Development
Large-sieve methods have been developed enough that they are applicable to small-sieve situations as well. By now, something is seen as related to the large sieve not necessarily in terms of whether it related to the kind situation outlined above, but, rather, if it involves one of the two methods of proof traditionally used to yield a large-sieve result:
- An approximate Plancherel inequality. If a set 'S' is ill-distributed modulo p (by virtue, for example, of being excluded from the congruence classes Ap) then the Fourier coefficients of the characteristic function fp of the set S mod p are in average large. These coefficients can be lifted to values of the Fourier transform of the characteristic function f of the set S (i.e., ). By bounding derivatives, we can see that must be large, on average, for all x near rational numbers of the form a/p. Large here means "a relatively large constant times |S|". Since, we get a contradiction with the Plancherel identity unless |S| is small. (In practice, to optimise bounds, people nowadays modify the Plancherel identity into an equality rather than bound derivatives as above.)
- The duality principle. One can prove a strong large-sieve result easily by noting the following basic fact from functional analysis: the norm of a linear operator (i.e., where A is an operator from a linear space V to a linear space W) equals the norm of its adjoint (i.e., ). This principle itself has come to acquire the name "large sieve" in some of the mathematical literature.
It is also possible to derive the large sieve from majorants in the style of Selberg (see Selberg, Collected Works, vol II, Lectures on sieves).
Read more about this topic: Large Sieve
Famous quotes containing the word development:
“This was the Eastham famous of late years for its camp- meetings, held in a grove near by, to which thousands flock from all parts of the Bay. We conjectured that the reason for the perhaps unusual, if not unhealthful development of the religious sentiment here, was the fact that a large portion of the population are women whose husbands and sons are either abroad on the sea, or else drowned, and there is nobody but they and the ministers left behind.”
—Henry David Thoreau (1817–1862)
“Good schools are schools for the development of the whole child. They seek to help children develop to their maximum their social powers and their intellectual powers, their emotional capacities, their physical powers.”
—James L. Hymes, Jr. (20th century)
“... work is only part of a man’s life; play, family, church, individual and group contacts, educational opportunities, the intelligent exercise of citizenship, all play a part in a well-rounded life. Workers are men and women with potentialities for mental and spiritual development as well as for physical health. We are paying the price today of having too long sidestepped all that this means to the mental, moral, and spiritual health of our nation.”
—Mary Barnett Gilson (1877–?)