spanningamplitude

Spanning amplitude is a term used in mathematics and signal processing to describe the maximal pointwise amplitude that can be produced by linear combinations of a given finite set of signals or basis functions, under a bound on the combination coefficients. The concept provides a way to quantify how much variation in amplitude a signal family can generate through linear combination.

Let s1, ..., sk be real-valued signals defined on a domain D. For a coefficient vector c = (c1, ..., ck), the combined signal is S_c(t) = sum_i c_i s_i(t). The spanning amplitude at a point t is the supremum of |S_c(t)| over all c subject to a fixed coefficient-norm bound, such as ||c||_1 ≤ 1 or ||c||_2 ≤ 1. The global spanning amplitude is the supremum over t in D of this quantity.

Two common cases illustrate the idea. If the constraint is ||c||_2 ≤ 1, then the spanning amplitude at t equals sqrt(sum_i s_i(t)^2). If the constraint is ||c||_1 ≤ 1, the spanning amplitude at t equals sum_i |s_i(t)|. In general, the choice of norm affects the resulting bound, reflecting different notions of how large coefficients are allowed to be.

Applications of spanning amplitude include assessing the expressive capacity of a dictionary in sparse representation, comparing sets of basis functions, and analyzing how amplitude can be controlled in signal synthesis. The term is not universally standardized and may be referred to alternatively as amplitude span or span-weighted amplitude in some contexts.

See also: linear span, amplitude, dictionary learning, sparse representation.