On symmetric basic sequences in lorentz sequence spaces II

It is shown that if {y _n} is a block of type I of a symmetric basis {x _n} in a Banach spaceX, then {y _n} is equivalent to {x _n} if and only if the closed linear span [y _n] of {y _n} is complemented inX. The result is used to study the symmetric basic sequences of the dual space of a Lorentz sequence spaced(a, p). Let {x _n,f _n} be the unit vector basis ofd(a, p), for 1≤p<+∞. It is shown that every infinite-dimensional subspace ofd(a, p) (respectively, [f _n] has a complemented subspace isomorphic tol _p (respectively,l _q, 1/p+1/q=1 when 1<p<+∞ andc ₀ whenp=1) and numerous other results on complemented subspaces ofd(a, p) and [f _n] are obtained. We also obtain necessary and sufficient conditions such that [f _n] have exactly two non-equivalent symmetric basic sequences. Finally, we exhibit a Banach spaceX with symmetric basis {x _n} such that every symmetric block basic sequence of {x _n} spans a complemented subspace inX butX is not isomorphic to eitherc ₀ orl _p, 1≤p<+∞.     

Characterization ofc 0 andl p among Banach spaces with symmetric basis

A Banach spaceX with symmetric basis {e _n} is isomorphic toc ₀ orl _p for some 1≦p<∞, if all symmetric basic sequences inX are equivalent to {e _n}, and all symmetric basic sequences in [f _n]≠X ^* are equivalent to {f _n} (wheref _n (e _j ) =δ _{n, j} ). The result proved in the paper is actually stronger, in the sense that it does not involve all symmetric basic sequences, but only the so called sequences generated by one vector. This is part of the author’s Ph.D. Thesis prepared at the Hebrew University of Jerusalem under the supervision of Professor L. Tzafriri. I wish to thank Professor Tzafriri for his interest and advice.     

A stability property of the unit vector basis ofl p

Let {e _n} be the unit vector basis ofl _p, l<p<∞, and letx _n=a_ne_n?b_nen+1. Necessary and sufficient conditions are given for the operatorT:l _p → span {x _n} defined byTe _i=x_i to be invertible.     

Eventually areally meanp-valent functions

Theorems concerning areally meanp-valent functions are extended to eventually areally meanp-valent functions. In particular, suppose is eventually areally meanp-valent in the unit disc,b, c are positive integers,a≧max {p−1, 0}. If |a _n|≦Cn ^α for alln=bm+c,m=1, 2, …, then |a _n|≦C′n ^α for alln. This is a marked extension of results due to Goluzin and to Hayman.     

Some geometrical properties of Banach spaces of polynomials

We investigate the asymmetry, gl constants and best factorization estimates of then-dimensional spaces of polynomialsH _p ⁿ =span{e ^ikx;k=1,2,…,n} equipped with theL _p norm for 1≦p≦∞. Supported in part by NSF grant # MCS-8109561.     

Nouvelle demonstration d’un theoreme de J. L. Krivine sur la finie representation del p dans un espace de Banach

We prove that if (x _n) is a sequence in a Banach space with infinite dimensional span, thenc _o orl _p for al<=p<∞ is block finitely represented in (x _n).     

Sums of almost equal prime cubes

We prove that almost all integers N satisfying some necessary congruence conditions are the sum of j almost equal prime cubes with j = 5; 6; 7; 8, i.e., N = p ₁³ + ... + p _j ³ with |p _i − (N/j)^1/3| ≦ $ N^{1/3 - \delta _j + \varepsilon } $ N^{1/3 - \delta _j + \varepsilon } (1 ≦ i ≦ j), for δ _j = 1/45; 1/30; 1/25; 2/45, respectively.     

An extension of a result of ribe

It is proved that for every 1≦p<∞, 1≦q<∞ and for every sequence {p _n}, 1≦p _n<∞,p _n→p, the spaceX=(Σ⊕l _{p n}) _q (resp.U=(Σ⊕L _{p n}(0, 1)) _q ) is uniformly homeomorphic toX⊕l _p (resp.U⊕L _p(0, 1)). This extends Ribe’s result from the casep=1 to generalp<∞ and thus provides examples of uniformly convex, uniformly homeomorphic Banach spaces which are not Lipschitz equivalent.     

Symmetric block bases of sequences with large average growth

We show that if 0<ε≦1, 1≦p<2 andx ₁, …,x _n is a sequence of unit vectors in a normed spaceX such thatE ‖∑ _l ⁿ ε_i x _l‖≧n ^1/p, then one can find a block basisy ₁, …,y _m ofx ₁, …,x _n which is (1+ε)-symmetric and has cardinality at leastγn ^2/p-1(logn)⁻¹, where γ depends on ε only. Two examples are given which show that this bound is close to being best possible. The first is a sequencex ₁, …,x _n satisfying the above conditions with no 2-symmetric block basis of cardinality exceeding 2n ^2/p-1. This sequence is not linearly independent. The second example is a sequence which satisfies a lowerp-estimate but which has no 2-symmetric block basis of cardinality exceedingCn ^2/p-1(logn)^4/3, whereC is an absolute constant. This applies when 1≦p≦3/2. Finally, we obtain improvements of the lower bound when the spaceX containing the sequence satisfies certain type-condition. These results extend results of Amir and Milman in [1] and [2]. We include an appendix giving a simple counterexample to a question about norm-attaining operators.     

Approximating common fixed points of asymptotically nonexpansive mappings by composite algorithm in Banach spaces

Let E be a uniformly convex Banach space and K a nonempty convex closed subset which is also a nonexpansive retract of E. Let T ₁, T ₂ and T ₃: K → E be asymptotically nonexpansive mappings with {k _n }, {l _n } and {j _n }. [1, ∞) such that Σ _n=1 ^∞ (k _n − 1) < ∞, Σ _n=1 ^∞ (l _n − 1) < ∞ and Σ _n=1 ^∞ (j _n − 1) < ∞, respectively and F nonempty, where F = {x ∈ K: T _1x = T _2x = T ₃ x} = x} denotes the common fixed points set of T ₁, _{T 2} and T ₃. Let {α _n }, {α′ _n } and {α″ _n } be real sequences in (0, 1) and ∈ ≤ {α _n }, {α′ _n }, {α″ _n } ≤ 1 − ∈ for all n ∈ N and some ∈ > 0. Starting from arbitrary x ₁ ∈ K define the sequence {x _n } by

Independent sequences in Banach spaces

In every ∞-dimensional separable Banach spaceX there is a fundamental sequence such that no subsequence of it, which is fundamental inX, is independent (“{x _n} is fundamental inX” meansX=span {x _n}).     

Tame Sets in the Complement of Algebraic Variety

Let A?? ^N be an algebraic variety with dim?A≤N?2. Given discrete sequences {a _j },{b _j }?? ^N \ A with slow growth ( $\sum_{j}{1\over|a_{j}|^{2}}<\infty,\sum_{j}{1\over |b_{j}|^{2}}<\inftyLet A⊂ℂ ^N be an algebraic variety with dim A≤N−2. Given discrete sequences {a _j },{b _j }⊂ℂ ^N \ A with slow growth ( ?_j[1/(|a_j|²)] < ￥,?_j[1/(|b_j|²)] < ￥\sum_{j}{1\over|a_{j}|^{2}}<\infty,\sum_{j}{1\over |b_{j}|^{2}}<\infty ) we construct a holomorphic automorphism F with F(z)=z for all z∈A and F(a _j )=b _j for all j∈ℕ. Additional approximation of a given automorphism on a compact polynomially convex set, fixing A, is also possible. Given unbounded analytic variety A there is a tame set E such that F(E)≠{(j,0 ^N−1):j∈ℕ} for all automorphisms F with F| _A =id. As an application we obtain an embedding of a Stein manifold into the complement of an algebraic variety in ℂ ^N with interpolation on a given discrete set.     

Commutators of BMO functions and degenerate Schr?dinger operators with certain nonnegative potentials

Let Lf(x)=-\frac1w?_i,j ?_i(a_i,j(·)?_jf)(x)+V(x)f(x){\mathcal{L}f(x)=-\frac{1}{\omega}\sum_{i,j} \partial_i(a_{i,j}(\cdot)\partial_jf)(x)+V(x)f(x)} with the non-negative potential V belonging to reverse H?lder class with respect to the measure ω(x)dx, where ω(x) satisfies the A ₂ condition of Muckenhoupt and a _i,j (x) is a real symmetric matrix satisfying l^-1w(x)|x|² ￡ ?ⁿ_i,j=1a_i,j(x)x_ix_j ￡ lw(x)|x|².{\lambda^{-1}\omega(x)|\xi|^2\le \sum^n_{i,j=1}a_{i,j}(x)\xi_i\xi_j\le\lambda\omega(x)|\xi|^2. } We obtain some estimates for V^aL^-a{V^{\alpha}\mathcal{L}^{-\alpha}} on the weighted L ^p spaces and we study the weighted L ^p boundedness of the commutator [b, V^a L^-a]{[b, V^{\alpha} \mathcal{L}^{-\alpha}]} when b ? BMO_w{b\in BMO_\omega} and 0 < α ≤ 1.     

Universal non-compact operators between super-reflexive Banach spaces and the existence of a complemented copy of Hilbert space

Suppose that 1<p≦2, 2≦q<∞. The formal identity operatorI:l _p→l _qfactorizes through any given non-compact operator from ap-smooth Banach space into aq-convex Banach space. It follows that ifX is a 2-convex space andY is an infinite dimensional subspace ofX which is isomorphic to a Hilbert space, thenY contains an isomorphic copy ofl ₂ which is complemented inX.     

On generalized shift bases for the wiener disc algebra

LetW(D) denote the set of functionsf(z)=Σ _n=0 ^∞ A _n Z ⁿ a _nzⁿ for which Σ_n=0 ^∞|a _n |<+∞. Given any finite set lcub;f _i (z)rcub; _i=1 ⁿ inW(D) the following are equivalent: (i) The generalized shift sequence lcub;f ₁(z)z ^kn ,f ₂(z)z ^kn+1, …,f _n (z)z ^(k+1)n−1rcub; _k=0 ^∞ is a basis forW(D) which is equivalent to the basis lcub;z ^m rcub; _m=0 ^∞ . (ii) The generalized shift sequence is complete inW(D), (iii) The function has no zero in |z|≦1, wherew=e ^2πiti /n.     

Asymptotic consistency of fixed-width sequential confidence intervals for a multiple regression function

Summary Letm _n (x) be the recursive kernel estimator of the multiple regression functionm(x)=E[Y|X=x]. For given α (0<α<1) andd>0 we define a certain class of stopping timesN=N(α,d, x) and takeI _N,d (x)=[m _N (x)−d, m _N (x)+d] as a 2d-width confidence interval form(x) at a given pointx. In this paper it is shown that the probability P{m(x)∈I _N,d (x)} converges to α asd tends to zero.     

Fourier and Hermite series estimates of regression functions

Summary In the paper we estimate a regressionm(x)=E {Y|X=x} from a sequence of independent observations (X ₁,Y ₁),…, (X _n, Y_n) of a pair (X, Y) of random variables. We examine an estimate of a type , whereN depends onn andϕ _N is Dirichlet kernel and the kernel associated with the hermite series. Assuming, that E|Y|<∞ and |Y|≦γ≦∞, we give condition for to converge tom(x) at almost allx, provided thatX has a density. if the regression hass derivatives, then converges tom(x) as rapidly asO(nC^{−(2s−1)/4s}) in probability andO(n ^{−(2s−1)/4s} logn) almost completely.     

Arithmetic progressions of zeros of the Riemann zeta function

If the Riemann zeta function vanishes at each point of the finite arithmetic progression {D+inp}_0<|n|<N (D?1/2, p>0), then N<13p if D=1/2, and N<p^1/D-1+o(1) in general.     

Graphs whose every transitive orientation contains almost every relation

Given a graphG onn vertices and a total ordering ≺ ofV(G), the transitive orientation ofG associated with ≺, denotedP(G; ≺), is the partial order onV(G) defined by settingx<y inP(G; ≺) if there is a pathx=x ₁ x ₂…x _r=y inG such thatx ₁ ≺x _j for 1≦i<j≦r. We investigate graphsG such that every transitive orientation ofG contains ₂ ⁿ −o(n ²) relations. We prove that almost everyG _n,p satisfies this requirement if , but almost noG _n,p satisfies the condition if (pn log log logn)/(logn log logn) is bounded. We also show that every graphG withn vertices and at mostcn logn edges has some transitive orientation with fewer than ₂ ⁿ −δ(c)n ² relations. Partially supported by MCS Grant 8104854.     

On Sums of Two Coprime k-th Powers(II)

For fixed k ≥ 3, let E_k(x) denote the error term of the sum ?_{n ￡ x}r_k(n)\sum_{n\le x}\rho_k(n) , where r_k(n) = ?_{n=|m|^k+|l|^k, g.c.d.(m,l)=1}\rho_k(n) = \sum_{n=|m|^k+|l|^k, g.c.d.(m,l)=1} 1. It is proved that if the Riemann hypothesis is true, then E₃(x) << x^331/1254+eE_3(x)\ll x^{331/1254+\varepsilon} , E₄(x) << x^37/184+eE_4(x)\ll x^{37/184+\varepsilon} . A short interval result is also obtained.