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, 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)
“Our mother gives us our earliest lessons in love—and its partner, hate. Our father—our “second other”Melaborates on them. Offering us an alternative to the mother-baby relationship . . . presenting a masculine model which can supplement and contrast with the feminine. And providing us with further and perhaps quite different meanings of lovable and loving and being loved.”
—Judith Viorst (20th century)
“The man who pretends that the distribution of income in this country reflects the distribution of ability or character is an ignoramus. The man who says that it could by any possible political device be made to do so is an unpractical visionary. But the man who says that it ought to do so is something worse than an ignoramous and more disastrous than a visionary: he is, in the profoundest Scriptural sense of the word, a fool.”
—George Bernard Shaw (1856–1950)