Sentence examples for a moment sequence from inspiring English sources

The proof is based on an intermediate result that asserts that if A is an operator on a reflexive real Banach space of dimension greater than one, and there exist non-zero vectors, u in the space and v in the dual space, such that {〈Anu, v〉}∞n=0 is a moment sequence of a finite positive Borel measure on a bounded interval on the real line, then A has a nontrivial invariant subspace.

Specifically, if f(n) is a real sequence with κ negative squares, then the main theorem asserts the existence of a minimal definitizing polynomial, i.e., a nonnegative polynomial q(x) = ∑i = 02κaixi of degree 2κ so that g(n) := ∑i = 02κaif(n + i) is a moment sequence, i.e., of the form g(n) = ∝ xndv with v ⩾ 0. The question of uniqueness of a minimal definitizing polynomial is discussed.

From this theorem, we know (see [1]) that a completely monotonic sequence is a moment sequence and is as follows.

Theorem 1 A sequence { μ n } n = 0 ∞ is a moment sequence if and only if there exists a constant L such that ∑ m = 0 k | λ k, m | < L, k ∈ N 0, (7).

Theorem 3 A necessary and sufficient condition that the sequence { μ n } n = 0 ∞ should be a moment sequence is that it should be the difference of two completely monotonic sequences.

A sequence { μ n } n = 0 ∞ is called a moment sequence if there exists a function α ( t ) of bounded variation on the interval [ 0, 1 ] such that μ n = ∫ 0 1 t n d α ( t ), n ∈ N 0. (1).

A linear operator S in a complex Hilbert space H for which the set D∞(S) of its C∞-vectors is dense in H and {‖Snf‖2}n="0∞ is a Stieltjes moment sequence for every f∈D∞(S) is said to generate Stieltjes moment sequences.

Let s be a non-vanishing Stieltjes moment sequence and let μ be a representing measure of it.

Figure 5 shows an example of MDCT 2-order Zernike moment sequence calculated from a 5-s clip of an MP3 song.

We derive necessary and sufficient conditions for an infinite sequence of generalized functions to be realizable on S, namely to be the moment sequence of a finite measure concentrated on S. Our approach combines the classical results about the moment problem on nuclear spaces with the techniques recently developed to treat the moment problem on basic semi-algebraic sets of Rd.

Then (i) the limits ({m_{k}}_{k=1}^{infty }) are the moment sequence of a distribution function, say F; (ii) if the limit F given by (i) is M-det, F n converges to F weakly as n→∞.