The maximal variation of a bounded martingale

Let \(\chi _0^n = \left\{ {X_t } \right\}_0^n \) be a martingale such that 0≦X_i≦1;i=0, …,n. For 0≦p≦1 denote by ? _p ⁿ the set of all such martingales satisfying alsoE(X₀)=p. Thevariation of a martingale χ ₀ ⁿ is denoted byV ₀ ⁿ and defined by \(V(\chi _0^n ) = E\left( {\sum {_{l = 0}^{n - 1} } \left| {X_{l + 1} - X_l } \right|} \right)\) . It is proved that $$\mathop {\lim }\limits_{n \to \infty } \left\{ {\mathop {Sup}\limits_{x_0^n \in \mathcal{M}_p^n } \left[ {\frac{1}{{\sqrt n }}V(\chi _0^n )} \right]} \right\} = \phi (p)$$ , where ?(p) is the well known normal density evaluated at itsp-quantile, i.e. $$\phi (p) = \frac{1}{{\sqrt {2\pi } }}\exp ( - \frac{1}{2}\chi _p^2 ) where \int_{ - \alpha }^{x_p } {\frac{1}{{\sqrt {2\pi } }}\exp ( - \frac{1}{2}\chi ^2 )} dx = p$$ . A sequence of martingales χ ₀ ⁿ ,n=1,2, … is constructed so as to satisfy \(\lim _{n \to \infty } (1/\sqrt n )V(\chi _0^n ) = \phi (p)\) .     

A nonclassical LIL for geometrically weighted series in Banach space

Let {X, Xn ; n ≥ 0} be a sequence of independent and identically distributed random variables, taking values in a separable Banach space (B,||·||) with topological dual B* . Considering the geometrically weighted series ξ(β) =∑∞n=0βnXn for 0 β 1, and a sequence of positive constants {h(n), n ≥ 1}, which is monotonically approaching infinity and not asymptotically equivalent to log log n, a limit result for(1-β2)1/2||ξ(β)||/(2h(1/(1-β2)))1/2 is achieved.     

On the laws of large numbers for double arrays of independent random elements in Banach spaces

For a double array of independent random elements {Vmn,m ≥ 1,n ≥ 1} in a real separable Banach space,conditions are provided under which the weak and strong laws of large numbers for the double sums mi=1 nj=1Vij,m ≥ 1,n ≥ 1 are equivalent.Both the identically distributed and the nonidentically distributed cases are treated.In the main theorems,no assumptions are made concerning the geometry of the underlying Banach space.These theorems are applied to obtain Kolmogorov,Brunk–Chung,and Marcinkiewicz–Zygmund type strong laws of large numbers for double sums in Rademacher type p(1 ≤ p ≤ 2) Banach spaces.     

A law of the iterated logarithm for geometrically weighted martingale difference sequences

We consider random variables ξ(β)=Σ _n=0 ^∞ β ⁿ Y _n for β<1. We prove that if the(Y _n)_n∈N is a stationary ergodic martingale difference sequence andE Y ₀ ² =1, then the following law of the iterated logarithm holds: $$\mathop {\lim \sup }\limits_{\beta \nearrow 1} \frac{{\sqrt {1 - \beta ^2 } }}{{\sqrt {2\log \log \frac{1}{{1 - \beta ^2 }}} }}\xi (\beta ) = 1 a.s.$$ We prove also the corresponding Central Limit Theorem. This generalizes a theorem by Bovier and Picco where the i.i.d. case was studied.     

Methods of summation and best approximation

Let λ={λ _k ⁿ } be a triangular method of summation,f ε L_p (1 ≤ p ≤ ∞), $$U_n (f,x,\lambda ) = \frac{{a_0 }}{2} + \sum _{k = 1}^n \lambda _k^n (a_k \cos kx + b_k \sin kx).$$ Consideration is given to the problem of estimating the deviations ∥f ? U_n (f, λ) ∥ L_p in terms of a best approximation E_n (f) L_p in abstract form (for a sequence of projectors in a Banach space). Various generalizations of known inequalities are obtained.     

How do random Fibonacci sequences grow?

We study the random Fibonacci sequences defined by ${F_1 = F_2 = \widetilde F_1 = \widetilde F_2 = 1}$ and for n ≥ 1, F _n+2 = F _n+1 ± F _n (linear case) and ${\widetilde F_{n+2} = |\widetilde F_{n+1} \pm \widetilde F_{n}|}$ (non-linear case), where each ± sign is independent and either + with probability p or ? with probability 1 ? p (0 < p ≤ 1). Our main result is that the exponential growth of F _n for 0 < p ≤ 1, and of ${\widetilde F_{n}}$ for 1/3 ≤ p ≤ 1 is almost surely given by $${\int \limits _0^\infty \log x\, d\nu_\alpha (x),}$$ where α is an explicit function of p depending on the case we consider, and ν _α is an explicit probability distribution on ${\mathbb{R}_+}$ defined inductively on Stern–Brocot intervals. In the non-linear case, the largest Lyapunov exponent is not an analytic function of p, since we prove that it is equal to zero for 0 < p ≤ 1/3. We also give some results about the variations of the largest Lyapunov exponent, and provide a formula for its derivative.     

Estimates for wave and Klein-Gordon equations on modulation spaces

We prove that the fundamental semi-group eit(m 2I+|Δ|)1/2(m = 0) of the Klein-Gordon equation is bounded on the modulation space M ps,q(Rn) for all 0 < p,q ∞ and s ∈ R.Similarly,we prove that the wave semi-group eit|Δ|1/2 is bounded on the Hardy type modulation spaces μsp,q(Rn) for all 0 < p,q ∞,and s ∈ R.All the bounds have an asymptotic factor tn|1/p 1/2| as t goes to the infinity.These results extend some known results for the case of p 1.Also,some applications for the Cauchy problems related to the semi-group eit(m2I+|Δ|)1/2 are obtained.Finally we discuss the optimum of the factor tn|1/p 1/2| and raise some unsolved problems.     

Commutator of hypersingular integral with rough kernels on Sobolev spaces

In this paper, the authors give the boundedness of the commutator of hypersingular integral T γ from the homogeneous Sobolev space Lpγ (Rn) to the Lebesgue space Lp(Rn) for 1p∞ and 0 γ min{ n/2 , n/p }.     

Convergence of biorthogonal series in the system of contractions and translations of functions in the spaces L p [0, 1]

We obtain conditions for the convergence in the spaces L ^p [0, 1], 1 ≤ p < ∞, of biorthogonal series of the form $$ f = \sum\limits_{n = 0}^\infty {(f,\psi _n )\phi _n } $$ in the system {? _n } _n≥0 of contractions and translations of a function ?. The proposed conditions are stated with regard to the fact that the functions belong to the space $ \mathfrak{L}^p $ of absolutely bundleconvergent Fourier-Haar series with norm $$ \left\| f \right\|_p^ * = \left| {f,\chi _0 } \right| + \sum\limits_{k = 0}^\infty {2^{k({1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} - {1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p})} } \left( {\sum\limits_{n = 2^k }^{2^{k + 1} - 1} {\left| {f,\chi _n } \right|^p } } \right)^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} , $$ where (f,χ _n ), n = 0, 1, ..., are the Fourier coefficients of a function f ? L ^p [0, 1] in the Haar system {χ _n } _n≥0. In particular, we present conditions for the system {? _n } _n≥0 of contractions and translations of a function ? to be a basis for the spaces L ^p [0, 1] and $ \mathfrak{L}^p $ .     

Convergence of weighted averages of noncommutative martingales

Let x = (x _n ) _n?1 be a martingale on a noncommutative probability space ( $\mathcal{M}$ , τ) and (w _n ) _n?1 a sequence of positive numbers such that $W_n = \sum\nolimits_{k = 1}^n {w_k \to \infty } $ as n → ∞. We prove that x = (x _n ) _n?1 converges bilaterally almost uniformly (b.a.u.) if and only if the weighted average (σ _n (x)) _n?1 of x converges b.a.u. to the same limit under some condition, where σ _n (x) is given by $\sigma _n (x) = \frac{1} {{W_n }}\sum\limits_{k = 1}^n {w_k x_k } ,n = 1,2,... $ Furthermore, we prove that x = (x _n ) _n?1 converges in L _p ( $\mathcal{M}$ ) if and only if (σ _n (x)) _n?1 converges in L _p ( $\mathcal{M}$ ), where 1 ? p < ∞. We also get a criterion of uniform integrability for a family in L ₁( $\mathcal{M}$ ).     

Commutators of Marcinkiewicz integral with rough kernels on Sobolev spaces

In this paper, the authors give the boundedness of the commutator [b, ??_??,?? ] from the homogeneous Sobolev space $\dot L_\gamma ^p \left( {\mathbb{R}^n } \right)$ to the Lebesgue space L ^p (? ⁿ ) for 1 < p < ??, where ??_??,?? denotes the Marcinkiewicz integral with rough hypersingular kernel defined by $\mu _{\Omega ,\gamma } f\left( x \right) = \left( {\int_0^\infty {\left| {\int_{\left| {x - y} \right| \leqslant t} {\frac{{\Omega \left( {x - y} \right)}} {{\left| {x - y} \right|^{n - 1} }}f\left( y \right)dy} } \right|^2 \frac{{dt}} {{t^{3 + 2\gamma } }}} } \right)^{\frac{1} {2}} ,$ , with ?? ?? L ¹(S ^n?1) for $0 < \gamma < min\left\{ {\frac{n} {2},\frac{n} {p}} \right\}$ or ?? ?? L(log⁺ L) ^?? (S ^n?1) for $\left| {1 - \frac{2} {p}} \right| < \beta < 1\left( {0 < \gamma < \frac{n} {2}} \right)$ , respectively.     

Erratum to: On the Crank�CNicolson scheme once again

Let ${\left(\tau_j\right)_{j\in\mathbb{N}}}$ be a sequence of strictly positive real numbers, and let A be the generator of a bounded analytic semigroup in a Banach space X. Put ${A_n=\prod_{j=1}^n\left(I+\frac{1}{2}\tau_jA\right)\left(I-\frac{1}{2}\tau_jA\right)^{-1}}$ , and let ${x\in X}$ . Define the sequence ${\left(x_n\right)_{n\in\mathbb{N}}\subset X}$ by the Crank?CNicolson scheme: x _n ?=?A _n x. In this erratum, it is proved that the Crank?CNicolson scheme is stable in the sense that ${\sup_{n\in\mathbb{N}}\left\Vert A_nx\right\Vert < \infty}$ provided that inequality (0.9) below holds.     

On the best polynomial approximation of entire transcendental functions in banach spaces. I

We study the behavior of the best approximationsE _n (?) _p of entire transcendental functions ?(z) of the order ρ=∞ by polynomials of at mostn th degree in the metric of the Banach space E′_p(Ω) of functions /tf(z) analytic in a bounded simply connected domain Ω with rectifiable Jordan boundary and such that $$\left\| f \right\|_{E'_p } = \left\{ {\iint_\Omega {\left| {f\left( z \right)} \right|^p }dxdy} \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}}< \infty $$ . In particular, we describe the relationship between the best approximationsE _n (?)_p and theq-order andq-type of the function ?(z).     

A note on maximal regular subsemigroups of the finite transformation semigroups \mathcal{T}(n,r)

Let $\mathcal{T}_{n}$ be the semigroup of all full transformations on the finite set X _n ={1,2,…,n}. For 1≤r≤n, set $\mathcal {T}(n, r)=\{ \alpha\in\mathcal{T}_{n} | \operatorname{rank}(\alpha)\leq r\}$ . In this note we show that, for 2≤r≤n?2, any maximal regular subsemigroup of the semigroup $\mathcal{T} (n,r)$ is idempotent generated, but this may not happen in the semigroup $\mathcal{T}(n, n-1)$ .     

Representations of analytic functions as infinite products and their application to numerical computations

Let D be an open disk of radius ≤1 in $\mathbb{C}$ , and let (? _n ) be a sequence of ±1. We prove that for every analytic function $f: D \to \mathbb{C}$ without zeros in D, there exists a unique sequence (α _n ) of complex numbers such that $f(z) = f(0)\prod_{n=1}^{\infty} (1+\epsilon_{n}z^{n})^{\alpha_{n}}$ for every z∈D. From this representation we obtain a numerical method for calculating products of the form ∏ _p prime f(1/p) provided f(0)=1 and f′(0)=0; our method generalizes a well-known method of Pieter Moree. We illustrate this method on a constant of Ramanujan $\pi^{-1/2}\prod _{p~\text{prime}} \sqrt{p^{2}-p}\ln(p/(p-1))$ . From the properties of the exponents α _n , we obtain a proof of the following congruences, which have been the subject of several recent publications motivated by some questions of Arnold: for every n×n integral matrix A, every prime number p, and every positive integer k we have $\operatorname{tr} A^{p^{k}} \equiv\operatorname{tr} A^{p^{k-1}} { \hbox {\rm { (mod\ $p^{k}$) }}}$ .     

Uniqueness in law for a class of degenerate diffusions with continuous covariance

We study the martingale problem associated with the operator $$ Lu(s, x) = \partial_su(s, x) + \frac{1}{2} \sum_{i,j=1}^{d_0} a^{ij}(s, x) \partial_{ij}u(s, x) + \sum_{i,j=1}^d B^{ij} x^j \partial_iu(s, x), $$ where d ₀ ≤  d. We show that the martingale problem is well-posed when the function a is continuous and strictly positive definite on ${\mathbb{R}^{d_0}}$ and the matrix B takes a particular lower-diagonal, block form. We then localize this result to show that the martingale problem remains well-posed when B is replaced by a sufficiently smooth vector field whose Jacobian matrix satisfies a nondegeneracy condition.     

Description of traces of functions in the Sobolev space with a Muckenhoupt weight

We characterize the trace of the Sobolev space W _p ^l (? ⁿ , γ) with 1 < p < ∞ and weight γ ∈ A _p ^loc (? ⁿ ) on a d-dimensional plane for 1 ≤ d < n. It turns out that for a function φ to be the trace of a function f ∈ W _p ^l (? ⁿ , γ), it is necessary and sufficient that φ belongs to a new Besov space of variable smoothness, $\overline B _p^l \left( {\mathbb{R}^d ,\left\{ {\gamma _{k,m} } \right\}} \right)$ , constructed in this paper. The space $\overline B _p^l \left( {\mathbb{R}^d ,\left\{ {\gamma _{k,m} } \right\}} \right)$ is compared with some earlier known Besov spaces of variable smoothness.     

ОБ ИНтЕРпОлИРОВАНИИ В кОМплЕксНОИ ОБлАст И

LetD be a simply connected domain, the boundary of which is a closed Jordan curveγ; \(\mathfrak{M} = \left\{ {z_{k, n} } \right\}\) , 0≦k≦n; n=1, 2, 3, ..., a matrix of interpolation knots, \(\mathfrak{M} \subset \Gamma ; A_c \left( {\bar D} \right)\) the space of the functions that are analytic inD and continuous on \(\bar D; \left\{ {L_n \left( {\mathfrak{M}; f, z} \right)} \right\}\) the sequence of the Lagrange interpolation polynomials. We say that a matrix \(\mathfrak{M}\) satisfies condition (B _m ), \(\mathfrak{M}\) ∈(B _m ), if for some positive integerm there exist a setB _m containingm points and a sequencen _{p p=1} ^∞ of integers such that the series \(\mathop \Sigma \limits_{p = 1}^\infty \frac{1}{{n_p }}\) diverges and for all pairsn _i ,n _j ∈{n _p } _p=1 ^∞ the set \(\left( {\bigcap\limits_{k = 0}^{n_i } {z_{k, n_i } } } \right)\bigcap {\left( {\bigcup\limits_{k = 0}^{n_j } {z_{k, n_j } } } \right)} \) is contained inB _m . The main result reads as follows. {Let D=z: ¦z¦ \(\Gamma = \partial \bar D\) and let the matrix \(\mathfrak{M} \subset \Gamma \) satisfy condition (B_m). Then there exists a function \(f \in A_c \left( {\bar D} \right)\) such that the relation $$\mathop {\lim \sup }\limits_{n \to \infty } \left| {L_n \left( {\mathfrak{M}, f, z} \right)} \right| = \infty $$ holds almost everywhere on γ.     

Entropy numbers of Besov classes of generalized smoothness on the sphere

We investigate the asymptotic behavior of the entropy numbers of Besov classes BBΩp,θ(Sd 1)of generalized smoothness on the sphere inL q(Sd 1)for 1≤p,q,θ≤∞,and get their asymptotic orders.We also obtain the exact orders of entropy numbers of Sobolev classesBWr p(Sd 1)inL q(Sd 1)whenpand/orqis equal to 1 or∞.This provides the last piece as far as exact orders of entropy numbers ofBWr p(Sd 1)inL q(Sd 1)are concerned.     

Fixed Points and Stability for Quadratic Mappings in β–Normed Left Banach Modules on Banach Algebras

The goal of the present paper is to investigate some new stability results by applying the alternative fixed point of generalized quadratic functional equation $$\begin{array}{ll}f\left(\sum\limits_{i=1}^{n}a_ix_i\right)+{\sum\limits_{i=1}^{n-1}}{\sum\limits_{j=i+1}^{n}}\left[f(a_ix_i+a_jx_j)+2f(a_ix_i-a_jx_j)\right]\\ \qquad \quad = (3n-2){\sum\limits_{i=1}^{n}}a^2_{i}f(x_{i})\end{array}$$ in β–Banach modules on Banach algebras, where ${a_{1},\dots,a_{n}\in \mathbb{Z}{\setminus}\{0\}}$ and some ${\ell\in\{1 , 2 ,\dots, n-1\},}$ a _? ?≠ ±1 and a _n ?=?1, where n is a positive integer greater or at least equal to two.