Relationship To The Dirac Delta Distribution
The normalized sinc function can be used as a nascent delta function, meaning that the following weak limit holds:
This is not an ordinary limit, since the left side does not converge. Rather, it means that
for any smooth function with compact support.
In the above expression, as a approaches zero, the number of oscillations per unit length of the sinc function approaches infinity. Nevertheless, the expression always oscillates inside an envelope of ±1/(π a x), and approaches zero for any nonzero value of x. This complicates the informal picture of δ(x) as being zero for all x except at the point x = 0 and illustrates the problem of thinking of the delta function as a function rather than as a distribution. A similar situation is found in the Gibbs phenomenon.
Read more about this topic: Sinc Function
Famous quotes containing the words relationship to the, relationship and/or distribution:
“Film music should have the same relationship to the film drama that somebody’s piano playing in my living room has to the book I am reading.”
—Igor Stravinsky (1882–1971)
“Strange and predatory and truly dangerous, car thieves and muggers—they seem to jeopardize all our cherished concepts, even our self-esteem, our property rights, our powers of love, our laws and pleasures. The only relationship we seem to have with them is scorn or bewilderment, but they belong somewhere on the dark prairies of a country that is in the throes of self-discovery.”
—John Cheever (1912–1982)
“The question for the country now is how to secure a more equal distribution of property among the people. There can be no republican institutions with vast masses of property permanently in a few hands, and large masses of voters without property.... Let no man get by inheritance, or by will, more than will produce at four per cent interest an income ... of fifteen thousand dollars] per year, or an estate of five hundred thousand dollars.”
—Rutherford Birchard Hayes (1822–1893)