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:
“The foregoing generations beheld God and nature face to face; we, through their eyes. Why should not we also enjoy an original relation to the universe? Why should not we have a poetry and philosophy of insight and not of tradition, and a religion by revelation to us, and not the history of theirs?”
—Ralph Waldo Emerson (1803–1882)
“The whole point of Camp is to dethrone the serious. Camp is playful, anti-serious. More precisely, Camp involves a new, more complex relation to “the serious.” One can be serious about the frivolous, frivolous about the serious.”
—Susan Sontag (b. 1933)
“Empirical science is apt to cloud the sight, and, by the very knowledge of functions and processes, to bereave the student of the manly contemplation of the whole.”
—Ralph Waldo Emerson (1803–1882)