Weighted shifts with periodic weight sequences

In two recent articles, we have examined a generalization of the binomial distribution associated with a sequence of positive numbers, involving asymmetric expressions of probabilities that break the symmetry win-loss. We present in this article another generalization (always associated with a sequence of positive numbers) that preserves the symmetry win-loss. This approach is also based on generating functions and presents constraints of non-negativeness, similar to those encountered in our previous articles.

We study the symbolic sequences associated with a two-dimensional nonlinear map describing a digital filter with 2's complement arithmetic, and the role of the filter parameter a = 2cos θ in determining admissibility of these sequences. We solve an appropriately specialized form of Chua and Lin's inequalities governing admissibility, and obtain a complete identification of the infinite class of periodic symbolic sequences admissible in an open interval having parameter value θ = 0 as left end-point. A key step in the argument, easily established by parity for sequences having odd period, requires a more penetrating analysis in the even case. Computer searches for sequences with periods up to 200 have revealed that this class accounts for the greater number of admissible periodic sequences associated with the map. Our solution provides an efficient construction for the periodic symbolic sequences admissible adjacent to zero, parametrized by the period and the number of +− digit pairs present. We also show that the precise interval of admissibility is (0, π/N), where N is the least period, which permits us to relate the sequences to the points of state space that give rise to them. We thereby obtain a bound on the measure of the subset of state space consisting of the points associated with periodic symbolic sequences, improving previously published bounds. A similar identification of the periodic sequences admissible at the far end of the parameter range, adjacent to θ = π, is deduced.

Let <TEX>$\mathcal{A}$</TEX> be the class of analytic functions f in the open unit disk <TEX>$\mathbb{U}$</TEX>={z : <TEX>${\mid}z{\mid}$</TEX> < 1} with the normalization conditions <TEX>$f(0)=f^{\prime}(0)-1=0$</TEX>. If <TEX>$f(z)=z+\sum_{n=2}^{\infty}a_nz^n$</TEX> and <TEX>${\delta}$</TEX> > 0 are given, then the <TEX>$T_{\delta}$</TEX>-neighborhood of the function f is defined as <TEX>$$TN_{\delta}(f)\{g(z)=z+\sum_{n=2}^{\infty}b_nz^n{\in}\mathcal{A}:\sum_{n=2}^{\infty}T_n{\mid}a_n-b_n{\mid}{\leq}{\delta}\}$$</TEX>, where <TEX>$T=\{T_n\}_{n=2}^{\infty}$</TEX> is a sequence of positive numbers. In the present paper we investigate some problems concerning <TEX>$T_{\delta}$</TEX>-neighborhoods of function in various classes of analytic functions with <TEX>$T=\{2^{-n}/n^2\}_{n=2}^{\infty}$</TEX>. We also find bounds for <TEX>$^{\delta}^*_T(A,B)$</TEX> defined by <TEX>$$^{\delta}^*_T(A,B)=jnf\{{\delta}</TEX><TEX>&</TEX><TEX>gt;0:B{\subset}TN_{\delta}(f)\;for\;all\;f{\in}A\}$$</TEX> where A, B are given subsets of <TEX>$\mathcal{A}$</TEX>.

In [1], J. Bruna studied the Beurling classes EM(I) of infinitely differentiable functions f defined on an interval I such that for every positive ε, there exists a constant Cε &gt; 0 with the property thatwhere M = {Mn} is a given sequence of positive numbers. With the hypothesis that the class EM(ℝ) is differentiable, he proved that it is inverse-closed (in the sense that if fεEM(ℝ) and if f is bounded away from zero on ℝ, then its inverse f-1 lies in EM(ℝ)) if and only if the associated sequence A = {An = (Mn/n!)1/n} is almost increasing (i.e. Am ≤ KAn for all m ≤ n, where K &gt; 0 is a constant independent of m and n).

Let be the collection of homogeneous product Moran sets determined by two sequences of positive integers , and two sequences of positive numbers , . We obtain the maximal and minimal values of the Hausdorff and packing dimensions of elements in .

Let $f = (f_1 , f_2 , \cdots ,f_n )$ be a sequence of positive numbers. We construct a class of piecewise linear, ergodic Markov maps $\mathcal{C}_f $ such that for each $\tau \in \mathcal{C}_f $, there exist an interval I and a partition $\{ I_i \} _{i = 1}^n $ of I with the property that $\mu (I_i ) = f_i $, where $\mu $ is the unique measure invariant under $\tau $. In $\mathcal{C}_f $ the trade-off between entropy (randomness) and the speed of numerically computing the orbit $\{ \tau ^j (x)\} _{j\geqq 0} $ can be assessed.

I. Notation. In this paper a Lusin set will be understood to be an uncountable subset of the interval I = [0, 1 ] such that if F is a first category subset of I, then Q F is, at most, a countable set or, equivalently, if { yi } is a countable dense subset of I and { Ei } is a sequence of positive numbers, then Q-UN(yi, Ei) is countable, where N(y, e) =In(y-e/2, y+E/2). By a universally measurable set we shall mean a subset E of R' such that if , is a nonatomic probability measure on the Borel subsets 63o of R', then ,u*(E) =,*(E). A set E is universally measurable if, and only if, every homeomorphism 4 of R' carries E onto a set q5(E) which is Lebesgue measurable. A universal null set is a universally measurable set which is mapped into a Lebesgue null set by all homneomorphisms. Since it imposes no real restriction to suppose that a probability measure on 6(B is nonatomic, henceforth we do so suppose. Finally, denote by (B the Borel subsets Q2lB, B-(Bo, of ?, i.e., 63=(3oI U.

An improved Vietoris sine inequality

Let $f$ be a density on $R^d$, and let $f_n$ be the kernel estimate of $f$, $f_n(x) = (nh^d)^{-1} \sum^n_{i=1} K((x - X_i)/h)$ where $h = h_n$ is a sequence of positive numbers, and $K$ is an absolutely integrable function with $\int K(x) dx = 1$. Let $J_n = \int |f_n(x) - f(x)| dx$. We show that when $\lim_nh = 0$ and $\lim_nnh^d = \infty$, then for every $\varepsilon > 0$ there exist constants $r, n_0 > 0$ such that $P(J_n \geq \varepsilon) \leq \exp(-rn), n \geq n_0$. Also, when $J_n \rightarrow 0$ in probability as $n \rightarrow \infty$ and $K$ is a density, then $\lim_nh = 0$ and $\lim_nnh^d = \infty$.

Let $\mathbf d=(d_j)_{j\in\mathbb I_m}\in\mathbb N^m$ be a finite sequence (of dimensions) and $\alpha=(\alpha_i)_{i\in\mathbb I_n}$ be a sequence of positive numbers (of weights), where $\mathbb I_k=\{1,\ldots,k\}$ for $k\in\mathbb N$. We introduce the $(\alpha\, , \,\mathbf d)$-designs i.e., $m$-tuples $\Phi=(\mathcal F_j)_{j\in\mathbb I_m}$ such that $\mathcal F_j=\{f_{ij}\}_{i\in\mathbb I_n}$ is a finite sequence in $\mathbb C^{d_j}$, $j\in\mathbb I_m$, and such that the sequence of non-negative numbers $(\|f_{ij}\|^2)_{j\in\mathbb I_m}$ forms a partition of $\alpha_i$, $i\in\mathbb I_n$. We characterize the existence of $(\alpha\, , \, \mathbf d)$-designs with prescribed properties in terms of majorization relations. We show, by means of a finite-step algorithm, that there exist $(\alpha\, , \, \mathbf d)$-designs $\Phi^{\rm op}=(\mathcal F_j^{\rm op})_{j\in\mathbb I_m}$ that are universally optimal; that is, for every convex function $\varphi:[0,\infty)\rightarrow [0,\infty)$ then $\Phi^{\rm op}$ minimizes the joint convex potential induced by $\varphi$ among $(\alpha\, , \, \mathbf d)$-designs, namely $$ \sum_{j\in\mathbb I_m}\text{P}_\varphi(\mathcal F_j^{\rm op})\leq \sum_{j\in \mathbb I_m}\text{P}_\varphi(\mathcal F_j) $$ for every $(\alpha\, , \, \mathbf d)$-design $\Phi=(\mathcal F_j)_{j\in\mathbb I_m}$, where $\text{P}_\varphi(\mathcal F)=tr(\varphi(S_{\mathcal F}))$; in particular, $\Phi^{\rm op}$ minimizes both the joint frame potential and the joint mean square error among $(\alpha\, , \, \mathbf d)$-designs. We show that in this case $\mathcal F_j^{\rm op}$ is a frame for $\mathbb C^{d_j}$, for $j\in\mathbb I_m$. This corresponds to the existence of optimal encoding-decoding schemes for multitasking devices with energy restrictions.

Let \(g_{1}, g_{2}, \ldots \) be a sequence of independent identically distributed standard normal random variables and \(\{x_{n}\}\) a sequence of positive numbers.

We introduce the notion of a-walk S(n) = a1X1 +… + anXn, based on a sequence of positive numbers a = (a1, a2,…) and a Rademacher sequence X1,X2,…. We study recurrence/ transience (properly defined) of such walks for various sequences of a.

Let\(\{ \beta (n)\} _{n = 0}^\infty \) be a sequence of positive numbers and 1≤p<∞. We consider the spaceHp(β) of all power series\(f(z) = \sum\limits_{n = 0}^\infty {\hat f(n)z^n } \) such that Σ|\(\Sigma |\hat f(n)|^p \beta (n)^p< \infty \)(n)|p β(np<∞. We investigate regions on which our formal power series represent bounded analytic functions.

Generalized frame called g-frame was first proposed using a sequence of adjointable operators to deal with all the existing frames as a united object. In fact, the g-frame is an extension of ordinary frames. Generalized frames with adjointable operators called K-g-frame is a generalization of a g-frame. It can be used to reconstruct elements from the range of a adjointable operator K. K-g-frames have a certain advantage compared with g-frames in practical applications. This paper is devoted to study some properties of K-g-frame in Hilbert C∗ -module, we characterize the concept of K-g-frame by quotient maps. Also discus some result of the dual K-g-Bessel sequences of K-g-frame in Hilbert C∗ -module. Our results are more general than those previously obtained. It is shown that the results we obtained can immediately lead to the existing corresponding results in Hilbert Spaces.

self adjoint operator has a unique positive self adjoint square root A1/2; and an arbitrary self adjoint operator has a unique representation = A+ ~ A~ with A+ and A self adjoint and positive and the range of each contained in the null space of the other. Write ( = (T*Γ) 1/2 and, for complex λ and r ^ 0, let E(\, r) be the null space projection of (| T — XI — r/) +. For compact subsets K of the complex plane E(K)= Λ and for any Borel set M, E{M) = V {E(K): K compact and K £ M) . It is shown that E is the unique spectral measure such that T = \xE(dX) .