Dyadic Space Research Articles

Coefficient Condition forL1-Convergence of Walsh–Fourier Series

In this paper we give a condition with respect to Walsh–Fourier coefficients that implies theL1-convergence of the corresponding Walsh–Fourier series. We show that theL1-convergence class induced by this condition contains each one of the previously known convergence classes as a proper subset. We also show that our condition implies not only theL1-convergence but also the convergence in the dyadic Hardy norm if the function represented by the series belongs to the dyadic Hardy space.

Dyadic scale space

In this paper, we first approximate the Gaussian function with any scale by the linear finite combination of Gaussian functions with dyadic scale; consequently, the scale space can be constructed much more efficiently: we only perform smoothing at these dyadic scales and the smoothed signals at other scales can be found by calculating linear combinations of these discrete scale signals. We show that the approximation error is so small that our approach can be used in most of the computer vision fields. We analyse the behavior of zero-crossing (ZC) across scales and show that features at any scale can be found efficiently by tracking from the dyadic scales, thus we show that the new representation is necessary and complete. In the case that the derivatives are calculated by a special multiscale filter, we show that all the derivative signals can be treated in the same way.

Some maximal lnequalities with respect to two-parameter dyadic derivative and cesàro summability

We consider two-dimensional convolution operators with general kernel functions and give a sufficient condition for the two-parameter maximal operator to be bounded from the dyadic martingale Hardy space Hp to Lp. Especially, the boundedness of the maximal operator of the twoparameter dyadic derivative of the dyadic integral function and the maximal operator of the two-parameter Cesaro means are verified. As a consequence we obtain that every function f ∊ L log+ L[O, 1)2 is Cesho summable and the dyadic integral of it is dyadically differentiable.

Some remarks on countably determined measures and uniform distribution of sequences

Two topics are investigated: countably determined (regular Borel probability) measures on compact Hausdorff spaces, and uniform distribution of sequences regarding mainly this kind of measures. We prove several characterizations of countably determined measures, and apply the results in order to show the existence of a well distributed sequence in the support of a countably determined measure. We also generalize a result of Losert on the existence of uniformly distributed sequences in compact dyadic spaces.

An Inverse Sidon Type Inequality for the Walsh System

Upper estimations for the integral norm of the linear combinations of Dirichlet kernels expressed only in terms of the coefficients are called Sidon type inequalities. They can be applied in various fields of harmonic analysis. Among them are integrability and L1 convergence problems and summation methods. Several improvements of Sidon′s original inequality have been proved, most of them invariant with respect to the rearrangements of the coefficients, but no nontrivial lower estimation has been given so far. In this paper we give the best rearrangement invariant expression, which corresponds to an Orlicz norm, of coefficients for a Sidon type inequality. Examples for applications for convergence in L1 and in the dyadic Hardy space are also given. These questions are considered for the Walsh system.

Special Points in Compact Spaces

Given a collection $\mathcal {C}$, of cardinality $\kappa$, of subsets of a compact space X, we prove the existence of a point x such that whenever $C \in \mathcal {C}$ and $X \in \bar C$, there exists a ${G_\lambda }$-set Z with $\lambda < \kappa$ and $x \in Z \subset \bar C$. We investigate the case when $\mathcal {C}$ is the collection of all cozerosets of X and also when X is a dyadic space. We apply this result to homogeneous compact spaces. Another application is a characterization of ${2^{{\omega _1}}}$ among dyadic spaces.

On almost Dugundji spaces and dyadic spaces

On almost Dugundji spaces and dyadic spaces

Special points in compact spaces

Given a collection C \mathcal {C} , of cardinality κ \kappa , of subsets of a compact space X, we prove the existence of a point x such that whenever C ∈ C C \in \mathcal {C} and X ∈ C ¯ X \in \bar C , there exists a G λ {G_\lambda } -set Z with λ &gt; κ \lambda &gt; \kappa and x ∈ Z ⊂ C ¯ x \in Z \subset \bar C . We investigate the case when C \mathcal {C} is the collection of all cozerosets of X and also when X is a dyadic space. We apply this result to homogeneous compact spaces. Another application is a characterization of 2 ω 1 {2^{{\omega _1}}} among dyadic spaces.

Rigid zerodimensional dyadic spaces

For each uncountable cardinal κ, it is shown that there are 2 κ pairwise nonhomeomorphic, rigid, zerodimensional or connected, dyadic spaces of weight κ.

Interpolation between dyadic Hardy spaces H p : The complex method

Interpolation between dyadic Hardy spaces H p : The complex method

On equivalence of rearrangements of the Haar system in dyadic Hardy and BMO spaces

ПустьH={h1,I∈ ℊ} — систе ма Хаара, занумерован ная множеством двоичных интервалов ℊ. Биекцияπ: ℊ→ℊ назыв ается сохраняющей ме ру, если для любогоI∈ℊ выполн ено равенство ¦π(I)¦=¦I¦, гд е ¦I¦ — мераI. Изучаетс я эквивалентность сис темH иH π ={hπ(I),I∈ℊ} в двоичных про странствах Харди и ВМ О в случае сохраняющих м еру биекцийπ. В частности, получе но явное представлен ие для нормы оператора, соот ветствующего перестановкеπ. Отсюд а получается один рез ультат Семенова об эквивале нтности в пространст вахL p .

Dyadic martingale Hardy and VMO spaces on the plane

Dyadic martingale Hardy and VMO spaces on the plane

A correction theorem and the dyadic space H1, ∞

It is shown that for every l∞-function f and for every ɛ, ɛ>0, there exists a function g such that mes {t=g} <ɛ, while the partial sums of the Fourier and Fourier-Walsh series of the function g are uniformly bounded by the number C log (e−1)∥f∥∞. In the proof we make use of the characterization of the dyadic space H1, ∞ in terms of atomic decompositions (it is, apparently, new).

Generalized dyadic spaces

Generalized dyadic spaces

Communicative Interaction: Mother Modification and Child Acquisition of American Sign Language

The communicative interaction in American Sign Language (ASL) of two deaf mothers with their deaf children was studied at 3-week intervals for 10 months to find what modification, if any, the mothers made in their language utterances addressed to the children (12–20 and 20–30 months old). As was hypothesized, and has been shown of hearing-speaking mothers’ language, modification in the direction of simplified and more linear language was found. Special attention was paid to POINTing behavior (i.e. pointing gestures constrained by the linguistic rules of ASL) and to verb “modulation” or inflection (changes from ASL citation forms to mark the sentential arguments of verbs). Phonological, semantic, syntactic, and pragmatic analyses were made of the data from these two mother-child dyads. The language of the children was found to be like that reported in previous studies of ASL acquisition. These show verb modulation to be absent from the earlier stages, making a first appearance at about two and one half years. Lacking productive inflection of verb signs to mark semantic or grammatical roles in an utterance, the two children in this study, like those in previous studies, were found to combine deictic POINTs with lexical signs. Nearly 100% of the signs in the data base from the younger child in early sessions (ca. 12 months) are POINTs, but only 66 % in the data base from later sessions with the older child (ca. 30 months). This study is the first to examine mothers’ language in conversations with their children for the use of the ASL verb inflecting system called Verb Modulation by Indexic Reference. In their conversations with their young deaf children the mothers do not use modulated ASL verbs. Most of their utterances contain lexical (i.e. citation) form signs and deictic POINTs. In the rare cases when modulation does occur, it is always in the presence of the verb-indicated referent, be it person, location, or object. There is only a single case of Indexic Reference in 2,035 of the mothers’ utterances; i.e. use of a point in signing space to establish an “Index” for a non-present noun referent. The mothers in the present study do not use the rich modulation system of adult ASL with their young deaf children but instead offer a model of ASL that separates out these highly analytic units into their simple components. Furthermore, they use several strategies to ensure comprehension during the course of development from these simpler forms of reference to the complex system of referencing non-present objects, locations, and persons by Indexic Reference and Verb Modulation. The strategies listed here are discussed more fully below: 1. Mothers bring the object they want to talk about (the referent object) directly into the dyadic conversational space; they show it. This strategy is the most explicit and does not even require the child to follow a pointing gesture to its target; it was used in an average of 20% of all the utterances of the mother of the younger child but in only 3 % of the utterances of the other. 2. Mothers use POINTing extensively, both as a phonological replacement for other handshapes in lexical items and to mark semantic/grammatical roles in an utterance; POINTing accounts for about 50% of all signs in the mothers’ corpus. 3. Mothers do not modulate verbs in their early conversations with their children. When modulation does occur, it is in the form of incorporating present objects, persons, and locations. As use of modulation increases so does the use of the redundant POINTing to mark grammatical/semantic roles. Locational verbs are the first to be modulated (moved toward or performed at the position of arbitrary or real Indexes).

On the Existence of Uniformly Distributed Sequences in Compact Topological Spaces. I

We prove the existence of uniformly distributed sequences for an arbitrary probability measure on a separable dyadic space, e.g. on a separable compact topological group. Some counterexamples for the nonexistence of u.d. sequences in certain dense subsets are given.

On the existence of uniformly distributed sequences in compact topological spaces. I

We prove the existence of uniformly distributed sequences for an arbitrary probability measure on a separable dyadic space, e.g. on a separable compact topological group. Some counterexamples for the nonexistence of u.d. sequences in certain dense subsets are given.

Sequential Continuity in Dyadic Spaces

A characterization of metrizability in dyadic spaces in terms of sequential continuity is given. Also, it is shown by examples that invariance under projections (composition) cannot be eliminated from the hypotheses of theorems of Mazur and Engelking concerning the continuity of sequentially continuous functions.

MAPPINGS AND IMBEDDINGS OF DYADIC SPACES

We prove that a dyadic space of weight contains the Cantor cube if and only if it cannot be represented as a countable union of closed subsets with weights less than . A similar result has been independently obtained by Gerlits. That solves a problem posed by Pelczynski. In the particular case when the dyadic space is, in addition, a Dugundji space, the problem has been recently solved by Haydon. Further, it follows that any dyadic space whose weight is not a sum of countably many smaller cardinals can be continuously mapped onto the Tihonov cube and contains the Cantor cube . This is true, in particular, when is a regular cardinal, as was proved by Hagler. By means of the methods developed in this paper we prove that the of a dyadic space is equal to its cardinality and is attained; this is a final solution of Arhangel'skii's problem about the depth of dyadic spaces.Bibliography: 19 titles.

The Baire set fixing property and uniform continuity in topological groups

Every topological group that satisfies theM-Baire set fixing property and is left separable isdense in a retract of a product of metrizable groups. Since compact groups satisfy MBSFP, it is a corollary that compact groups are dyadic spaces (Kuz'minov). IfG satisfies MBSFP, then the left uniformity is separable if and only ifG satisfies the countable chain condition. Dense subgroups of left separable MBSFP groups satisfy MBSFP, and any product of left separable MBSFP groups satisfies MBSFP. A left separable group satisfies MBSFP if and only if the left uniformity is weak generated by natural maps to metrizable quotient groups. If for each α,Aα is a denseC-embedded set in the left separable MBSFP groupGα, then the real-compactification νΠAα = ΠνAα. It is a corollary that the product of pseudocompact groups is pseudocompact (Comfort and Ross). A locally compact group satisfies MBSFP if and only if it is metrizable or sigma compact (Ross and Stromberg, forG Abelian).