Cauchy Product - Relation To Convolution of Functions

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:

    Any relation to the land, the habit of tilling it, or mining it, or even hunting on it, generates the feeling of patriotism. He who keeps shop on it, or he who merely uses it as a support to his desk and ledger, or to his manufactory, values it less.
    Ralph Waldo Emerson (1803–1882)

    Among the most valuable but least appreciated experiences parenthood can provide are the opportunities it offers for exploring, reliving, and resolving one’s own childhood problems in the context of one’s relation to one’s child.
    Bruno Bettelheim (20th century)

    Mark the babe
    Not long accustomed to this breathing world;
    One that hath barely learned to shape a smile,
    Though yet irrational of soul, to grasp
    With tiny finger—to let fall a tear;
    And, as the heavy cloud of sleep dissolves,
    To stretch his limbs, bemocking, as might seem,
    The outward functions of intelligent man.
    William Wordsworth (1770–1850)