation point, so Carlson needs growth bounds the identity theorem does not.
Exponential type: A measure of how rapidly an entire function can grow, defined through its asymptotic maximum modulus. The type bound limits growth enough for integer values to force uniqueness.
Maximum modulus principle: A theorem stating that a nonconstant holomorphic function cannot attain a local maximum of its modulus inside its domain. It underlies many bounds used to control holomorphic functions from boundary behavior.
Carlson's 1914 paper: Fritz Carlson’s paper introducing the uniqueness theorem for entire functions from their integer values. The theorem’s original formulation appeared in this work.
Newton series: A series expansion in falling factorial polynomials, with coefficients given by finite differences. It interpolates sequences, while Carlson-type conditions can establish uniqueness of an entire interpolant.
Müntz–Szász theorem: A theorem characterizing when powers with selected exponents span a dense subspace of continuous functions on an interval. It concerns completeness of exponentials or powers, rather than uniqueness from integer samples under entire-function growth.