Semianalytiska

Semianalytiska refers to a class of functions in mathematics that exhibit properties between analytic and general measurable functions. These functions are not necessarily analytic in the strict sense, which would require them to be representable by a convergent power series in a neighborhood of every point. However, they possess more regularity than arbitrary measurable functions, often through specific summability or integrability conditions.

The study of semianalytic functions is important in various areas of mathematics, including real analytic geometry, control theory, and the theory of differential equations. In real analytic geometry, semianalytic sets, which are defined by equalities and inequalities involving real analytic functions, play a crucial role. These sets possess a rich geometric structure that is more manageable than arbitrary subsets of Euclidean space.

A common definition of a semianalytic function involves its behavior on a domain. For instance, a function might be considered semianalytic if it is continuous and its graph is a semianalytic set. Alternatively, a function can be semianalytic if it can be locally represented as the solution to a system of polynomial equations or inequalities. The precise definition can vary depending on the context and the specific mathematical field. These functions often arise as images or preimages of semianalytic sets under analytic mappings, preserving certain structural properties. Their investigation provides a bridge between purely analytic objects and more broadly defined mathematical structures.