Folgeoperation

Folgeoperation is a term used in mathematics and computer science to denote any operation that acts on sequences (Folgen) and produces either another sequence, a single value, or another mathematical object. In formal terms, an operation O is a (possibly partial) function that takes as input a sequence (a_n) and returns O((a_n)). The input sequences are typically indexed by natural numbers and may take values in a set such as the real or complex numbers.

There are several kinds of Folgeoperation. Unary operations act on a single sequence, while binary operations take two sequences as input. In many common situations, Folgeoperationes are linear and defined termwise or through a rule that relates the terms of the input sequences to the terms of the output. The output may be a transformed sequence or a scalar, depending on the operation.

Typical examples include:

- Pointwise (termwise) operations: given two sequences (a_n) and (b_n), the operation yields (a_n + b_n), (a_n − b_n), or (c a_n) for a scalar c.

- Shifts and truncations: shifting a sequence (a_{n}) to (a_{n+1}) or truncating to the first N terms.

- Reversal and concatenation: reversing a finite sequence or concatenating two finite sequences.

- Convolution: (a * b)_n = sum_{k=0}^n a_k b_{n−k}, producing a new sequence.

- Accumulation: prefix sums (S a)_n = sum_{k=1}^n a_k, producing a transformed sequence.

In functional analysis and sequence spaces (such as l^p), Folgeoperationes are viewed as linear operators on spaces of sequences. Properties of interest include linearity, boundedness, and continuity. Applications range from solving recurrence relations and generating new sequences to time-series processing and signal analysis. See also sequence, operator, and convolution.