Approximative characteristics of the isotropic classes of periodic functions of many variables

Exact-order estimates are obtained for the best orthogonal trigonometric approximations of the Besov (B _p,θ ^r ) and Nukol’skii (H _p ^r ) classes of periodic functions of many variables in the metric of L _q , 1 ≤ p, q ≤ ∞. We also establish the orders of the best approximations of functions from the same classes in the spaces L ₁ and L _∞ by trigonometric polynomials with the corresponding spectrum.     

Trigonometric widths of the classes B p,θ r of functions of many variables in the space L q

We obtain estimates exact in order for the trigonometric widths of the Besov classes B _p,θ^r of periodic functions of many variables in the space L _q for 1 ≤ p ≤ 2 < q < p/(p - 1). Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 8, pp. 1089–1097, August, 1998.     

Best m-ter m trigono metric approxi mation for the classes $ B_{p,{{\uptheta }}}^r $ of functions of low s moothness

We obtain an exact-order estimate for the best m-term trigonometric approximation of the Besov classes B_p,\uptheta ^r B_{p,{{\uptheta }}}^r of periodic functions of many variables of low smoothness in the space L _q , 1 < p ≤ 2 < q < ∞.     

The Wirtinger-Steklov inequality between the norm of a periodic function and the norm of the positive cutoff of its derivative

We study the sharp constant in the inequality between the L _p -mean (p ≥ 0) of a 2π-periodic function with zero mean value and the L _q -norm (q ≥ 1) of the positive cutoff of its derivative. We obtain estimates of the constant from below for 0 ≤ p ≤ ∞ and from above for 1 ≤ p ≤ ∞ for an arbitrary 1 ≤ q ≤ ∞. We write out the values of the sharp constant in the cases p = 2, 1 ≤ q ≤ ∞ and p = ∞, 1 ≤ q ≤ ∞.     

Average Widths of Sobolev-Wiener Classes with Mixed Smoothness in L q (ℝ d )

In this paper, the average σ-Κ width of Sobolev-Wiener classes S ^r _pq W with mixed smoothness in L _q (ℝ ^d ) is studied for 1 < q≤p < ∞, and the weak asymptotical behaviour of these quantities is obtained. Received July 21, 1998, Accepted May 31, 1999     

Best approximation by ridge functions in <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-spaces

We study the approximation of the classes of functions by the manifold R _n formed by all possible linear combinations of n ridge functions of the form r(a · x)): It is proved that, for any 1 ≤ q ≤ p ≤ ∞, the deviation of the Sobolev class W ^r _p from the set R _n of ridge functions in the space L _q (B ^d ) satisfies the sharp order n ^-r/(d-1).     

Isolation and Component Structure in Spaces of Composition Operators

We establish a condition that guarantees isolation in the space of composition operators acting between H^p(B_N) and H^q(B_N), for 0 < p ≤ ∞, 0 < q < ∞, and N ≥ 1. This result will allow us, in certain cases where 0 < q < p ≤ ∞, completely to characterize the component structure of this space of operators.     

Approximation to the Sobolev classesW q r of functions of several variables by bilinear forms inL p for 2≤q≤p≤∞

We study the approximation of functions of several variables by bilinear forms that are the pairwise products of functions of fewer variables. The order of approximation of Sobolev classesW _q ^r by bilinear forms inL _p for 2≤q≤p≤∞ is found. Translated by N. K. Kulman Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 18–34, July, 1997.     

Coefficient multipliers of mixed norm space in the ball

In the paper, we characterize the coefficient multiplier spaces of mixed norm spaces (H _p,q (φ₁), H _u,v (φ₂)) for the values of p, q, u, v in three cases: (i) 0 < p ≤ u ≤ ∞, 0 < q ≤ min(1, v) ≤ ∞. (ii) v = ∞, 0 < p ≤ u ≤ ∞, 1 ≤ u, q ≤ ∞. (iii) 1 ≤ v ≤ 2 ≤ q ≤ ∞, and 0 < p ≤ u ≤ ∞ or 1 ≤ p, u ≤ ∞. The first case extends the result of Blasco, Jevtić, and Pavlović in one variable. The third case generalizes partly the results of Jevtić, Jovanović, and Wojtaszczyk to higher dimensions. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

Kolmogorov and linear widths of classes of <Emphasis Type="Italic">s</Emphasis>-monotone integrable functions

Let s ∈ ℕ and let Δ ₊ ^s be the set of functions x: I ↦ ℝ on a finite interval I such that the divided differences [x; t ₀, ..., t _s ] of order s of these functions are nonnegative for all collections of s + 1 different points t ₀, ..., t _s ∈ I. For the classes Δ ₊ ^s B _p : = Δ ₊ ^s ∩ B _p , where B _p is the unit ball in L _p , we determine the orders of Kolmogorov and linear widths in the spaces _Lq for 1 ≤ q > p ≤ ∞. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 12, pp. 1633–1652, December, 2005.     

Greedy Bases for Besov Spaces

We prove that the Banach space (?_n=1^￥l_pⁿ)_lq(\bigoplus_{n=1}^{\infty}\ell_{p}^{n})_{\ell_{q}}, which is isomorphic to certain Besov spaces, has a greedy basis whenever 1≤p≤∞ and 1<q<∞. Furthermore, the Banach spaces (?_n=1^￥l_pⁿ)_l1(\bigoplus_{n=1}^{\infty}\ell _{p}^{n})_{\ell_{1}}, with 1<p≤∞, and (?_n=1^￥l_pⁿ)_c0(\bigoplus_{n=1}^{\infty}\ell_{p}^{n})_{c_{0}}, with 1≤p<∞, do not have a greedy basis. We prove as well that the space (?_n=1^￥l_pⁿ)_lq(\bigoplus_{n=1}^{\infty}\ell _{p}^{n})_{\ell_{q}} has a 1-greedy basis if and only if 1≤p=q≤∞.     

Finite groups in whichp′-classes haveq′-length

IfG is a finite group in which every element ofp′-order centralizes aq-Sylow subgroup ofG, wherep andq are distinct primes, it is shown thatO ^q′ (G) is solvable,l _q (G)≤1 andl _p (O ^q′ (G)) ≤2. Further, the structure ofG is determined to some extent. Work partially supported by MURST research program “Teoria dei gruppi ed applicazioni”.     

On the Fréchet space L p −

This paper contains a study of the structure of the Fréchet space L _p ⁻, 1< p ≤∞, defined as the intersection of L _q [0,1] for q<p, and endowed with the projective topology. The main topics covered are: normable, Schwartz and nuclear subspaces of L _p ⁻; construction of uncomplemented copies of ?₂ inside L _p ⁻ for p<2; construction of Montel non-Schwartz subspaces; the space L _p ⁻ is primary. Received: 30 October 1996 / Revised version: 1 February 1998     

Strongly singular convolution operators on the weighted Herz-type Hardy spaces

Let 0<p≤1<q<0, andw ₁ ,w ₂ ∈ A ₁ (Muckenhoupt-class). In this paper the authors prove that the strongly singular convolution operators are bounded from the homogeneous weighted Herz-type Hardy spacesH K^{α, p} _q(w₁; w₂) to the homogeneous weighted Herz spacesK ^{α, p} _q (w₁; w₂), provided α=n(1−1/q). Moreover, the boundedness of these operators on the non-homogeneous weighted Herz-type Hardy spacesH K ^{α, p} _q (w ₁;w ₂) is also investigated. Supported by the National Natural Science Foundation of China     

Compositions of isometric immersions¶in higher codimension

Given a submanifold M ⁿ of Euclidean space ℝ ^{n + p} with codimension p≤6, under generic conditions on its second fundamental form, we show that any other isometric immersion of M ⁿ into ℝ ^{n + p + q} , 0≤q≤n− 2p−1 and 2q≤n+ 1 if q≥ 5, must be locally a composition of isometric immersions. This generalizes several previous results on rigidity and compositions of submanifolds. We also provide conditions under which our result is global. 14 March 2001     

The weighted Herz-type Hardy spaces hK q α,p (ω1,ω2)

Let 1<q<∞, n(1−1/q)≤α<∞, 0<p<∞ and ω₁,ω₂ ɛA ₁(R ⁿ ) (the Muckenhoupt class). In this paper, the author introduce the weighted Herz-type Hardy spaces hk _q ^α,p (gw₁,ω₂) and present their atomic decomposition. Using the atomic decomposition, the author find out their dual spaces, establish the boundedness on these spaces of the pseudo-differential operators of order zero and show thatD(R ⁿ ), the class of C^∞(Rⁿ)-functions with compactly support, is dense inhK _q ^α,p (ω₁,ω₂) and there is a subsequence, which converges in distrbutional sense to some distribution ofhK _q ^α,p (ω₁,ω₂), of any bounded sequence inhK _q ^α,p (ω₁,ω₂). In addition, the author also set up the boundedness of some non-linear quantities in compensated compactness. Supported by the NECF and the NECF and the NNSF of China.     

Trigonometric and orthoprojection widths of classes of periodic functions of many variables

We obtain exact order estimates for trigonometric and orthoprojection widths of the Besov classes B ^r _p,θ and Nikol’skii classes Hr p of periodic functions of many variables in the space L _q for certain relations between the parameters p and q.     

Test elements for endomorphisms of free groups and algebras

LetE be a bounded Borel subset of ℝⁿ,n≥2, of positive Lebesgue measure andP _E the corresponding ‘Pompeiu transform”. We prove thatP _E is injective onL ^p(ℝⁿ) if 1≤p≤2n/(n-1). We explore the connection between this problem and a Wiener-Tauberian type theorem for theM(n) action onL ^q(ℝⁿ) for various values ofq. We also take up the question of whenP _E is injective in caseE is of finite, positive measure, but is not necessarily a bounded set. Finally, we briefly look at these questions in the contexts of symmetric spaces of compact and non-compact type.     

On the Minimum Length of some Linear Codes of Dimension 5

In this paper, we shall prove that the minimum length n_q(5,d) is equal to g_q(5,d) +1 for q⁴−2q²−2q+1≤ d≤ q⁴ − 2q² − q and 2q⁴ − 2q³ − q² − 2q+1 ≤ d ≤ 2q⁴−2q³−q²−q, where g_q(5,d) means the Griesmer bound . Communicated by: J.D. Key     

A Note on the Mean Value of Numbersof the Solutions of x^α≡ 1（modn）

Let T = T(p, q, α) be the number of solutions of the congruence x^α ≡ 1 (mod p^ηq^θ). Let A and B be sets of primes satisfying x₁ < p ≤ x₂ and y₁ < q ≤ y₂, respectively. A mean value estimation of is given. Supported by National Natural Science Foundation of China (No. 19971024) and Zhejiang Provincial Natural Science Foundation of China (No. 199047)