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:
“We must get back into relation, vivid and nourishing relation to the cosmos and the universe. The way is through daily ritual, and is an affair of the individual and the household, a ritual of dawn and noon and sunset, the ritual of the kindling fire and pouring water, the ritual of the first breath, and the last.”
—D.H. (David Herbert)
“Unaware of the absurdity of it, we introduce our own petty household rules into the economy of the universe for which the life of generations, peoples, of entire planets, has no importance in relation to the general development.”
—Alexander Herzen (1812–1870)
“Adolescents, for all their self-involvement, are emerging from the self-centeredness of childhood. Their perception of other people has more depth. They are better equipped at appreciating others’ reasons for action, or the basis of others’ emotions. But this maturity functions in a piecemeal fashion. They show more understanding of their friends, but not of their teachers.”
—Terri Apter (20th century)