Examples
Simplest examples of (linear) sequence transformations include shifting all elements, (resp. = 0 if n + k < 0) for a fixed k, and scalar multiplication of the sequence.
A little less trivial generalization would be the discrete convolution with a fixed sequence. A particularly basic form is the difference operator, which is convolution with the sequence and is a discrete analog of the derivative. The binomial transform is another linear transformation of a still more general type.
An example of a nonlinear sequence transformation is Aitken's delta-squared process, used to improve the rate of convergence of a slowly convergent sequence. An extended form of this is the Shanks transformation. The Möbius transform is also a nonlinear transformation, only possible for integer sequences.
Read more about this topic: Sequence Transformation
Famous quotes containing the word examples:
“There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
—Bernard Mandeville (1670–1733)
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (1896–1966)
“In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
—Michel de Montaigne (1533–1592)