Completeness of Integer Translates in Function Spaces on

We show that each of the Banach spacesC₀( ) andL^p( ), 2<p<∞, contains a function whose integer translates are complete. This function can also be chosen so that one of the following additional conditions hold: (1) Its non-negative integer translates are already complete. (2) Its integer translates form an orthonormal system inL²( ). (3) Its integer translates form a minimal system. A similar result holds for the corresponding Sobolev space, for certain weightedL²spaces, and in the multivariate setting. We also prove some results in the opposite direction.     

Commutators for Fourier multipliers on Besov Spaces

If T is any bounded linear operator on Besov spaces B_p^σ_j,q_j(Rⁿ)(j=0,1, and 0<σ₁<σ<σ₀), it is proved that the commutator [T,T_μ]=TT_μ−T_μT is bounded on B_p^σ,q(Rⁿ), if T_μ is a Fourier multiplier such that μ is any (possibly unbounded) symbol with uniformly bounded variation on dyadic coronas.     

Schauder-type expansions of continuous functions on the unit interval

Let the space of continuous functions on [0, 1] which vanish at 0 be denoted by C. It will be shown that for any complete orthonormal set of functions {α_i(s)} of bounded variation and such that α_i(1) = 0, there is a simply described linear combination of the continuous functions {∝₀^tα_i(s) ds} which converges uniformly to x(t) for almost all x ε C (“almost all” in the sense of Wiener measure).     

Uniform embeddings of Banach spaces

It is proved that for 1<-p≤2,L _p(0,1) andl _p are uniformly equivalent to bounded subsets of themselves. It is also shown that for 1<=p<=2, 1≦q<∞,L _p is uniformly equivalent to a subset ofl _q. This is a part of the author’s Ph. D. thesis prepared at the Hebrew University of Jerusalem under the supervision of Professor J. Lindenstrauss. The author wishes to thank Professor Lindenstrauss for his guidance.     

Uncertainty principles for orthonormal sequences

The aim of this paper is to provide complementary quantitative extensions of two results of H.S. Shapiro on the time-frequency concentration of orthonormal sequences in L²(R). More precisely, Shapiro proved that if the elements of an orthonormal sequence and their Fourier transforms are all pointwise bounded by a fixed function in L²(R) then the sequence is finite. In a related result, Shapiro also proved that if the elements of an orthonormal sequence and their Fourier transforms have uniformly bounded means and dispersions then the sequence is finite. This paper gives quantitative bounds on the size of the finite orthonormal sequences in Shapiro's uncertainty principles. The bounds are obtained by using prolate spheroïdal wave functions and combinatorial estimates on the number of elements in a spherical code. Extensions for Riesz bases and different measures of time-frequency concentration are also given.     

On cross products of orthonormal systems of functions

The continual analog of an orthonormal system of functions is an orthonormal kernel. In this article the concept of cross product of orthonormal systems of functions is introduced, and it is shown that the cross product of any two orthonormal systems which are complete in L₂ is a complete orthonormal kernel with respect to Lebesgue measure on half-axes. The properties of the cross product of two orthonormal systems which are complete in L₂, each of which is uniformly bounded, are studied, as are the properties of the cross product of a Haar system on an orthonormal system of functions, complete in L₂, which are uniformly bounded.Translated from Matematicheskie Zametki, Vol. 13, No. 3, pp. 469–480, March, 1973.     

On the asymptotic behaviour of a semigroup of linear operators

Let T = {T(t)}_{t ≥ 0} be a C₀-semigroup on a Banach space X. In this paper, we study the relations between the abscissa ω_Lp(T) of weak p-integrability of T (1 ≤ p < ∞), the abscissa ω^p_R(A) of p-boundedness of the resolvent of the generator A of T (1 ≤ p ≤ ∞), and the growth bounds ω_β(T), β ≥ 0, of T. Our main results are as follows.

1. (i) Let T be a C₀-semigroup on a B-convex Banach space such that the resolvent of its generator is uniformly bounded in the right half plane. Then ω_{1 − ε}(T) < 0 for some ε > 0.

2. (ii) Let T be a C₀-semigroup on L^p such that the resolvent of the generator is uniformly bounded in the right half plane. Then ω_β(T) < 0 for all β>¦1/p − 1/p′¦, 1/p + 1/p′ = 1.

3. (iii) Let 1 ≤ p ≤ 2 and let T be a weakly L^p-stable C₀-semigroup on a Banach space X. Then for all β>1/p we have ω_β(T) ≤ 0.

Further, we give sufficient conditions in terms of ω^q_R(A) for the existence of L^p-solutions and W^1,p-solutions (1 ≤ p ≤ ∞) of the abstract Cauchy problem for a general class of operators A on X.     

A multiplier theorem using the Schechter's method of interpolation

Let m be a measurable bounded function and let us assume that there exists a bounded functions S so that m(ξ)S(ξ)^it−1 is a Fourier multiplier on L^p uniformly in . Then, using the analytic interpolation theorem of Stein, one can show that necessarily m is a L^p multiplier. The purpose of this work is to show that under the above conditions, it holds that, for every , m(log S)^kM_p. The technique is based on the Schechter's interpolation method.     

Multivariate Lp-error estimates for positive linear operators via the first-order τ-modulus

Let M(I) {ƒ:ƒ is a real-valued function that is bounded and measurable on an m-dimensional compact interval I} and let L: M(I) → M(I) be a multivariate positive linear operator. The aim of this paper is to give estimates for the approximation error's L_p-norm ƒ − Lƒ_p using the so-called averaged modulus of smoothness or τ-modulus of first order.     

Completeness of Orthonormal Wavelet Systems for Arbitrary Real Dilations

It is shown that the discrete Calderón condition characterizes completeness of orthonormal wavelet systems, for arbitrary real dilations. That is, if a>1,b>0, and the system Ψ={a^j/2ψ(a^jx−bk):j,k } is orthonormal in L²( ), then Ψ is a basis for L²( ) if and only if ∑_j | (a^jξ)|²=b for almost every ξ . A new proof of the Second Oversampling Theorem is found, by similar methods.     

Efficiency of weak greedy algorithms for m-term approximations

We investigate the efficiency of weak greedy algorithms for m-term expansional approximation with respect to quasi-greedy bases in general Banach spaces.We estimate the corresponding Lebesgue constants for the weak thresholding greedy algorithm(WTGA) and weak Chebyshev thresholding greedy algorithm.Then we discuss the greedy approximation on some function classes.For some sparse classes induced by uniformly bounded quasi-greedy bases of L_p,1p∞,we show that the WTGA realizes the order of the best m-term approximation.Finally,we compare the efficiency of the weak Chebyshev greedy algorithm(WCGA) with the thresholding greedy algorithm(TGA) when applying them to quasi-greedy bases in L_p,1≤p∞,by establishing the corresponding Lebesgue-type inequalities.It seems that when p2 the WCGA is better than the TGA.     

Properties of convergence for -Bernstein polynomials

In this paper, we discuss properties of the ω,q-Bernstein polynomials introduced by S. Lewanowicz and P. Woźny in [S. Lewanowicz, P. Woźny, Generalized Bernstein polynomials, BIT 44 (1) (2004) 63–78], where fC[0,1], ω,q>0, ω≠1,q⁻¹,…,q⁻ⁿ⁺¹. When ω=0, we recover the q-Bernstein polynomials introduced by [G.M. Phillips, Bernstein polynomials based on the q-integers, Ann. Numer. Math. 4 (1997) 511–518]; when q=1, we recover the classical Bernstein polynomials. We compute the second moment of , and demonstrate that if f is convex and ω,q(0,1) or (1,∞), then are monotonically decreasing in n for all x[0,1]. We prove that for ω(0,1), q_n(0,1], the sequence converges to f uniformly on [0,1] for each fC[0,1] if and only if lim_n→∞q_n=1. For fixed ω,q(0,1), we prove that the sequence converges for each fC[0,1] and obtain the estimates for the rate of convergence of by the modulus of continuity of f, and the estimates are sharp in the sense of order for Lipschitz continuous functions.     

On convergence of Weak Thresholding Greedy Algorithm in

For any 0<t<1 we construct a Weak Thresholding Greedy Algorithm with weakness parameter t which converges in L¹(0,1) with respect to the Haar system, i. e. the Haar system is a ‘good non quasi-greedy basis’.     

Approximation ofk-Monotone Functions

It is shown that an algebraic polynomial of degree k−1 which interpolates ak-monotone functionfatkpoints, sufficiently approximates it, even if the points of interpolation are close to each other. It is well known that this result is not true in general for non-k-monotone functions. As an application, we prove a (positive) result on simultaneous approximation of ak-monotone function and its derivatives inL_p, 0<p<1, metric, and also show that the rate of the best algebraic approximation ofk-monotone functions (with bounded (k−2)nd derivatives inL_p, 1<p<∞, iso(n^−k/p).     

Relativistic Lamé functions: Completeness vs. polynomial asymptotics

In earlier work we introduced and studied two commuting generalized Lamé operators, obtaining in particular joint eigenfunctions for a dense set in the natural parameter space. Here we consider these difference operators and their eigenfunctions in relation to the Hilbert space L²((0, π/r), w(x)dx), with r > 0 and the weight function w(x) a ratio of elliptic gamma functions. In particular, we show that the previously known pairwise orthogonal joint eigenfunctions need only be supplemented by finitely many new ones to obtain an orthogonal base. This completeness property is derived by exploiting recent results on the large-degree Hilbert space asymptotics of a class of orthonormal polynomials. The polynomials p_n(cos(rx)), n ε , that are relevant in the Lamé setting are orthonormal in L²((0, π/r), w_P(x)dx), with w_p(x) closely related to w(x).     

Best polynomial and durrmeyer approximation in Lp(S)

On a simplex SR^d, the best polynomial approximation is E_n()_Lp(S)=Inf{P_n−_Lp(S): P_n of total degree n}. The Durrmeyer modification, M_n, of the Bernstein operator is a bounded operator on L_p(S) and has many “nice” properties, most notably commutativity and self-adjointness. In this paper, relations between M_n−z.dfnc;_Lp(S) and E_[√n]()_Lp(S) will be given by weak inequalities will imply, for 0<α<1 and 1≤p≤∞, E_n()_Lp(S)=O(n^-2α)M_n−z.dfnc;_Lp(S)=O(n^-α). We also see how the fact that P(D)εL_p(S) for the appropriate P(D) affects directional smoothness.     

A discrete Korovkin theorem

In this paper we give a sufficient condition for the pointwise Korovkin property on B(X), the space of bounded real valued functions on an arbitrary countable set X = {x_l,…, x_j,…}. Our theorem follows from its L_p(X, μ) analogue (and conversely); here 1 p < ∞ and μ is a positive finite measure on X such that μ({x_j}) > 0 for all j.     

Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces

Let (E,H,μ) be an abstract Wiener space and let D_V:=VD, where D denotes the Malliavin derivative and V is a closed and densely defined operator from H into another Hilbert space . Given a bounded operator B on , coercive on the range , we consider the operators A:=V^*BV in H and in , as well as the realisations of the operators and in L^p(E,μ) and respectively, where 1<p<∞. Our main result asserts that the following four assertions are equivalent:

(1) with for ;

(2) admits a bounded H^∞-functional calculus on ;

(3) with for ;

(4) admits a bounded H^∞-functional calculus on .

Moreover, if these conditions are satisfied, then . The equivalence (1)–(4) is a non-symmetric generalisation of the classical Meyer inequalities of Malliavin calculus (where , V=I, ). A one-sided version of (1)–(4), giving L^p-boundedness of the Riesz transform in terms of a square function estimate, is also obtained. As an application let −A generate an analytic C₀-contraction semigroup on a Hilbert space H and let −L be the L^p-realisation of the generator of its second quantisation. Our results imply that two-sided bounds for the Riesz transform of L are equivalent with the Kato square root property for A. The boundedness of the Riesz transform is used to obtain an L^p-domain characterisation for the operator L.

Wavelets and Bidemocratic Pairs in Weighted Norm Spaces

A complete characterization of weight functions for which the higher-rank Haar wavelets are greedy bases in weighted L^p spaces is given. The proof uses the new concept of a bidemocratic pair for a Banach space and also pairs (Φ, Φ), where Φ is an orthonormal system of bounded functions in the spaces L^p, p≠2.     

Ridge Functions and Orthonormal Ridgelets

Orthonormal ridgelets provide an orthonormal basis for L²(R²) built from special angularly-integrated ridge functions. In this paper we explore the relationship between orthonormal ridgelets and true ridge functions r(x₁ cos θ+x₂ sin θ). We derive a formula for the ridgelet coefficients of a ridge function in terms of the 1-D wavelet coefficients of the ridge profile r(t). The formula shows that the ridgelet coefficients of a ridge function are heavily concentrated in ridge parameter space near the underlying scale, direction, and location of the ridge function. It also shows that the rearranged weighted ridgelet coefficients of a ridge function decay at essentially the same rate as the rearranged weighted 1-D wavelet coefficients of the 1-D ridge profile r(t). In short, the full ridgelet expansion of a ridge function is in a certain sense equally as sparse as the 1-D wavelet expansion of the ridge profile. It follows that partial ridgelet expansions can give good approximations to objects which are countable superpositions of well-behaved ridge functions. We study the nonlinear approximation operator which “kills” coefficients below certain thresholds (depending on angular- and ridge-scale); we show that for approximating objects which are countable superpositions of ridge functions with 1-D ridge profiles in the Besov space B^1/p_p, p(R), 0<p<1, the thresholded ridgelet approximation achieves optimal rates of N-term approximation. This implies that appropriate thresholding in the ridgelet basis is equally as good, for certain purposes, as an ideally-adapted N-term nonlinear ridge approximation, based on perfect choice of N-directions.