Definition
A subset E of the circle is called a set of uniqueness, or a U-set, if any trigonometric expansion
which converges to zero for is identically zero; that is, such that
- c(n) = 0 for all n.
Otherwise E is a set of multiplicity (sometimes called an M-set or a Menshov set). Analogous definitions apply on the real line, and in higher dimensions. In the latter case one needs to specify the order of summation, e.g. "a set of uniqueness with respect to summing over balls".
To understand the importance of the definition it is important to get out of the Fourier mind-set. In Fourier analysis there is no question of uniqueness, since the coefficients c(n) are derived by integrating the function. Hence in Fourier analysis the order of actions is
- Start with a function f.
- Calculate the Fourier coefficients using
- Ask: does the sum converge to f? In which sense?
In the theory of uniqueness the order is different:
- Start with some coefficients c(n) for which the sum converge in some sense
- Ask: does this means that they are the Fourier coefficients of the function?
In effect, it is usually sufficiently interesting (as in the definition above) to assume that the sum converges to zero and ask if that means that all the c(n) must be zero. As is usual in analysis, the most interesting questions arise when one discusses pointwise convergence. Hence the definition above, which arose when it became clear that neither convergence everywhere nor convergence almost everywhere give a satisfactory answer.
Read more about this topic: Set Of Uniqueness
Famous quotes containing the word definition:
“The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
—William James (1842–1910)
“It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.”
—François, Duc De La Rochefoucauld (1613–1680)
“Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)