Wavelet approximation and Fourier widths of classes of periodic functions of several variables. I

We obtain characterizations (and prove the corresponding equivalence of norms) of function spaces B _pq ^sm ($ \mathbb{I} $ \mathbb{I} ^k ) and L _pq ^sm ($ \mathbb{I} $ \mathbb{I} ^k ) of Nikol’skii-Besov and Lizorkin-Triebel types, respectively, in terms of representations of functions in these spaces by Fourier series with respect to a multiple system $ \mathcal{W}_m^\mathbb{I} $ \mathcal{W}_m^\mathbb{I} of Meyer wavelets and in terms of sequences of the Fourier coefficients with respect to this system. We establish order-sharp estimates for the approximation of functions in B _pq ^sm ($ \mathbb{I} $ \mathbb{I} ) and L _pq ^sm ($ \mathbb{I} $ \mathbb{I} ^k ) by special partial sums of these series in the metric of L _r ($ \mathbb{I} $ \mathbb{I} ^k ) for a number of relations between the parameters s, p, q, r, and m (s = (s ₁, ..., s _n ) ∈ ℝ₊ ⁿ , 1 ≤ p, q, r ≤ ∞, m = (m ₁, ..., m _n ) ∈ ℕ ⁿ , k = m ₁ +... + m _n , and $ \mathbb{I} $ \mathbb{I} = ℝ or $ \mathbb{T} $ \mathbb{T} ). In the periodic case, we study the Fourier widths of these function classes.     

Linear maps that strongly preserve regular matrices over the Boolean algebra

The set of all m × n Boolean matrices is denoted by $ \mathbb{M} $ \mathbb{M} _m,n . We call a matrix A ∈ $ \mathbb{M} $ \mathbb{M} _m,n regular if there is a matrix G ∈ $ \mathbb{M} $ \mathbb{M} _n,m such that AGA = A. In this paper, we study the problem of characterizing linear operators on $ \mathbb{M} $ \mathbb{M} _m,n that strongly preserve regular matrices. Consequently, we obtain that if min{m, n} ⩽ 2, then all operators on $ \mathbb{M} $ \mathbb{M} _m,n strongly preserve regular matrices, and if min{m, n} ⩾ 3, then an operator T on $ \mathbb{M} $ \mathbb{M} _m,n strongly preserves regular matrices if and only if there are invertible matrices U and V such that T(X) = UXV for all X ε $ \mathbb{M} $ \mathbb{M} _m,n , or m = n and T(X) = UX ^T V for all X ∈ $ \mathbb{M} $ \mathbb{M} _n .     

Strict topology over the space of measures with continuous translations on a locally compact foundation semigroup

Let $ \mathfrak{S} $ \mathfrak{S} be a locally compact semigroup, ω be a weight function on $ \mathfrak{S} $ \mathfrak{S} , and M _a ($ \mathfrak{S} $ \mathfrak{S} , ω) be the weighted semigroup algebra of $ \mathfrak{S} $ \mathfrak{S} . Let L ₀^∞ ($ \mathfrak{S} $ \mathfrak{S} ; M _a ($ \mathfrak{S} $ \mathfrak{S} , ω)) be the C*-algebra of all M _a ($ \mathfrak{S} $ \mathfrak{S} , ω)-measurable functions g on $ \mathfrak{S} $ \mathfrak{S} such that g/ω vanishes at infinity. We introduce and study a strict topology β ¹($ \mathfrak{S} $ \mathfrak{S} , ω) on M _a ($ \mathfrak{S} $ \mathfrak{S} , ω) and show that the Banach space L ₀^∞ ($ \mathfrak{S} $ \mathfrak{S} ; M _a ($ \mathfrak{S} $ \mathfrak{S} , ω)) can be identified with the dual of M _a ($ \mathfrak{S} $ \mathfrak{S} , ω) endowed with β ¹($ \mathfrak{S} $ \mathfrak{S} , ω). We finally investigate some properties of the locally convex topology β ¹($ \mathfrak{S} $ \mathfrak{S} , ω) on M _a ($ \mathfrak{S} $ \mathfrak{S} , ω).     

Trace class Toeplitz operators with unbounded symbols on weighted Bergman spaces

In this note we construct a function φ in L2（Bn,dμ） which is unbounded on any neighborhood of each boundary point of Bn such that Tφ is a trace class operator on weighted Bergman space Lα2（Bn,dμ） for several complex variables.     

Integral-type operators from a mixed norm space to a bloch-type space on the unit ball

Let $ \mathbb{B} $ \mathbb{B} be the unit ball in ℂ ⁿ and let H($ \mathbb{B} $ \mathbb{B} ) be the space of all holomorphic functions on $ \mathbb{B} $ \mathbb{B} . We introduce the following integral-type operator on H($ \mathbb{B} $ \mathbb{B} ):

Spaces with fibered approximation property in dimension <Emphasis Type="Italic">n</Emphasis>

A metric space M is said to have the fibered approximation property in dimension n (briefly, M ∈ FAP(n)) if for any ɛ > 0, m ≥ 0 and any map g: $ \mathbb{I} $ \mathbb{I} ^m × $ \mathbb{I} $ \mathbb{I} ⁿ → M there exists a map g′: $ \mathbb{I} $ \mathbb{I} ^m × $ \mathbb{I} $ \mathbb{I} ⁿ → M such that g′ is ɛ-homotopic to g and dim g′ ({z} × $ \mathbb{I} $ \mathbb{I} ⁿ ) ≤ n for all z ∈ $ \mathbb{I} $ \mathbb{I} ^m . The class of spaces having the FAP(n)-property is investigated in this paper. The main theorems are applied to obtain generalizations of some results due to Uspenskij [11] and Tuncali-Valov [10].     

On a semilattice of numberings

We study some properties of a $ \mathfrak{c} $ \mathfrak{c} -universal semilattice $ \mathfrak{A} $ \mathfrak{A} with the cardinality of the continuum, i.e., of an upper semilattice of m-degrees. In particular, it is shown that the quotient semilattice of such a semilattice modulo any countable ideal will be also $ \mathfrak{c} $ \mathfrak{c} -universal. In addition, there exists an isomorphism $ \mathfrak{A} $ \mathfrak{A} such that $ {\mathfrak{A} \mathord{\left/ {\vphantom {\mathfrak{A} {\iota \left( \mathfrak{A} \right)}}} \right. \kern-\nulldelimiterspace} {\iota \left( \mathfrak{A} \right)}} $ {\mathfrak{A} \mathord{\left/ {\vphantom {\mathfrak{A} {\iota \left( \mathfrak{A} \right)}}} \right. \kern-\nulldelimiterspace} {\iota \left( \mathfrak{A} \right)}} will be also $ \mathfrak{c} $ \mathfrak{c} -universal. Furthermore, a property of the group of its automorphisms is obtained. To study properties of this semilattice, the technique and methods of admissible sets are used. More exactly, it is shown that the semilattice of mΣ-degrees $ L_{m\Sigma }^{\mathbb{H}\mathbb{F}\left( S \right)} $ L_{m\Sigma }^{\mathbb{H}\mathbb{F}\left( S \right)} on the hereditarily finite superstructure $ \mathbb{H}\mathbb{F} $ \mathbb{H}\mathbb{F} (S) over a countable set S will be a $ \mathfrak{c} $ \mathfrak{c} -universal semilattice with the cardinality of the continuum.     

On additive properties of product sets in an arbitrary finite field

It is proved that for any two subsets A and B of an arbitrary finite field $ \mathbb{F}_q $ \mathbb{F}_q such that |A||B| > q, the identity 10AB = $ \mathbb{F}_q $ \mathbb{F}_q holds. Under the assumption |A||B| ⩾2q, this improves to 8AB = $ \mathbb{F}_q $ \mathbb{F}_q .     

The best extension of algebraic polynomials from the unit circle

We study the class $ \mathfrak{P}_n $ \mathfrak{P}_n of algebraic polynomials P _n (x, y) in two variables of total degree n whose uniform norm on the unit circle Γ₁ centered at the origin is at most 1: $ \left\| {P_n } \right\|_{C(\Gamma _1 )} $ \left\| {P_n } \right\|_{C(\Gamma _1 )} ≤ 1. The extension of polynomials from the class $ \mathfrak{P}_n $ \mathfrak{P}_n to the plane with the least uniform norm on the concentric circle Γ _r of radius r is investigated. It is proved that the values θ _n (r) of the best extension of the class $ \mathfrak{P}_n $ \mathfrak{P}_n satisfy the equalities θ _n (r) = r ⁿ for r > 1 and θ _n (r) = r ⁿ⁻¹ for 0 < r < 1.     

Local limit theorem for the first crossing time of a fixed level by a random walk

Let X,X(1),X(2),... be independent identically distributed random variables with mean zero and a finite variance. Put S(n) = X(1) + ... + X(n), n = 1, 2,..., and define the Markov stopping time η _y = inf {n ≥ 1: S(n) ≥ y} of the first crossing a level y ≥ 0 by the random walk S(n), n = 1, 2,.... In the case $ \mathbb{E} $ \mathbb{E} |X|³ < ∞, the following relation was obtained in [8]: $ \mathbb{P}\left( {\eta _0 = n} \right) = \frac{1} {{n\sqrt n }}\left( {R + \nu _n + o\left( 1 \right)} \right) $ \mathbb{P}\left( {\eta _0 = n} \right) = \frac{1} {{n\sqrt n }}\left( {R + \nu _n + o\left( 1 \right)} \right) as n → ∞, where the constant R and the bounded sequence ν _n were calculated in an explicit form. Moreover, there were obtained necessary and sufficient conditions for the limit existence $ H\left( y \right): = \mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _y = n} \right) $ H\left( y \right): = \mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _y = n} \right) for every fixed y ≥ 0, and there was found a representation for H(y). The present paper was motivated by the following reason. In [8], the authors unfortunately did not cite papers [1, 5] where the above-mentioned relations were obtained under weaker restrictions. Namely, it was proved in [5] the existence of the limit $ \mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _y = n} \right) $ \mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _y = n} \right) for every fixed y ≥ 0 under the condition $ \mathbb{E} $ \mathbb{E} X ² < ∞ only; In [1], an explicit form of the limit $ \mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _0 = n} \right) $ \mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _0 = n} \right) was found under the same condition $ \mathbb{E} $ \mathbb{E} X ² < ∞ in the case when the summand X has an arithmetic distribution. In the present paper, we prove that the main assertion in [5] fails and we correct the original proof. It worth noting that this corrected version was formulated in [8] as a conjecture.     

The Hasse-Witt invariant in some towers of function fields over finite fields

In this article we investigate the p-rank of function fields in several good towers. To do this we first recall and establish some properties of the behaviour of the p-rank under extensions. Then we compute the p-ranks of function fields in several optimal towers over a quadratic field $ \mathbb{F}_{q^2 } $ \mathbb{F}_{q^2 } , as well as for a specific good tower over a cubic field $ \mathbb{F}_{q^3 } $ \mathbb{F}_{q^3 } , which was introduced by Bassa, Garcia and Stichtenoth.     

Σ-definability of uncountable models of <Emphasis Type="Italic">c</Emphasis>-simple theories

We show that each c-simple theory with an additional discreteness condition has an uncountable model Σ-definable in ℍ$ \mathbb{H} $ \mathbb{H} ($ \mathbb{L} $ \mathbb{L} ), where $ \mathbb{L} $ \mathbb{L} is a dense linear order. From this we establish the same for all c-simple theories of finite signature that are submodel complete.     

Explicit sum-product theorems for large subsets of $$ \mathbb{F} $$p

In this note, we use ‘classical’ methods to obtain sum-product theorems for subsets A⊂$ \mathbb{F} $ \mathbb{F} _p .     

On equivalences of derived and singular categories

Let X and Y be two smooth Deligne-Mumford stacks and consider a pair of functions f: X → $ \mathbb{A}^1 $ \mathbb{A}^1 , g:Y → $ \mathbb{A}^1 $ \mathbb{A}^1 . Assuming that there exists a complex of sheaves on X × $ \mathbb{A}^1 $ \mathbb{A}^1 Y which induces an equivalence of D ^b (X) and D ^b (Y), we show that there is also an equivalence of the singular derived categories of the fibers f ⁻¹(0) and g ⁻¹(0). We apply this statement in the setting of McKay correspondence, and generalize a theorem of Orlov on the derived category of a Calabi-Yau hypersurface in a weighted projective space, to products of Calabi-Yau hypersurfaces in simplicial toric varieties with nef anticanonical class.     

On $$ \mathfrak{F}_h $$-normal subgroups of finite groups

Let G be a finite group, and let $ \mathfrak{F} $ \mathfrak{F} be a formation of finite groups. We say that a subgroup H of G is $ \mathfrak{F}_h $ \mathfrak{F}_h -normal in G if there exists a normal subgroup T of G such that HT is a normal Hall subgroup of G and (H ∩ T)H _G /H _G is contained in the $ \mathfrak{F} $ \mathfrak{F} -hypercenter $ Z_\infty ^\mathfrak{F} $ Z_\infty ^\mathfrak{F} (G/H _G ) of G/H _G . In this paper, we obtain some results about the $ \mathfrak{F}_h $ \mathfrak{F}_h -normal subgroups and then use them to study the structure of finite groups.     

Cardinalities of definable sets in superstructures over models

We introduce the notion of a superstructure over a model. This is a generalization of the notion of the hereditarily finite superstructure ℍ$ \mathbb{F}\mathfrak{M} $ \mathbb{F}\mathfrak{M} over a model $ \mathfrak{M} $ \mathfrak{M} . We consider the question on cardinalities of definable (interpretable) sets in superstructures over λ-homogeneous and λ-saturated models.     

Sharp estimates for derivatives of functions in the Sobolev classes $$ \mathop {W_2^r }\limits^ \circ $$(−1,1)

Explicit formulas are obtained for the maximum possible values of the derivatives f ^(k)(x), x ∈ (−1, 1), k ∈ {0, 1, ..., r − 1}, for functions f that vanish together with their (absolutely continuous) derivatives of order up to ≤ r − 1 at the points ±1 and are such that $ \left\| {f^{\left( r \right)} } \right\|_{L_2 ( - 1,1)} \leqslant 1 $ \left\| {f^{\left( r \right)} } \right\|_{L_2 ( - 1,1)} \leqslant 1 . As a corollary, it is shown that the first eigenvalue λ _1,r of the operator (−D ²) ^r with these boundary conditions is $ \sqrt 2 $ \sqrt 2 (2r)! (1 + O(1/r)), r → ∞.     

Completely nonmeasurable unions

Assume that no cardinal κ < 2 ^ω is quasi-measurable (κ is quasi-measurable if there exists a κ-additive ideal $ \mathbb{I} $ \mathbb{I} of X contains uncountably many pairwise disjoint subfamilies $ \mathbb{I} $ \mathbb{I} -Bernstein unions ∪ $ \mathbb{I} $ \mathbb{I} -Bernstein if A and X \ A meet each Borel $ \mathbb{I} $ \mathbb{I} -positive subset B ⊆ X). This result is a generalization of the Four Poles Theorem (see [1]) and results from [2] and [4].     

Smooth maps of a foliated manifold in a symplectic manifold

Let M be a smooth manifold with a regular foliation $ \mathcal{F} $ \mathcal{F} and a 2-form ω which induces closed forms on the leaves of $ \mathcal{F} $ \mathcal{F} in the leaf topology. A smooth map f: (M, $ \mathcal{F} $ \mathcal{F} ) → (N, σ) in a symplectic manifold (N, σ) is called a foliated symplectic immersion if f restricts to an immersion on each leaf of the foliation and further, the restriction of f*σ is the same as the restriction of ω on each leaf of the foliation. If f is a foliated symplectic immersion then the derivative map Df gives rise to a bundle morphism F: TM → T N which restricts to a monomorphism on T $ \mathcal{F} $ \mathcal{F} ⊆ T M and satisfies the condition F*σ = ω on T $ \mathcal{F} $ \mathcal{F} . A natural question is whether the existence of such a bundle map F ensures the existence of a foliated symplectic immersion f. As we shall see in this paper, the obstruction to the existence of such an f is only topological in nature. The result is proved using the h-principle theory of Gromov.