Relation To Convolution of Functions
One can also define the Cauchy product of doubly infinite sequences, thought of as functions on . In this case the Cauchy product is not always defined: for instance, the Cauchy product of the constant sequence 1 with itself, is not defined. This doesn't arise for singly infinite sequences, as these have only finite sums.
One has some pairings, for instance the product of a finite sequence with any sequence, and the product . This is related to duality of Lp spaces.
Read more about this topic: Cauchy Product
Famous quotes containing the words relation to, relation and/or functions:
“It would be disingenuous, however, not to point out that some things are considered as morally certain, that is, as having sufficient certainty for application to ordinary life, even though they may be uncertain in relation to the absolute power of God.”
—René Descartes (1596–1650)
“Our sympathy is cold to the relation of distant misery.”
—Edward Gibbon (1737–1794)
“The mind is a finer body, and resumes its functions of feeding, digesting, absorbing, excluding, and generating, in a new and ethereal element. Here, in the brain, is all the process of alimentation repeated, in the acquiring, comparing, digesting, and assimilating of experience. Here again is the mystery of generation repeated.”
—Ralph Waldo Emerson (1803–1882)