Analogues of the “zero-two” law for positive linear contractions inL p andC(X)

LetT be a positive linear contraction inL ^p (1≦p<∞), then we show that lim ‖T ^pf −T ⁿ⁺¹ f‖ _p ≦(1 − ε)2^1/p (f∈L _p ⁺ , ε>0 independent off) implies already lim_n n→∞ ‖T ⁿf −T ⁿ⁺¹ n+1f ‖_p p=0. Several other related results as well as uniform variants of these are also given. Finally some similar results inLsu/t8 andC(X) are shown.     

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.     

Matrix transformations of lq（X） to lP（Y）

Let X and Y be Banach spaces,0 ＜ q ＜ +∞,1 ＜ p ＜ +∞.In this paper,we characterize matrix transformations of lq(X) to lp(Y).     

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.     

n-Widths of Sobolev spaces inL p

LetW _p ^(r) ={f:f∈C ^r?1[0, 1],f ^(r?1) abs.cont., ∥f ^(r)∥ _p <∞}, and setB _p ^(r) ={f:f∈W _p ^(r) ,∥f ^(r)∥ _p ≤1}. We find the exact Kolmogorov, Gel'fand, linear, and Bernsteinn-widths ofB _p ^(r) inL ^p for allp∈(1, ∞). For the Kolmogorovn-width we show that forn≥r there exists an optimal subspace of splines of degreer?1 withn?r fixed simple knots depending onp.     

Uniform boundedness inL p (p=p(x)) of some families of convolution operators

Suppose that a measurable 2π-periodic essentially bounded function (the kernel) κ_λ =κ_λ(x) is given for any realλ≥1. We consider the following linear convolution operator inL ^p: $$\kappa _\lambda = \kappa _\lambda f = (\kappa _\lambda f)(x) = \int_{ - \pi }^\pi {f(t)} k_\lambda (t - x) dt.$$ Uniform boundedness of the family of operators {Κ_λ}_λ≥1 is studied. Conditions on the variable exponentp=p(x) and on the kernel κ_λ that ensure the uniform boundedness of the operator family {Κ_λ}_λ≥1 inL ^p are obtained. The condition on the exponentp=p(x) is given in its final form.     

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 the fractional integral of Weyl inL p

This work has been partially supported by D.G.C.Y.T., grant PS88-0115, Spain     

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.     

Walsh equiconvergence and (l,p) distinguished points

Here we study some simple properties of (l, p) distinguished points for functions in the classA _p (p>1) in the context of Walsh equiconvergence. We give a condition which is necessary and sufficient for a set Z of points in C to be an (l, p) distinguished set. This condition seems to complement the one given earlier in [2].     

Inequalities of Jackson type and multipliers inL p

This research was supported by the ISF Grant No. MC 5000.     

The negative discrete spectrum of the operator (-τ) l -αV inL 2(R d ) ford even and 2l≥dinL 2(R d ) ford even and 2l≥d

We study the asymptotic behaviour ofN(α)—the number of negative eigenvalues of the operator (-τ) ^l -αV inL ₂(R ^d ) for an evend and2l≥d. This is the only case where the previously known results were far from being complete. In order to describe our results we introduce an auxiliary ordinary differential operator (system) on the semiaxis. Depending on the spectral properties of this operator we can distinguish between three cases whereN(α) is of the Weyl-type,N(α) is of the Weyl-order but not the Weyl-type coefficient and finally whereN(α)=O(α^q) withq>d/2l.