Estimates
The main thrust of the subject is that a sum
is trivially estimated by the number N of terms. That is, the absolute value
by the triangle inequality, since each summand has absolute value 1. In applications one would like to do better. That involves proving some cancellation takes place, or in other words that this sum of complex numbers on the unit circle is not of numbers all with the same argument. The best that is reasonable to hope for is an estimate of the form
which signifies, up to the implied constant in the big O notation, that the sum resembles a random walk in two dimensions.
Such an estimate can be considered ideal; it is unattainable in many of the major problems, and estimates
have to be used, where the o(N) function represents only a small saving on the trivial estimate. A typical 'small saving' may be a factor of log(N), for example. Even such a minor-seeming result in the right direction has to be referred all the way back to the structure of the initial sequence xn, to show a degree of randomness. The techniques involved are ingenious and subtle.
A variant of 'Weyl differencing' investigated by Weyl involving a generating exponential sum
Was previously studied by Weyl himself, he developed a method to express the sum as the value, where 'G' can be defined via a linear differential equation similar to Dyson equation obtained via summation by parts.
Read more about this topic: Exponential Sum
Famous quotes containing the word estimates:
“And, by the way, who estimates the value of the crop which nature yields in the still wilder fields unimproved by man? The crop of English hay is carefully weighed, the moisture calculated, the silicates and the potash; but in all dells and pond-holes in the woods and pastures and swamps grows a rich and various crop only unreaped by man.”
—Henry David Thoreau (1817–1862)
“A State, in idea, is the opposite of a Church. A State regards classes, and not individuals; and it estimates classes, not by internal merit, but external accidents, as property, birth, etc. But a church does the reverse of this, and disregards all external accidents, and looks at men as individual persons, allowing no gradations of ranks, but such as greater or less wisdom, learning, and holiness ought to confer. A Church is, therefore, in idea, the only pure democracy.”
—Samuel Taylor Coleridge (1772–1834)
“Writing a book I have found to be like building a house. A man forms a plan, and collects materials. He thinks he has enough to raise a large and stately edifice; but after he has arranged, compacted and polished, his work turns out to be a very small performance. The authour however like the builder, knows how much labour his work has cost him; and therefore estimates it at a higher rate than other people think it deserves,”
—James Boswell (1740–1795)