Formal name for functions that use the OVER clause
I believe the term you are looking for is window functions. Most databases have support for these types of functions.

Window functions belong to a type of function known as a ‘set function’, which means a function that applies to a set of rows. The word ‘window’ is used to refer to the set of rows that the function works on.
Windowing functions were added to the standard SQL:2003 that is managed by the ISO and it was specified in more detail in SQL:2008 For some time, other DBMSs such as Oracle, Sybase and DB2 have had support for window functions. Even the open source RDBMS PostgreSQL has a full implementation. SQL Server has had only a partial implementation up to now, but it is coming in SQL 2012.

Source
A window function is one of:

A rank function.
A distribution function.
The row number function.
A window aggregate function.
The ntile function.
The lead function.
The lag function.
The first-value function.
The last-value function.
The nth-value function.