Rademacher Sequence Research Articles

Saeki’s improvement of the Vitali-Hahn-Saks-Nikodým theorem holds precisely for Banach spaces having cotype

We prove that a Banach space X X has nontrivial cotype if and only if given any σ \sigma -field Σ \Sigma and any sequence μ n : Σ → X {\mu _n}:\Sigma \to X of strongly additive vector measures such that for some γ ≥ 1 , lim sup n → ∞ ‖ μ n ( E ) ‖ ≤ γ lim inf n → ∞ ‖ μ n ( E ) ‖ > ∞ \gamma \geq 1,\lim {\sup _{n \to \infty }}\left \| {{\mu _n}\left ( E \right )} \right \| \leq \gamma \lim {\inf _{n \to \infty }}\left \| {{\mu _n}\left ( E \right )} \right \| > \infty for each E ∈ Σ E \in \Sigma then { μ n : n ∈ N } \left \{ {{\mu _n}:n \in \mathbb {N}} \right \} is uniformly strongly additive.

RUC-systems and Besselian systems in Banach spaces

Let (rj) be a Rademacher sequence of random variables – that is, a sequence of independent random variables, with , for each j. A biorthogonal system in a Banach space X is called an RUC-system[l] if for every x in [ej] (the closed linear span of the vectors ej), the seriesconverges for almost every ω. A basis which, together with its coefficient functionals, forms an RUC-system is called an RUC-basis. A biorthogonal system is an RLTC-svstem if and only if there exists 1 ≤ K < ∞ such thatfor each x in [ej]: the RUC-constant of the system is the smallest constant K satisfying (1) (see [1], proposition 1.1).

Spread Spectrum and Time Division Multiple Access Satellite Communications

Although time division multiple access (TDMA) satellite communication provides the highest traffic capacity per satellite and offers efficient transmission of a wide variety of services, it suffers from network timing and ranging requirements and message security. On the other hand, spread spectrum multiple access (SSMA) satellite communication is suited for reliable random access and tactical transmission systems. Orthogonal functions such as Rademacher, Haar, and Walsh, or pseudonoise sequence coding of amplitude and phase of the analog message or code division of baseband signals, frequency and time hopping are among the several methods employed for spread spectrum communications. A combination of TDMA and SSMA can be used for asynchronous and reliable transmission of digital or analog signals.

Uniform convergence of random Fourier series

The sufficient part of this theorem was obtained by Salem and Zygmund [7] in the case where (b, is a real number and {q.} a Rademacher sequence (a Rademacher sequence is a sequence of independent random variables {e.} where e.-+_ 1 each with probability 1/2) and extended to independent symmetric {q.e i~-} by Kahane [4]. In fact for sufficiency neither the condition a.~ nor l i m i n f . ~ EIr/.l>0 is needed, on the other hand symmetry is not needed for necessity. Theorem 1.1 was obtained for random trigonometric series that are also stationary Gaussian