Averaging operator in the Lp spaces for 0<p<1

Translated from Matematicheskie Zametki, Vol. 43, No. 1, pp. 38–43, January, 1988.     

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

u . uuu u     

Polynomial approximation inL p (0<p<1)

We prove that forf∈L _p , 0<p<1, andk a positive integer, there exists an algebraic polynomialP _n of degree ≤n such that $$\left\| {f - P_n } \right\|_p \leqslant C\omega _k^\varphi \left( {f,\frac{1}{n}} \right)_p $$ whereω _k ^? (f,t)p is the Ditzian-Totik modulus of smoothness off inL _p , andC is a constant depending only onk andp. Moreover, iff is nondecreasing andk≤2, then the polynomialP _n can also be taken to be nondecreasing.     

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.     

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.     

On the duals of Lp spaces with 0<p<1

Given a measure space < Ω,m,μ >, a locally bounded, Hausdorff topological linear space < X, τ > and a real number 0<p<1, one can define the space L_p(Ω,m,μ,X), which is, under certain assumptions, a Fréchet space if endowed with a suitable topology. M.M. Day [1] has given a necessary and sufficient condition, in terms of the properties of the measure space < Ω,m,μ >, for the dual of L_p(Ω,m,μ,C) to be trivial. In this paper a different proof along with a slight generalization is given for this result, using standard and elementary measure theoretic arguments. This revised version was published online in June 2006 with corrections to the Cover Date.     

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).     

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.     

Semigroups of linear operators on p-Fréchet spaces, 0 < p < 1

We develop the beginning of a theory of semigroups of linear operators on p-Fréchet spaces, 0 < p < 1 (which are non-locally convex F-spaces), and give some applications.     

p-Helson sets, 1<p<2

Ap-Helson set is defined to be a closed subsetE of the circle groupT with the property that every continuous function onE can be extended to the full circle in such a way that this extension has its sequence of Fourier coefficients inl ^p. For 1<p<2, the union of two such sets is again ap-Helson set. It is shown that thep-Helson sets (p>1) differ from the Helson sets and also that the notion really depends on the indexp. An analogue of H. Helson’s result is given: ap-Helson set supports no nonzero measure with Fourier-Stieltjes transform inl ^q, 1/p+1/q=1.     

Lower bound for matrix operators on the Euler weighted sequence space e_{w,p}^{\theta} (0<p<1)

Let A=(a _n,k ) _n,k≥0 be a non-negative matrix. Denote by \(L_{l_{p} (w),~e_{w,q}^{\theta}}(A)\) the supremum of those L, satisfying the following inequality: