sequenceors

Sequenceors are a theoretical class of mathematical objects conceived to act on sequences, producing new sequences. Formally, a sequenceor is a map that assigns to each input sequence x = (x_n) from a given set S another sequence y = (y_n) in S, possibly of a different length or type. Sequenceors may be deterministic or stochastic, and may possess additional structure such as linearity, time-invariance, or locality. In the linear case, a sequenceor can be represented by a family of linear functionals or by a convolution with a fixed kernel, acting on sequence spaces such as l^p or c0.

Common examples include the prefix-sum sequenceor y_n = sum_{i≤n} x_i; the finite-difference sequenceor y_n = x_n - x_{n-1}; and the reversal sequenceor y_n = x_{N-n+1} for sequences of length N. Shift sequenceors move each term forward or backward by a fixed lag. More general moving-average sequenceors apply weighted sums over windows.

Operationally, sequenceors can be implemented by recurrence relations, by convolution with kernels, or by stateful machines that read input symbols and emit output symbols. They can be composed, forming a monoid under composition, with an identity sequenceor that returns the input sequence unchanged. In analysis, sequenceors are used to study transformations of sequence spaces, convergence, and stability properties, with applications in time-series processing and signal analysis.

Related concepts include operators on sequence spaces, linear operators, and convolution. The term sequenceor is primarily descriptive and used in theoretical discussions rather than as standard terminology in mainstream courses.