Riesz bases of exponentials and sine-type functions

Completeness of systems of complex exponentials and the Lambert 𝑊 functions

“…where both K (1) n and K (2) n are real and can be bounded with a bound independent of n. Moreover, K (1) n → (a + b) > 0 as n → ∞. Noting that the denominator of the last expression contains a term in n 2 , it follows that for any given x, the infinite product L(n, x) converges.…”

“…Choose N > 0, independently of x, such that when n > N, we have −2α −n −K (1) n > 0 and −2ω n − 2α −n x + K (3) n > 1. Then for sufficiently large M > 0 and x < −M , …”

“…[26]) or nearby the zeroes of a function of sine type (e.g. [1]). These results did not apply to our case since for the sequences we wish to consider, the λ n 's are located in a curvilinear strip.…”

Exact controllability of a Rayleigh beam with a single boundary control

“…Simultaneous controllability problems were first introduced by [24,34]. Controllability and stabilizability of the beam/plate with a control applied to a point/a curve in the beam/plate cases were investigated by a number of researchers including [7][8][9]16,21,41], and references therein. By using a generalization of Ingham's inequality (with a weakened gap condition) (i.e., see [23]) and Diophantine's approximations [14], exact controllability (observability) in finite time, and stabilizability are obtained depending on the Diophantine approximation properties of the joints in the beam case, and how strategic the controlled curve is in the plate case.…”

Construction of Sampling and Interpolating Sequences for Multi-Band Signals. The Two-Band Case

“…(Lyubarskii and Seip, 1997) remark that the method of (Kohlenberg, 1953) can be extended to the case when the intervals comprising E have commensurable lengths. The results of Seip (1995) are free of arithmetic restrictions on the lengths of the intervals comprising the set E; in particular, starting from the "1/4 in the mean" theorem (Avdonin, 1979), he gives a construction of at least one real sampling and interpolating sequence for an arbitrary E consisting of two intervals.…”