supportswithout

Supportswithout is a term used in mathematical and computational contexts to describe the operation of obtaining the support of a function after removing a specified subset of its domain. In discrete or finite settings, it is often treated as a simple set-theoretic modification of the function’s support.

Formally, if f is a function from a set X to a field (typically R or C), with finite support supp(f) = {x in X : f(x) ≠ 0}, and A is a subset of X, then supportswithout(f, A) is defined as supp(f) \ A. Equivalently, this can be viewed as the support of the function restricted to X \ A, preserving nonzero values outside A.

Basic properties include monotonicity with respect to the exclusion set: if A1 ⊆ A2, then supportswithout(f, A2) ⊆ supportswithout(f, A1). It is also invariant under removing elements outside the original support, since supportswithout(f, A) = supportswithout(f, A ∩ supp(f)). In practice, the concept is used to model sparsity and to study how supports change under suppression or filtering of certain indices.

Example: let X = {1,2,3,4} and f with f(1) = 2, f(3) = 5, and f(2) = f(4) = 0. Then supp(f) = {1,3}. supportswithout(f, {1}) = {3}, and supportswithout(f, {2,4}) = {1,3}.

Usage and relation: supportswithout relates to restricting a function and to notions of conditioned or filtered supports in data analysis and signal processing. It is not a universally standard term, but appears in niche discussions and algorithmic contexts where selective suppression of indices is relevant. See also: support, restriction, and sparse representations.