Uniqueness of uniform norm and C*-norm in L p (G, ω)

Let G be a locally compact abelian group with a fixed Haar measure and ω be a weight on G. For 1 < p < ∞, we study uniqueness of uniform and C*-norm properties of the invariant weighted algebra L ^p (G, ω).     

On uniqueness sets for multiple Walsh series

We study uniqueness sets for multiple Walsh series under ρ-regular (or bounded) convergence in rectangles. We prove that a countable set is a uniqueness set for such a series under this convergence. We construct a class of perfect uniqueness sets for multiple Walsh series under this convergence. We show that the notion of index of a perfect set does not solve the problem of whether this set belongs to the class of uniqueness sets. We note that the results of this paper remain valid for several rearranged multiple Walsh series.     

Contractive projections onl 1

Two characterizations of ranges of contractive (nonlinear) projections on a finite or infinite dimensionall ¹ space are given. One is related to best co-approximation, the other deals with intersections of sets having a simple structure. Concerning the latter the notion of a cylinder set is introduced which serves to replace a certain type of closed half-spaces that are known to be used in characterizing contractive retracts ofl ^p spaces, 1<p<∞,p≠2.     

Generalized Absolute Convergence of Series of Fourier Coefficients With Respect to Haar Type Systems

For the orthogonal systems ofHaar type, introduced by Vilenkin in 1958, we study absolute convergence of series composed from positive powers of Fourier coefficients with multiplicators from the Gogoladze–Meskhia class. The conditions for convergence of the series are given in terms of either best approximations of functions in L ^p spaces by polynomials with respect to Haar type systems or fractional modulus of continuity for functions from the Wiener spaces V _p , p > 1. We establish the sharpness of the obtained results.     

Vector-valued walsh-paley martingales and geometry of banach spaces

The concept of Rademacher typep (1≤p≤2) plays an important role in the local theory of Banach spaces. In [3] Mascioni considers a weakening of this concept and shows that for a Banach spaceX weak Rademacher typep implies Rademacher typer for allr<p. As with Rademacher typep and weak Rademacher typep, we introduce the concept of Haar typep and weak Haar typep by replacing the Rademacher functions by the Haar functions in the respective definitions. We show that weak Haar typep implies Haar typer for allr<p. This solves a problem left open by Pisier [5]. The method is to compare Haar type ideal norms related to different index sets.     

Fractal Haar system

The Haar system is an alternative to the classical Fourier bases, being particularly useful for the approximation of discontinuities. The article tackles the construction of a set of fractal functions close to the Haar set. The new system holds the property of constitution of bases of the Lebesgue spaces of p-integrable functions on compact intervals. Likewise, the associated fractal series of a continuous function is uniformly convergent. The case p=2 owns some peculiarities and is studied separately.     

Hörmander multipliers on two-dimensional dyadic Hardy spaces

In this paper we are interested in conditions on the coefficients of a two-dimensional Walsh multiplier operator that imply the operator is bounded on certain of the Hardy type spaces Hp, 0<p<∞. We consider the classical coefficient conditions, the Marcinkiewicz-Hörmander-Mihlin conditions. They are known to be sufficient for the trigonometric system in the one and two-dimensional cases for the spaces Lp, 1<p<∞. This can be found in the original papers of Marcinkiewicz [J. Marcinkiewicz, Sur les multiplicateurs des series de Fourier, Studia Math. 8 (1939) 78-91], Hörmander [L. Hörmander, Estimates for translation invariant operators in Lp spaces, Acta Math. 104 (1960) 93-140], and Mihlin [S.G. Mihlin, On the multipliers of Fourier integrals, Dokl. Akad. Nauk SSSR 109 (1956) 701-703; S.G. Mihlin, Multidimensional Singular Integrals and Integral Equations, Pergamon Press, 1965]. In this paper we extend these results to the two-dimensional dyadic Hardy spaces.     

Uniqueness sets for multiple Haar series

Criteria are found for the membership of multidimensional sets in the class of M-sets for multiple Haar series with various conditions imposed on the coefficients. Several generalizations of the uniqueness theorem are established.Translated from Matematicheskie Zametki, Vol. 14, No. 6, pp. 789–798, December, 1973.     

Greedy bases in L p spaces

We consider a weighted L ^p space L ^p (w) with a weight function w. It is known that the Haar system H _p normalized in L ^p is a greedy basis of L ^p , 1 < p < ∞. We study a question of when the Haar system H _p ^w normalized in L ^p (w) is a greedy basis of L ^p (w), 1 < p < ∞. We prove that if w is such that H _p ^w is a Schauder basis of L ^p (w), then H _p ^w is also a greedy basis of L ^p (w), 1 < p < ∞. Moreover, we prove that a subsystem of the Haar system obtained by discarding finitely many elements from it is a Schauder basis in a weighted norm space L ^p (w); then it is a greedy basis.     

On the uniqueness of generalized monosplines of least Lp-norm

The uniqueness of monosplines and perfect splines of leastL _p-norm is treated in the framework of generalized monosplines and total positivity. The analysis is based on the invariance properties of the degree of a certain mapping and on a new composition result for totally positive kernels. For theL _p-case 1<p<∞, uniqueness is shown under the same extra conditions as were previously shown to be needed in theL _p-case. The uniqueness in theL _∞-case is obtained without any restrictions.     

Abstract time-dependent transport equations

We consider the abstract time-dependent linear transport equation as an initial-boundary value evolution problem in the Banach spaces L_p, 1 ⩽ p < ∞, or on a space of measures on a (possibly time-dependent) kinetic phase space. Existence, uniqueness, dissipativity, and positivity results are proved for very general, possibly time-dependent, transport operators and boundary conditions. When the phase space, boundary conditions, and transport operator are independent of time, corresponding results are obtained for the associated semigroup.     

The smallest enclosing ball problem and the smallest intersecting ball problem: existence and uniqueness of solutions

In this paper we study the following problems and their further generalizations: given a finite number of nonempty closed subsets of a normed space, find a ball with the smallest radius that encloses all of the sets, and find a ball with the smallest radius that intersects all of the sets. These problems can be viewed as generalized versions of the smallest enclosing circle problem introduced in the nineteenth century by Sylvester which asks for the circle of smallest radius enclosing a given set of finite points in the plane. We will focus on sufficient conditions for the existence and uniqueness of an optimal solution for each problem, while the study of optimality conditions and numerical implementation will be addressed in our next projects.     

A wavelet theory for local fields and related groups

Let G be a locally compact abelian group with compact open subgroup H. The best known example of such a group is G = ℚ_p, the field of padic rational numbers (as a group under addition), which has compact open subgroup H = ℤ_p, the ring of padic integers. Classical wavelet theories, which require a non trivial discrete subgroup for translations, do not apply to G, which may not have such a subgroup. A wavelet theory is developed on G using coset representatives of the discrete quotient Ĝ/H^⊥ to circumvent this limitation. Wavelet bases are constructed by means of an iterative method giving rise to socalled wavelet sets in the dual group Ĝ. Although the Haar and Shannon wavelets are naturally antipodal in the Euclidean setting, it is observed that their analogues for G are equivalent.     

Sufficient conditions for the Lebesgue integrability of Fourier transforms

We give sufficient conditions for the Lebesgue integrability of the Fourier transform of a function f ∈ L ^p (?) for some 1 < p ≤ 2. These sufficient conditions are in terms of the L ^p integral modulus of continuity of f; in particular, they apply for functions in the integral Lipschitz class Lip(α, p) and for functions of bounded s-variation for some 0 < s < p. Our theorems are nonperiodic versions of the classical theorems of Bernstein, Szász, Zygmund and Salem, and recent theorems of Gogoladze and Meskhia on the absolute convergence of Fourier series.     

On the spectral representation of symmetric stable processes

The so-called spectral representation theorem for stable processes linearly imbeds each symmetric stable process of index p into L^p (0 < p ≤ 2). We use the theory of L^p isometries for 0 < p < 2 to study the uniqueness of this representation for the non-Gaussian stable processes. We also determine the form of this representation for stationary processes and for substable processes. Complex stable processes are defined, and a complex version of the spectral representation theorem is proved. As a corollary to the complex theory we exhibit an imbedding of complex L^q into real or complex L^p for 0 < p < q ≤ 2.     

p-Adic Multiresolution Analysis and Wavelet Frames

We study p-adic multiresolution analyses (MRAs). A complete characterization of test functions generating an MRA (scaling functions) is given. We prove that only 1-periodic test functions may be taken as orthogonal scaling functions and that all such scaling functions generate the Haar MRA. We also suggest a method for constructing sets of wavelet functions and prove that any set of wavelet functions generates a p-adic wavelet frame.     

Some results concerning Lp(μ)-spaces

It is proved that L^p(μ) does not have an unconditional basis if the cardinality of L^p(μ) is sufficiently large and μ is a finite measure. It is also shown that L^p(μ) has a weaker kind of basis for arbitrary μ and 1 < p < ∞. A new truncation lemma concerning sequences in L^p equivalent to the usual l^p-basis is given. This lemma is used in solving the problem of when l^p(Γ) imbeds in L^r(μ) for uncountable sets Γ and finite measures μ. It may also be used to give a nonprobabilistic proof of the fact (due to Schwartz-Kwapien) that there exist non-q-absolutely summing operators from L^∞to L^qfor 2 < q < ∞. It is again used in proving that basic sequences in L^p equivalent to the usual l^p-basis admit subsequences with a complemented linear span. Other applications of the techniques introduced are also given.     

Local sidon sets and uniform convergence of fourier series

For compact groups several necessary and sufficient conditions for a set to be local Sidon are given; these conditions are expressed in functional-analytic terms. An immediate corollary is the existence of a large class of compact groupsG, including all the connected non-Abelian Lie groups, which support functions not inA(G) but having Fourier series that are both unformly and absolutely convergent. This has been shown for the special case ofSU(2) by Mayer. Also several necessary and sufficient conditions for a set to be local Λ (p) are given.     

Nonlinear semigroup approach to transport equations with delayed neutrons

This paper deal with a nonlinear transport equation with delayed neutron and general boundary conditions. We establish, via the nonlinear semigroups approach, the existence and uniqueness of the mild solution, weak solution, strong solution and local solution on L~p-spaces(1 ≤ p +∞). Local and non local evolution problems are discussed.     

Finite prime-field characteristic sets for planar configurations

Examples are constructed of planar matroids with finite prime-field characteristic sets (i.e. matroids representable over a finite set of prime fields but over fields of no other characteristic). In particular, for any n>3, a projectively unique integer matrix is constructed with 2lsqblog₂nrsqb+6 columns which often gives nonsingleton characteristic sets and, when n is prime, has characteristic set {n}. Many finite subsets of primes are shown to be characteristic sets, including {23,59} (the smallest pair found using these methods), all pairs of primes {p, p′:67?p<p′?293}, and the seventeen largest five-digit primes. Probabilistic arguments are presented to support the conjecture that prime-field characteristic sets exist of every finite cardinality. For p>3, AG(2,p) is shown to be a subset of PG(2, q) only for q=p^s. Another general construction technique suggests that when $\text{P}\text{={p}{}_1\text{,…,p}{}_{\mathrm{k}}\text{}}$ are the primitive prime divisors of 2ⁿ±1 (n sufficiently large), then there is a matroid with O (log n) points whose characteristic set is $\text{P}$. We remark that although only one finite nonsingleton characteristic set (due to R. Reid) was known prior to this paper, a new technique by J. Kahn has shown that every finite set of primes forms a (non-prime-field) characteristic set.