On dyadic analysis based on the pointwise dyadic derivative

В пРЕДыДУЩИх РАБОтАх АВтОРы В ОсНОВНОМ РАж ВИВАлИ ДВОИЧНыИ АНАлИж, ОсНО ВАННыИ НА пОНьтИИ сИльНОИ ДВ ОИЧНОИ пРОИжВОДНОИ Д ль ФУНкцИИ, ОпРЕДЕлЕННых НА ДИАД ИЧЕскОИ ГРУппЕ ИлИ НА [0,1) с пЕРИО ДОМ 1. цЕльУ НАстОьЩЕИ Р АБОты ьВльЕтсь пОстРОЕНИЕ ДВОИЧНОгО ДИФФЕРЕНцИАльНОгО И ИНтЕгРАльНОгО ИсЧИс лЕНИИ НА ОсНОВЕ БОлЕЕ слОжНОг О, НО жАтО И БОлЕЕ клАссИЧЕскОгО пОНьт Иь ДВОИЧНОИ пРОИжВОД НОИ В тОЧкЕ. ИсслЕДУУтсь тЕ пРОст РАНстВА ФУНкцИИ, Дль кОтОРых пРИМЕНИМ ДВОИЧНыИ АНАлИж, А тАк жЕ ОпРЕДЕльУтсь гРАНИц ы ЕгО пРИМЕНИМОстИ. тАк ОкАжАлОсь, ЧтО пРО стРАНстВОL ^p(0, l), 1≦∞, ьВльЕ тсь БОлЕЕ ЕстЕстВЕННыМ п РОстРАН стВОМ Дль пОстРОЕНИь ДВОИЧНОгО АНАлИжА, ЧЕ М клАссИЧЕскОЕ пРОстР АНстВОс[0,1]. НАпРИМЕР, ЕслИ пЕРВАь ДВОИЧНАь пРОИжВОДНАь пРИНАДл ЕжИтс[0,1], тОf=const. с ДРУгОИ стОРОНы, ЕслИfεс[0,1], тО ДВОИЧНыИ ИНтЕгРАл, пОстРОЕННы И Дльf, НЕ пРИНАДлЕжИтс[0,1]. Уст АНОВлЕНО тАкжЕ, ЧтО сИльНАь ДВО ИЧНАь пРОИжВОДНАь И Д ВОИЧНАь пРОИжВОДНАь В тОЧкЕ с ОВпАДАУт пОЧтИ ВсУДУ Дль ФУНкцИИ, пРИ НАДлЕжАЩИх ОпРЕДЕлЕ ННОМУ пОДклАссУL ^p[0, 1]. пОлУЧЕННыЕ РЕжУльтА ты пРИМЕНьУтсь к пОЧл ЕННОМУ ДИФФЕРЕНцИРОВАНИУ И ИНтЕгРИРОВАНИУ РьДОВ пО сИстЕМЕ УОлш А, к ОцЕНкАМ ВЕлИЧИН кОЁФФИцИЕНтОВ ФУРьЕ-УОлшА, к ДОкАжАтЕльст ВУ АНАлОгА ОсНОВНОИ тЕО РЕМы О НАИлУЧшЕМ пРИБ лИжЕНИИ Дль пОлИНОМОВ пО сИстЕМЕ УОлшА, А тАкжЕ к РЕшЕНИ У ДВОИЧНОгО ВОлНОВОг О УРАВНЕНИь.     

nth order Stieltjes differential boundary operators and Stieltjes differential boundary systems

This article discusses linear differential boundary systems, which include nth-order differential boundary relations as a special case, in $\text{L}$_n^p[0,1] × $\text{L}$_n^p[0,1], 1 ? p < ∞. The adjoint relation in $\text{L}$_n^q[0,1] × $\text{L}$_n^q[0,1], $\text{1}\text{p}\text{+}\text{1}\text{q}\text{= 1}$, is derived. Green's formula is also found. Self-adjoint relations are found in $\text{L}$_n²[0,1] × $\text{L}$_n²[0,1], and their connection with Coddington's extensions of symmetric operators on subspaces of $\text{L}$_n^p[0,1] × $\text{L}$_n²[0,1] is established.     

Inverse spectral problems for Sturm-Liouville operators with singular potentials. Part III: Reconstruction by three spectra

We solve the inverse spectral problem of recovering the singular potential from W⁻¹₂(0,1) of a Sturm-Liouville operator by its spectra on the three intervals [0,1], [0,a], and [a,1] for some a∈(0,1). Necessary and sufficient conditions on the spectral data are derived, and uniqueness of the solution is analyzed.     

Dual frames in connected with generalized sampling in shift-invariant spaces

The aim of this article is to derive stable generalized sampling in a shift-invariant space by using some special dual frames in L²(0,1). These sampling formulas involve samples of filtered versions of the functions in the shift-invariant space. The involved samples are expressed as the frame coefficients of an appropriate function in L²(0,1) with respect to some particular frame in L²(0,1). Since any shift-invariant space with stable generator is the image of L²(0,1) by means of a bounded invertible operator, our generalized sampling is derived from some dual frame expansions in L²(0,1).     

On Jodeit’s multiplier extension theorems

Let Δ(x) = max {1 - ¦x¦, 0} for all x ∈ ?, and let ξ_[0,1) be the characteristic function of the interval 0 ≤x < 1. Two seminal theorems of M. Jodeit assert that A and ξ_[0,1) act as summability kernels convertingp-multipliers for Fourier series to multipliers forL ^P (?). The summability process corresponding to Δ extendsL ^P (T)-multipliers from ? to ? by linearity over the intervals [n, n + 1],n ∈ ?, when 1 ≤p < ∞, while the summability process corresponding to ξ_[0,1) extends L^P(T)-multipliers by constancy on the intervals [n, n + 1),n ∈ ?, when 1 <p < ∞. We describe how both these results have the following complete generalization: for 1 ≤p < ∞, an arbitrary compactly supported multiplier forL ^P (?) will act as a summability kernel forL ^P (T)-multipliers, transferring maximal estimates from L^P(T) to L^P(?). In particular, specialization of this maximal theorem to Jodeit’s summability kernel ξ_{[0, 1)} provides a quick structural way to recover the fact that the maximal partial sum operator on L^P(?), 1 <p < ∞, inherits strong type (p,p)-boundedness from the Carleson-Hunt Theorem for Fourier series. Another result of Jodeit treats summability kernels lacking compact support, and we show that this aspect of multiplier theory sets up a lively interplay with entire functions of exponential type and sampling methods for band limited distributions.     

Coreflective hull of finite strong L-topological spaces

This paper is concerned with the interaction between the logic features of the table of truth values and categorical properties of L-topological spaces and L-co-topological spaces. On one hand, it is shown that for each unital quantale L, the category of Alexandroff strong L-co-topological spaces is the coreflective hull of finite strong L-co-topological spaces. On the other hand, in the case that the quantale L is the unit interval [0,1] equipped with a continuous t-norm, it is shown that the category of Alexandroff strong [0,1]-topological spaces is the coreflective hull of finite strong [0,1]-topological spaces if and only if the continuous t-norm is an ordinal sum of the ?ukasiewicz t-norm whose set of idempotent elements is a well-ordered subset of [0,1] under the usual order.     

Скорость рациональн ой аппроксимации и дифференциальные с войства функций

The paper deals with the order of best rational approximation of some classes of functions, depending on their differentiability properties. Improvements and generalizations of some results by P. P. Petrushev, V. A. Popov and the author are obtained. The proofs are based on the author's direct rational approximation theorems received recently. One of the results reads as follows. LetR _n (f,L _p ) denote the value of the best approximation of a functionf inL _p ,f∈L _p [0,1], by rational fractions of degree not exceedingn, n≧1. Suppose that 0<p≦∞,s∈NU{0}, andp≠∞ fors=0. Iff is thes-th primitive of some function of bounded variation on [0,1], then $$\sum\limits_{n = 1}^\infty {\frac{1}{n}(n^{s + 1} R_n (f,L_p ))^2< \infty } $$ . This statement is exact. Namely, for everys, s∈NU {0}, and every sequence {a _n } _n=1 ^∞ , $$a_n \geqq a_{n + 1} and \sum n^{ - 1} (n^{s + 1} a_n )^2< \infty ,$$ , there exists a functiong of the classC ^s+1 [0,1] satisfying the inequalities $$R_n (g, L_p ) \geqq c(p)a_{12} , n = 1, 2, \ldots ,$$ , for everyp, p∈(0, ∞).     

Global existence and asymptotic convergence of weak solutions for the one-dimensional Navier-Stokes equations with capillarity and nonmonotonic pressure

We construct global weak solution of the Navier-Stokes equations with capillarity and nonmonotonic pressure. The volume variable v₀ is initially assumed to be in H¹ and the velocity variable u₀ to be in L² on a finite interval [0,1]. We show that both variables become smooth in positive time and that asymptotically in time u→0 strongly in L²([0,1]) and v approaches the set of stationary solutions in H¹([0,1]).     

On the structure of fractional degree vortices in a spinor Ginzburg-Landau model

We consider a Ginzburg-Landau functional for a complex vector order parameter Ψ=(ψ₊,ψ₋), whose minimizers exhibit vortices with half-integer degree. By studying the associated system of equations in R² which describes the local structure of these vortices, we show some new and unconventional properties of these vortices. In particular, one component of the solution vanishes, but the other does not. We also prove the existence and uniqueness of equivariant entire solutions, and provide a second proof of uniqueness, valid for a large class of systems with variational structure.     

Irregular wavelet frames

For the non-band-limited functionΨ, a sufficient condition is presented under which $\{ \sqrt {s_j } \psi (s_j \cdot - kb)\} $ is a frame for L²(R). The stability of these frames is studied. For the wavelets frequently used in signal processing, some concrete results are given.     

Schauder estimates for a degenerate second order elliptic operator on a cube

In the present article we are concerned with a class of degenerate second order differential operators LA,b defined on the cube d[0,1], with d?1. Under suitable assumptions on the coefficients A and b (among them the assumption of their Hölder regularity) we show that the operator LA,b defined on C²(d[0,1]) is closable and its closure is m-dissipative. In particular, its closure is the generator of a C₀-semigroup of contractions on C(d[0,1]) and C²(d[0,1]) is a core for it. The proof of such result is obtained by studying the solvability in Hölder spaces of functions of the elliptic problem λu(x)−LA,bu(x)=f(x), x∈d[0,1], for a sufficiently large class of functions f.     

On the fourier-vilenkin coefficients

In this article, we prove the following statement that is true for both unbounded and bounded Vilenkin systems: for any ε∈(0, 1), there exists a measurable set E [0, 1)of measure bigger than 1-ε such that for any function f ∈ L~1[0, 1), it is possible to find a function g ∈ L~1[0, 1) coinciding with f on E and the absolute values of non zero Fourier coefficients of g with respect to the Vilenkin system are monotonically decreasing.     

A note on a problem of Erdös and Hajnal

In [5], Erdös and Hajnal formulate the following proposition, which we shall refer to as Φ: If ? is an order-type such that $\text{|?| = ω}{}_2\text{but}\text{ω}{}_2\text{, ω}{}_2{}^1\text{? ?}$, there is ψ ? ?, |ψ| = ω₁, such that $\text{ω}{}_1\text{, ω}{}_1{}^1\text{? ψ}$. In [2], we showed that if V = L, then ?Φ. We do not know if the assumption V = L can be weakened to CH, or if, in fact, Φ is consistent with CH. However, in this note we show that, relative to a certain large cardinal assumption, Φ is consistent with 2^ω = ω₂, so that ?Φ is not provable in ZFC alone. Our proof has an interesting model-theoretic consequence, which we mention at the end.     

Analytic Semigroups on C [0,1] Generated by Some Classes of Second Order Differential Operators

C [0,1], α > 0 in (0,1) and α(1), we consider the second order differential operator on C[0,1] defined by Au: = αu″ + βu′, where D(A) may include Wentzell boundary conditions. Under integrability conditions involving √α and β/√α, we prove the analyticity of the semigroup generated by (A,D(A)) on Co[0,1], Cπ[0,1] and on C[0,1], where Co[0,1]: {u∈ C[0,1]|u (1)} and Cπ[0,1]: = {u∈ C[0,1]| u (0) = u (1)}. We also prove different characterizations of D(A) related to some results in [1], where β≡ 0, exhibiting peculiarities of Wentzell boundary conditions. Applications can be derived for the case αx = x ^k (1 - x )^kγ(x )(k≥j/2, x∈ [0,1], γ∈ C{0,1}).     

Spectra of Bochner-Riesz Means on Lp

In this article, the Bochner-Riesz means are shown to have either the unit interval [0,1] or the whole complex plane as their spectra on L^p,1≤p<∞.     

Generalized monotone approximation inL p Space

Letf(x) ∈L _p[0,1], 1?p? ∞. We shall say that functionf(x)∈Δ^k (integerk?1) if for anyh ∈ [0, 1/k] andx ∈ [0,1?kh], we have Δ _h ^k f(x)?0. Denote by ∏ _n the space of algebraic polynomials of degree not exceedingn and define $$E_{n,k} (f)_p : = \mathop {\inf }\limits_{\mathop {P_n \in \prod _n }\limits_{P_n^{(\lambda )} \geqslant 0} } \parallel f(x) - P_n (x)\parallel _{L_p [0,1]} .$$ We prove that for any positive integerk, iff(x) ∈ Δ ^k ∩ L _p[0, 1], 1?p?∞, then we have $$E_{n,k} (f)_p \leqslant C\omega _2 \left( {f,\frac{1}{n}} \right)_p ,$$ whereC is a constant only depending onk.     

Operator-valued Fourier Haar multipliers

Criteria are given to ensure the boundedness of Fourier Haar multiplier operators from Lp([0,1],X) to Lq([0,1],Y) where the Fourier Haar multiplier sequences come not from R, as in the classical setting, but rather from the space of bounded linear operators from a Banach space X into a Banach space Y.     

Shrinking target problems for beta-dynamical system

For any β>1,let([0,1],Tβ) be the beta dynamical system.For a positive function ψ:N→R+ and a real number x0 ∈[0,1],we define D(Tβ,ψ,x0) the set of ψ-well approximable points by x0as {x∈[0,1]:|Tβnx-x0|<ψ(n) for infinitely many n∈N}.In this note,by proving a structure lemma that any ball B(x,r) contains a regular cylinder of comparable length with r,we determine the Hausdorff dimension of the set D(Tβ,ψ,x0) completely for any β>1 and any positive function ψ.     

Besov regularity for the generalized local time of the indefinite Skorohod integral

Let (t∈[0,1]) be the indefinite Skorohod integral on the canonical probability space (Ω,F,P), and let Lt(x) (t∈[0,1], x∈R) be its the generalized local time introduced by Tudor in [C.A. Tudor, Martingale-type stochastic calculus for anticipating integral processes, Bernoulli 10 (2004) 313-325]. We prove that the generalized local time, as function of x, has the same Besov regularity as the Brownian motion, as function of t, under some conditions imposed on the anticipating integrand u.     

On the L2n 1– Extension Properties

We have proved that for all compact linear operator u from R into an L_p ([0,1], ν) (0 < p < 1) extends to L ₁ ([0,1], ν), where R denotes the closed linear subspace in L ₁ ([0,1], ν) of the Rademacher functions {r_n }_{n ? N}. In this paper, we study this type of extension for E_n ? L_2n ¹ where E_n is the n–dimensional subspace which appears in Kasin's theorem such that L_2n ¹ = E_n ⊕ E ^⊥ _n and the L_2n ¹ , L_2n ² norms are universally equivalent on both E_n , E ^⊥ _n. We show that, the precedent extension fails for the pair (E_n , L_2n ¹ ) and we generalize this to any E in an L ₁(Ω, A, P) by giving some conditions on E.