Polynomial approximation inL p(S) forp>0

For a simple polytopeS inR ^d andp>0 we show that the best polynomial approximationE _n(f)_p≡E_n(f)L_p(S) satisfies $$E_n \left( f \right)_p \leqslant C\omega _S^r \left( {f,\frac{1}{n}} \right)p,$$ where ω _S ^r is a measure of smoothness off. This result is the best possible in the sense that a weak-type converse inequality is shown and a realization of ω _S ^r (f,t)_p via polynomial approximation is proved.     

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).     

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.     

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.     

On subsequences of the Haar system inL p [0, 1], (1<p<∞)

We show that ifX is the closed linear span inL _p [0,1] of a subsequence of the Haar system, thenX is isomorphic either tol _p or toL _p [0,1], [1<p<∞]. We give criteria to determine which of these cases holds; for a given subsequence, this is independent ofp. This is part of the second author's Ph.D. dissertation, written at the University of Alberta under the supervision of J. L. B. Galmen. The first author's research was partially supported by NRC A7552.     

A remark on unconditional basic sequences inL p (1<p<∞)

We prove that every separable ? _p space (1<p<∞), with an unconditional basis is isomorphic to a complemented subspace ofL _p which is spanned by a block basis of the Haar system.     

Moduli of smoothness and best approximation in the spacesL p , 0<p<1

L ^p , 0<<1, .     

Subspaces ofL p (G) spanned by characters: 0<p<1

LetG be an infinite compact abelian group,μ a Borel measure onG with spectrumE, and 0<p<1. We show that ifμ is not absolutely continuous with respect to Haar measure, thenL _E ^P (G), the closure inL ^p (G) of theE-trigonometric polynomials, does not have enough continuous linear functionals to separate points. Ifμ is actually singular, thenL _E ^p (G) does not have any nontrivial continuous linear functionals at all. Our methods recover the classical F. and M. Riesz theorem, and a related several variable result of Bochner; they reveal the existence of small sets of characters that spanL ^P (T), where T is the unit circle; and they show that theH ^p spaces of the “big disc algebra” have one-dimensional dual.     

On the span of some three valued martingale difference sequences inL p (1<p<∞) andH 1

LetX ^p denote the span of a three valued martingale difference sequence with nested supports inL ^p. The spacesL ^{p, l p , l 2 , l p} ⊗l ², (Σ ⊗l ²) _p are the only isomorphic types which can be obtained in this way (Theorem 2). This result is obtained by interpolation from the correspondingH ¹ result (Theorem 1).     

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

u . uuu u     

Shape-preserving approximation inL p

We prove a direct theorem for shape preservingL _p -approximation, 0p, in terms of the classical modulus of smoothnessw ₂(f, t _p ¹ ). This theorem may be regarded as an extension toL _p of the well-known pointwise estimates of the Timan type and their shape-preserving variants of R. DeVore, D. Leviatan, and X. M. Yu. It leads to a characterization of monotone and convex functions in Lipschitz classes (and more general Besov spaces) in terms of their approximation by algebraic polynomials.Communicated by Ron DeVore.     

Onp-hyponormal operators for 0<p<1

The distance formula Tt-I)^–1=[Dist(, (T)]^–1, (T), for hyponormal operators, is generalized top-hyponormal operators for 0<p<1. Several other results involving eigenspaces ofU and |T|, the joint point spectrum, and the spectral radius are also otained, where |T|=(T ^* T)^1/2 andU is the unitary operator in the polar decomposition of thep-hyponormal operatorT=U|T|.     

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.     

Generalized monotone approximation inL p Space

Letf(x) ∈L _p[0,1], 1?p? ∞. We shall say that functionf(x)∈Δ^k (integerk?1) if for anyh ∈ [0, 1/k] andx ∈ [0,1?kh], we have Δ _h ^k f(x)?0. Denote by ∏ _n the space of algebraic polynomials of degree not exceedingn and define $$E_{n,k} (f)_p : = \mathop {\inf }\limits_{\mathop {P_n \in \prod _n }\limits_{P_n^{(\lambda )} \geqslant 0} } \parallel f(x) - P_n (x)\parallel _{L_p [0,1]} .$$ We prove that for any positive integerk, iff(x) ∈ Δ ^k ∩ L _p[0, 1], 1?p?∞, then we have $$E_{n,k} (f)_p \leqslant C\omega _2 \left( {f,\frac{1}{n}} \right)_p ,$$ whereC is a constant only depending onk.