Wavelet characterization of growth spaces of harmonic functions

We consider the space h _ν ^∞ of harmonic functions in R ₊ ⁿ⁺¹ with finite norm ‖u‖ _ν = sup |u(x, t)|/v(t), where the weight ν satisfies the doubling condition. Boundary values of functions in h _ν ^∞ are characterized in terms of their smooth multiresolution approximations. The characterization yields the isomorphism of Banach spaces h _ν ^∞ ～ l ^∞. The results are also applied to obtain the law of the iterated logarithm for the oscillation of functions in h _ν ^∞ along vertical lines.     

An embedding theorem for a weighted space of Sobolev type and correct solvability of the Sturm-Liouville equation

We consider the weighted space W ₁ ⁽²⁾ (?,q) of Sobolev type $$W_1^{(2)} (\mathbb{R},q) = \left\{ {y \in A_{loc}^{(1)} (\mathbb{R}):\left\| {y''} \right\|_{L_1 (\mathbb{R})} + \left\| {qy} \right\|_{L_1 (\mathbb{R})} < \infty } \right\} $$ and the equation $$ - y''(x) + q(x)y(x) = f(x),x \in \mathbb{R} $$ Here f ε L ₁(?) and 0 ? q ∈ L ₁ ^loc (?). We prove the following:

1. The problems of embedding W ₁ ⁽²⁾ (?q) ? L ₁(?) and of correct solvability of (1) in L ₁(?) are equivalent
2. an embedding W ₁ ⁽²⁾ (?,q) ? L ₁(?) exists if and only if $$\exists a > 0:\mathop {\inf }\limits_{x \in R} \int_{x - a}^{x + a} {q(t)dt > 0} $$

     

A local version of Hardy spaces associated with operators on metric spaces

Let(X,d,μ) be a metric measure space endowed with a distance d and a nonnegative Borel doubling measure μ.Let L be a second order self-adjoint positive operator on L2(X).Assume that the semigroup e tL generated by L satisfies the Gaussian upper bounds on L 2(X).In this article we study a local version of Hardy space h1L(X) associated with L in terms of the area function characterization,and prove their atomic characters.Furthermore,we introduce a Moser type local boundedness condition for L,and then we apply this condition to show that the space h1L(X) can be characterized in terms of the Littlewood-Paley function.Finally,a broad class of applications of these results is described.     

Multi-neighboring grids schemes for solving PDE eigen-problems

Instead of most existing postprocessing schemes,a new preprocessing approach,called multineighboring grids(MNG),is proposed for solving PDE eigen-problems on an existing grid G(Δ).The linear or multi-linear element,based on box-splines,are taken as the frst stage Kh1Uh=λh1Mh1Uh.In this paper,the j-th stage neighboring-grid scheme is defned asKh jUh=λh j Mh jUh,where Kh j:=Mh j 1Kh1and Mh jUh is to be found as a better mass distribution over the j-th stage neighboring-gridG(Δ),and Kh jcan be seen as an expansion of Kh1on the j-th neighboring-grid with respect to the(j 1)-th mass distribution Mh j 1.It is shown that for an ODE model eigen-problem,the j-th stage scheme with 2j-th order B-spline basis can reach2j-th order accuracy and even(2j+2)-th order accuracy by perturbing the mass matrix.The argument can be extended to high dimensions with separable variable cases.For Laplace eigen-problems with some 2-D and 3-D structured uniform grids,some 2j-th order schemes are presented for j 3.     

An example of asymmetry of convolution operators

Esistono un gruppo compatto non commutativoG ^～ ed un operatore di convoluzioneT tale che: perp∈[2,4] e perq∈[1,2),T∈L _p ^p (G ^～) eT?L _q ^q (G ^～).     

Connection between random curves,changes of time,and regenerative times of stochastic processes

The product of spaces Φ × D is considered, where Φ is the set of all continuous, nondecreasing functions ?:[0,∞)→(0,∞), ?(0)=0, ?(t)→∞(t→∞), and D is the set of all right continuous functions ξ:(0,∞)→X; here X is some metric space. Two mappings are defined: the first is the projection q(?,ξ)=ξ, and the second is the change of time U(?,ξ)=ξº?. The following equivalence relation is defined on D: $$\xi _1 \sim \xi _2 \Leftrightarrow \exists _{\varphi _1 , \varphi _1 } \in \Phi :\xi _1 ^\circ \varphi _1 = \xi _2 ^\circ \varphi _2 $$ . Let? be the set of all equivalence classes, and let L be the mapping ξ₄～ξ₂, Lξ is called the curve corresponding to ξ. The following theorem is proved: two stochastic processes with probability measures P¹ and P² on D possess identical random curves (i.e.,P¹ºL^?1=P²ºL^?1) if and only if there exist two changes of time (i.e., probability measures Q¹ and Q² on ?×D for which P¹=Q¹ºq^?1, P²=Q²ºq^?1 which take these two processes into a process with measure \(\tilde P\) (i.e., Q¹ºu^?1=Q²ºu^?1,=～P) If (P _x ¹ )_x∈X and (P _x ² )_x∈X are two families of probability measures for which P _x ¹ ºL^?1=P _x ² ºL^?1?x∈X then for each x ε X the corresponding measures Q _X ¹ andQ _X ² can be found in the following manner. The set of regenerative times of the family \(\left( {\tilde P_x } \right)_{x \in X} \) contains all stopping times which are simultaneously regenerative times of the families (p _x ¹ )_x∈X and (P _x ² )_x∈X and possess a certain special property of first intersection.     

L p-approximation by Kantorovič operators

Оператор Канторович а дляf∈L _p(I), I=[0,1], определяе тся соотношением $$P_n (f,x) = (n + 1)\sum\limits_{k = 0}^n {\left( {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right)} x^k (1 - x)^{n - 1} \int\limits_{I_k } {f(t)dt,} $$ гдеI _k=[k/(n}+1),(k+1)/(n+ 1)],n∈N. Доказывается, что есл ир>1 иf∈W _p ² (I), т.е.f абсол ютно непрерывна наI иf″∈L _p(I), то $$\left\| {P_n f - f} \right\|_p = O(n^{ - 1} ).$$ Далее, установлено, чт о еслиf∈L _p(I),p>1 и ∥P _n f-f∥_р=О(n ^?1), тоf∈S, гдеS={f∶f аб-солютно непрерывна наI, x(1?x)f′(x)=∝ ₀ ^x h(t)dt, гдеh∈L _p(I) и ∝ ₀ ¹ h(t)dt=0}. Если жеf∈L_p(I),p>1, то из условия ∥P _n(f)?f∥_pL=o(n^?1) вытекает, чтоf постоянна почти всюду.     

The local and global existence of solutions for a generalized Camassa-Holm equation

A fully nonlinear generalization of the Camassa-Holm equation is investigated. Using the pseudoparabolic regularization technique, its local well-posedness in Sobolev space Hs(R) with s3/2 is established via a limiting procedure. Provided that the initial momentum (1-x2)u0 satisfies the sign condition, u0∈Hs(R)(s3/2) and u0∈L1(R),the existence and uniqueness of global solutions for the equation are shown to be true in the space C([0,∞); Hs(R))∩C1([0,∞);Hs-1(R)).     

Asymptotics of the eigenvalues of the Sturm-Liouville problem with discrete self-similar weight

We study the asymptotics of the spectrum of the boundary-value problem $$ - y'' - \lambda \rho y = 0,y(0) = y(1) = 0 $$ for the case in which the weight ρ ∈ W? ₂ ^?1 [0, 1] is the generalized (in the sense of distributions) derivative of a self-similar function P ∈ L ₂[0, 1] of zero spectral order.     

Vector measures: where are their integrals?

Let ν be a vector measure with values in a Banach space Z. The integration map $I_\nu: L^1(\nu)\to Z$ , given by $f\mapsto \int f\,d\nu$ for f ∈ L ¹(ν), always has a formal extension to its bidual operator $I_\nu^{**}: L^1(\nu)^{**}\to Z^{**}$ . So, we may consider the “integral” of any element f ^** of L ¹(ν)^** as I _ν ^** (f ^**). Our aim is to identify when these integrals lie in more tractable subspaces Y of Z ^**. For Z a Banach function space X, we consider this question when Y is any one of the subspaces of X ^** given by the corresponding identifications of X, X′′ (the Köthe bidual of X) and X′^* (the topological dual of the Köthe dual of X). Also, we consider certain kernel operators T and study the extended operator I _ν ^** for the particular vector measure ν defined by ν(A) := T(χ _A ).     

Statistical limit of Lebesgue measurable functions at ∞ with applications in Fourier Analysis and Summability

This is basically a survey paper on recent results indicated in the title. A function s: [a, ∞) → ?, measurable in Lebesgue’s sense, where a ≥ 0, is said to have statistical limit ? at ∞ if for every ? > 0, $\mathop {\lim }\limits_{b \to \infty } (b - a)^{ - 1} |\{ \nu \in (a,b):|s(\nu ) - \ell | > \varepsilon \} | = 0$ . We briefly summarize the main properties of this new concept of statistical limit at ∞. Then we demonstrate its applicability in Fourier Analysis. For example, the classical inversion formula involving the Fourier transform $\hat s$ of a function s ∈ L ¹(?) remains valid even in the general case when $\hat s\not \in L^1 (\mathbb{R})$ . We also present Tauberian conditions, under which the ordinary limit of a function s ∈ L _loc ¹ [1,∞) follows from the existence of the statistical limit of its logarithmic mean at ∞.     

Extensions of certain operators to spaces of abstract integrable functions

In this paper we discuss the extension of operators onL ₁ ^R spaces to operators onL ₁ ^E andP ₁ ^E spaces (see Section 1), whereE is a Banach space. A necessary and sufficient condition for the existence of the extension to a spaceP ₁ ^E is given (see Section 3) whenE has the weak Radon-Nikodym property. The paper contains certain applications to ergodic theory and a theorem giving a characterization of weakly conditionally compact sets.     

On the cosets of the 2q-power group in the unit group modulop

Letp be a prime number ≡ 3 mod 4,G ^p the unit group of ?/p?, andg a generator ofG _p. Letq be an odd divisor ofp - 1 andG _p ^2q = {t ^2q;t ∈G _pthe subgroup of index2q inG _p. The groupG _p ² / _p ^2q consists of the classes \(\bar g^{2j} \) ,j = 0,...,q – 1. In this paper we study the ’excesses’ of the classes \(\bar g^{2j} \) in {l,...,(_p–l)/2}, i.e., the numbers \(\Phi _j = \left| {\left\{ {k;1 \leqslant k \leqslant \left( {p - 1} \right)/2,\bar k \in \bar g^{2j} } \right\}} \right| - \left| {\left\{ {k;\left( {p - 1} \right)/2 \leqslant k \leqslant p - 1,\bar k \in \bar g^{2j} } \right\}} \right|\) ,j = 0.....q — 1. First we express therelative class number h _2q of the subfieldK _2q? ?(e^2#x03C0;i/p ) of degree [K _2q: ?] =2q in terms of these excesses. We use this formula to establish certaincongruences for the Ф^j. E.g., ifq ∈ {3,5,11}, each number Фj is congruent modulo 4 to each other iff 2 dividesh _2q ^- . Finally we study thevariance of the excesses, i.e., the number \(\sigma ^2 = ((\Phi _0 - \hat \Phi )^2 + \ldots + (\Phi _{q - 1} - \hat \Phi )^2 )/(q - 1)\) , where \(\hat \Phi \) is the mean value of the numbers Ф_j. We obtain an explicit lower bound for σ² in terms ofh _2q ^- /h ₂ ^- . Moreover, we show that log σ² is asymptotically equal to 21og(h _2q ^- h ₂ ^- )/(q - 1) forp→∞. Three tables illustrate the results.     

Inversion complexity of self-correcting circuits for a certain sequence of boolean functions

The asymptotics L _k ^? (f ₂ ⁿ ) ?? n min{k+1, p} is obtained for the sequence of Boolean functions $f_2^n \left( {x_1 , \ldots ,x_n } \right) = \mathop \vee \limits_{1 \leqslant i < j \leqslant n}$ for any fixed k, p ?? 1 and growing n, here L _k ^? (f ₂ ⁿ ) is the inversion complexity of realization of the function f ₂ ⁿ by k-self-correcting circuits of functional elements in the basis B = {&, ?}, p is the weight of a reliable invertor.     

On quasinormal,subnormal, and hyponormal Toeplitz operators

Let ? be a non-constant function inL ^∞(D) such thatφ=φ ₁+φ ₂, whereφ ₁ is an element in the Bergman spaceL _a ² (D), and \(\phi _2 \in \overline {L_a^2 (D)} \) , the space of all complex conjugates of functions inL _a ² (D). In this paper, it is shown that if 1 is an element in the closure of the range of the self-commutator ofT _?, \(T_{\bar \phi } T_\phi - T_\phi T\phi \) , then the Toeplitz operatorT _? defined ofL _a ² (D) is not quasinormal. Moreover, if \(\phi = \psi + \lambda \bar \psi \) , whereψ∈ H ^∞(D), and λεC, it is proved that ifT _? is quasinormal, thenT _? is normal. Also, the spectrum of a class of pure hyponormal Toeplitz operators is determined.     

Truncation and aliasing errors for Whittaker-Kotelnikov-Shannon sampling expansion

Let B _?? ^p , 1 ?? p < ??, be the space of all bounded functions from L _p (?) which can be extended to entire functions of exponential type ??. The uniform error bounds for truncated Whittaker-Kotelnikov-Shannon series based on local sampling are derived for functions f ?? B _?? ^p without decay assumption at infinity. Then the optimal bounds of the aliasing error and truncation error of Whittaker-Kotelnikov-Shannon expansion for non-bandlimited functions from Sobolev classes U(W _p ^r (?)) are determined up to a logarithmic factor.     

Explicit solution of the inverse kinematic problem in the non-Herglotz case

The inverse kinematic problem is solved in the half space R ₊ ^ν+1 ={(x,z)|z?0,x∈R^ν, ν?1 under the assumption that the index of refraction can be represented in the form $$n^2 (x,z) = K^2 (z) + \sum\limits_{j = 1}^\nu {\Phi _j^2 (x_j ),} n_z< 0.$$ . The solution obtained is a generalization of the Herglotz-Wiechert formula. A formula is presented for the solution of the inverse kinematic problem in the general case of separation of variables in the eikonal equation.     

Asymptotic expansions of solutions of equations with a deviating argument in Banach spaces

For the equation $$Lu = \frac{1}{i}\frac{{du}}{{dt}}\sum\nolimits_{j = 0}^m {A_j u} (l - h_j^0 - h_j^1 (t)) = f(t),$$ whereh ₀ ^o =0,h ₀ ¹ =0 (t) ≡ 0,h _j ^o = const > 0,h ₁ ^j (t),j= 1, ...,m are nonnegative continuously differentiable functions in [0, ∞), Aj are bounded linear operators, under conditions on the resolvent and on the right hand sidef(t), we have obtained an asymptotic formula for any solution u(t) from L₂ in terms of the exponential solutions u_k(t), k=1, ..., n, of the equation $$\frac{1}{i}\frac{{du}}{{dt}} - A_0 u - \sum\nolimits_{j = 0}^m {A_j u} (t - h_j^0 ) = 0,$$ connected with the poles λ_k, k=1, ..., n, of the resolvent R_λ in a certain strip.     

Multipliers for the Weyl transform and Laguerre expansions

Let P_k denote the projection of L²(R ^R ) onto the kth eigenspace of the operator (-δ+?x?² andS _N ^α =(1/A _N ^α )Σ _k ^N =0A _N?k ^α P _k . We study the multiplier transformT _N ^α for the Weyl transform W defined byW(T _N ^αf )=S _n ^αW(f) . Applications to Laguerre expansions are given.     

Bounds for the spectral abscissa of an element in a Banach algebra

For an arbitrary element x with spectrum sp(x) in a Banach algebra with identity e ≠ 0 we define the upper (lower) spectral abscissa \(\mathop {\sigma + (x)}\limits_{( - )} = \mathop {\max }\limits_{(\min )} \operatorname{Re} \lambda ,\lambda \in sp(x)\) . With the aid of the spectral radius \(\rho (x) = \mathop {\max }\limits_{\lambda \in sp(x)} \left| \lambda \right| = \mathop {\lim }\limits_{n \to + \infty } \parallel x^n {{1 - } \mathord{\left/ {\vphantom {{1 - } n}} \right. \kern-0em} n}\) we prove the following bounds: γ_?(x)?σ_?(x)?Γ_?(x)?₊(x)?σ₊(x)?γ₊(x), Γ_(±)(x)=(2δ_(±))^?1 (ρ _δ ² )_(±)?δ _(±) ² ?ρ ₀ ² )(δ_(±)≠0), γ_(±)(x)= (±)ρ_δ(±)?δ_(±), δ₊?0, δ_??0 ρ _(±) ^δ = ρ(x+eδ_(±)). We mention a case where equality is achieved, some corollaries,and discuss the sharpness of the bounds: for every ? > 0 there is a δ: ¦δ¦ ≥ρ ₀ ² /2?, such that Δ: = ¦γ_(±) x?Γ_(±) x¦?ε and conversely, if the bounds are computed for some δ ≠ 0, then △ ≤ρ ₀ ² /2 ¦δ¦. An example is considered.