Nonuniform Sampling and Recovery of?Multidimensional Bandlimited Functions by?Gaussian Radial-Basis Functions

Let S⊂ℝ ^d be a bounded subset with positive Lebesgue measure. The Paley-Wiener space associated to S, PW _S , is defined to be the set of all square-integrable functions on ℝ ^d whose Fourier transforms vanish outside S. A sequence (x _j :j∈ℕ) in ℝ ^d is said to be a Riesz-basis sequence for L ₂(S) (equivalently, a complete interpolating sequence for PW _S ) if the sequence (e^-iáx_j,·?:j ? \mathbb N)(e^{-i\langle x_{j},\cdot \rangle }:j\in \mathbb {N}) of exponential functions forms a Riesz basis for L ₂(S). Let (x _j :j∈ℕ) be a Riesz-basis sequence for L ₂(S). Given λ>0 and f∈PW _S , there is a unique sequence (a _j ) in ℓ ₂ such that the function

On limitedness in locally convex spaces

Archiv der Mathematik -     

Asymptotic properties of Banach spaces under renormings

It is shown that a separable Banach space can be given an equivalent norm with the following properties: If is relatively weakly compact and , then converges in norm. This yields a characterization of reflexivity once proposed by V.D. Milman. In addition it is shown that some spreading model of a sequence in is 1-equivalent to the unit vector basis of (respectively, ) implies that contains an isomorph of (respectively, ).

     

A Banach space block finitely universal for monotone bases

A reflexive Banach space with a basis is constructed having the property that every monotone basis is block finitely representable in each block basis of .

     

Greedy Bases for Besov Spaces

We prove that the Banach space (?_n=1^￥l_pⁿ)_lq(\bigoplus_{n=1}^{\infty}\ell_{p}^{n})_{\ell_{q}}, which is isomorphic to certain Besov spaces, has a greedy basis whenever 1≤p≤∞ and 1<q<∞. Furthermore, the Banach spaces (?_n=1^￥l_pⁿ)_l1(\bigoplus_{n=1}^{\infty}\ell _{p}^{n})_{\ell_{1}}, with 1<p≤∞, and (?_n=1^￥l_pⁿ)_c0(\bigoplus_{n=1}^{\infty}\ell_{p}^{n})_{c_{0}}, with 1≤p<∞, do not have a greedy basis. We prove as well that the space (?_n=1^￥l_pⁿ)_lq(\bigoplus_{n=1}^{\infty}\ell _{p}^{n})_{\ell_{q}} has a 1-greedy basis if and only if 1≤p=q≤∞.     

Shift invariant preduals of ℓ 1(ℤ)

The Banach space ? ₁(?) admits many non-isomorphic preduals, for example, C(K) for any compact countable space K, along with many more exotic Banach spaces. In this paper, we impose an extra condition: the predual must make the bilateral shift on ? ₁(?) weak*-continuous. This is equivalent to making the natural convolution multiplication on ? ₁(?) separately weak*-continuous and so turning ? ₁(?) into a dual Banach algebra. We call such preduals shift-invariant. It is known that the only shift-invariant predual arising from the standard duality between C ₀(K) (for countable locally compact K) and ? ₁(?) is c ₀(?). We provide an explicit construction of an uncountable family of distinct preduals which do make the bilateral shift weak*-continuous. Using Szlenk index arguments, we show that merely as Banach spaces, these are all isomorphic to c ₀. We then build some theory to study such preduals, showing that they arise from certain semigroup compactifications of ?. This allows us to produce a large number of other examples, including non-isometric preduals, and preduals which are not Banach space isomorphic to c ₀.     

The distortion of Hilbert space

The unit sphere of Hilbert space, ₂, is shown to contain a remarkable sequence of nearly orthogonal sets. Precisely, there exist a sequence of sets of norm one elements of ₂, (C _i ) _i=1 , and reals _i 0 so that a) each setC _i has nonempty intersection with every infinite dimensional closed subspace of ₂ and b) forij,xC, andyC _j , |x, y|<_{min(i, j)} E. Odell was partially supported by NSF and TARP. Th. Schlumprecht was partially supported by NSF and LEQSF.     

Coefficient Quantization in Banach Spaces

Let (e _i ) be a dictionary for a separable infinite-dimensional Banach space X. We consider the problem of approximation by linear combinations of dictionary elements with quantized coefficients drawn usually from a ‘finite alphabet’. We investigate several approximation properties of this type and connect them to the Banach space geometry of X. The existence of a total minimal system with one of these properties, namely the coefficient quantization property, is shown to be equivalent to X containing c ₀. We also show that, for every ε>0, the unit ball of every separable infinite-dimensional Banach space X contains a dictionary (x _i ) such that the additive group generated by (x _i ) is (3+ε)⁻¹-separated and 1/3-dense in X.      

A universal reflexive space for the class of uniformly convex Banach spaces

We show that there exists a separable reflexive Banach space into which every separable uniformly convex Banach space isomorphically embeds. This solves problems raised by J. Bourgain [B] in 1980 and by W. B. Johnson in 1977 [Jo]. We also give intrinsic characterizations of separable reflexive Banach spaces which embed into a reflexive space with a block q-Hilbertian and/or a block p-Besselian finite dimensional decomposition. Dedicated to the memory of V. I. Gurarii Research supported by the National Science Foundation.