Представление функций классовL p [0, 1], 0<p<1, ортогональными рядами

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Nikol’skii-Stechkin inequality for trigonometric polynomials in L 0

In this paper, the Nikol’skii-Stechkin inequality for the trigonometric polynomials is generalized to the space L ₀. The resulting estimates are final.     

Inequality of the Turan type for trigonometric polynomials and conjugate trigonometric polynomials in <Emphasis Type="Italic">L</Emphasis><Subscript>0</Subscript>

We study inequalities of the Turan type for trigonometric polynomials and conjugate trigonometric polynomials in the quasinorm of L ₀ and derivatives of any order. We present expressions for constants in these inequalities and obtain double-sided estimates for them.     

Isometries on the Quasi-Banach Spaces Lp （0〈p〈1）

We study the extension of isometries between the unit spheres of quasi-Banach spaces Lp for 0〈p〈1. We give some sufficient conditions such that an isometric mapping from the the unit sphere of Lp（μ） into that of another LP（ν） can be extended to be a linear isometry defined on the whole space.     

Approximation by Kantorovich-Bernstein polynomials inL p(0<p<1)-metric

The purpose of the present paper is to evaluate the error of the approximation of the function f∈L₁[0,1] by Kantorovich-Bernstein polynomials in L_p-metric (0<p<1).     

On Bernstein inequalities for multivariate trigonometric polynomials in Lp, 0 ⩽ p ⩽ ∞

Czechoslovak Mathematical Journal - Let $${{mathbb{T}}_n}$$ be the space of all trigonometric polynomials of degree not greater than n with complex coefficients. Arestov extended the result of...     

A characterization of L p(ℝ) by local trigonometric bases with 1<p<∞

We show that the local trigonometric bases introduced by Malvar, Coifman and Meyer constitute bases, but not unconditional bases, for L^p(ℝ) with 1<p<∞, p≠2. In addition, we characterize the functions in L^p(ℝ) for 1<p<∞ in terms of their local trigonometric basis coefficients. Dedicated to Dr. Charles A. Micchelli for his 60th birthday Mathematics subject classification (2000) 42C15. Supported by Prof. Y. Xu under his grant in program of “One Hundred Distinguished Chinese Scientists” of the Chinese Academy of Sciences, the National Natural Science Foundation of China (No. 10371122), and the second author is supported by Tianyuan Fund for Mathematics (No. A0324648).     

Russian text ignored Lp, 0 < p < 1

Certain weak forms of normality are discussed and are shown to coincide with normality in an appropriate setting by devising a strong form of semilocal connectedness. It is shown that every compact Hausdorff locally connected space is strongly semilocally connected. This improves a result of Mrówka and Pervin. Properties of s-continuous mappings are studied and in the process improvements of certain results of Pervin and Levine, Friedler, and Kohli are obtained.     

On translation and dilation invariant subspaces of L p (ℝn), 0 < p < 1

We prove that each translation and dilation invariant subspace X ⊂ L ^p (ℝⁿ), X ≠ L ^p (ℝⁿ), is contained in a maximal translation and dilation invariant subspace of L ^p (ℝⁿ). Moreover, we prove that the set of all maximal translation and dilation invariant subspaces of L ^p (ℝⁿ) has the power of continuum. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 345, 2007, pp. 5–21.     

H p -spaces of martingales, 0<p≦1

For 0 <p≦1 the dual Banach space to the space of H _p -martingales is characterized as a Lipschitz class of order α=l/p?1. When p=1 and α=0 this reduces to BMO being the dual of H ₁ . In the course of the investigation a hierarchy of martingale spaces _a L _p is introduced with the property that the index p is in the range ?∞ <1/p<∞. For l≦b≦q<∞ the space L _q is a subspace of _b L _q (equality when b=q) while for q<0 the space _b L _q corresponds to a Lipschitz class of order α=?1/q.     

Perturbation of the trigonometric system in L 1(0, π)

In this paper, we study a special perturbation of the function system obtained from the Fejér kernel. It is shown how this relates to the stability of bases and complete systems as well to the stability of the trigonometric system. An approximation algorithm in systems resulting from a perturbation of the original system is given.     

Moduli of smoothness andK-functionals inL p, 0<p<1

It is shown that forL _p, 0p<1, the=">K-functional betweenL _p andW _p ^r is identically zero. Useful measures that are equivalent to the moduli of smoothness are found. The equivalence results that are given are valid for 0p.Communicated by Vilmos Totik.     

Embeddings of non-commutative Lp-spaces into non-commutative L1-spaces, 1 < p < 2

It will be shown that for 1 < p < 2 the Schatten p-class is isometrically isomorphic to a subspace of the predual of a von Neumann algebra. Similar results hold for non-commutative L_p(N, t) L_p(N, \tau) -spaces defined by a finite trace on a finite von Neumann algebra. The embeddings rely on a suitable notion of p-stable processes in the non-commutative setting.     

Boundedness of trigonometric integrals and polynomials in the L metric

Conditions are derived for the uniform boundedness of special trigonometric integrals and sums in the L metric.Translated from Matematicheskie Zametki, Vol. 8, No. 6, pp. 811–822, December, 1970.     

Besov spaces andK-functionals inL p, 0<p<1*

We investigate Besov spaces and their connection with trigonometric polynomial approximation inL _p[−π,π], algebraic polynomial approximation inL _p[−1,1], algebraic polynomial approximation inL _p(S), and entire function of exponential type approximation inL _p(R), and characterizeK-functionals for certain pairs of function spaces including (L _p[−π,π],B _s ^a(L _p[−π,π])), (L _p(R),_s ^a(L_p(R))), , and , where 0<s≤∞, 0<p<1,S is a simple polytope and 0<α<r. This project is supported by the National Science Foundation of China.     

Embedding Subspaces of Lp into 𝓁Np, 0 < p < 1

It is shown that if X is an n – dimensional subspace of L_p, 0 < p < 1, then there exists a subspace Y of 𝓁^N_p such that d(X, Y) ≤ 1 + ε and N ≤ C(ε, p)n(log n)³.     

Primarite DeL p(L r), 1<p,r<∞

In this article we show thatL ^p(L ^r) is primary forp andr in ]1,+∞[. If (h _k) _k≧1 denote the Haar basis, we begin with a study of the sequence (h _k ⊗h _i) and, in particular, the space generated by a subsequence of this sequence. In the first part we study the base ofL ^p(L ^r) and in the second part we show that this space is primary.