Hardy Spaces for Laguerre Expansions

Let be the standard Laguerre functions of type a. We denote . Let and be the semigroups associated with the orthonormal systems and . We say that a function f belongs to the Hardy space associated with one of the semigroups if the corresponding maximal function belongs to . We prove special atomic decompositions of the elements of the Hardy spaces.     

Atomic Decomposition of Hardy Spaces Associated with Certain Laguerre Expansions

Let L _n ^a (x), n=0,1,…, be the Laguerre polynomials of order a>−1. Denote ℓ _n ^a (x)=(n!/Γ(n+a+1))^1/2 L _n ^a (x)e ^−x/2. Let

Hardy's Inequalities for Hermite and Laguerre Expansions

The well-known inequality of Hardy for Fourier coefficientsof functions in the real Hardy space is . We shall establish analogues of this inequality for the Hermite function expansionsand also for the Laguerre function expansions. 1991 MathematicsSubject Classification 42C10, 33C45.     

Laguerre and Disk Polynomial Expansions with Nonnegative Coefficients

We establish Wiener type theorems and Paley type theorems for Laguerre polynomial expansions and disk polynomial expansions with nonnegative coefficients.     

Laplace Type Multipliers for Laguerre Expansions of Hermite Type

We investigate Laplace transform type and Laplace-Stieltjes type multipliers associated to the multi–dimensional Laguerre function expansions of Hermite type. We prove that, under the assumption α _i ≥ ?1/2, α _i ? (?1/2, 1/2), these operators are Calderón-Zygmund operators. Consequently, their mapping properties follow by the general theory.     

Averages Using Translation Induced by Laguerre and Jacobi Expansions

Approximation by averages of the generalized translation induced by Laguerre and Jacobi expansions will be shown to satisfy a strong converse inequality of type B with the appropriate K -functional. April 9, 1998. Date revised: February 22, 1999. Date accepted: March 5, 1999.     

Approximation in Sobolev Spaces by Kernel Expansions

For interpolation of smooth functions by smooth kernels having an expansion into eigenfunctions (e.g., on the circle, the sphere, and the torus), good results including error bounds are known, provided that the smoothness of the function is closely related to that of the kernel. The latter fact is usually quantified by the requirement that the function should lie in the “native” Hilbert space of the kernel, but this assumption rules out the treatment of less smooth functions by smooth kernels. For the approximation of functions from “large” Sobolev spaces W by functions generated by smooth kernels, this paper shows that one gets at least the known order for interpolation with a less smooth kernel that has W as its native space.     

Min-Max Spaces and Complexity Reduction in Min-Max Expansions

Idempotent methods have been found to be extremely helpful in the numerical solution of certain classes of nonlinear control problems. In those methods, one uses the fact that the value function lies in the space of semiconvex functions (in the case of maximizing controllers), and approximates this value using a truncated max-plus basis expansion. In some classes, the value function is actually convex, and then one specifically approximates with suprema (i.e., max-plus sums) of affine functions. Note that the space of convex functions is a max-plus linear space, or moduloid. In extending those concepts to game problems, one finds a different function space, and different algebra, to be appropriate. Here we consider functions which may be represented using infima (i.e., min-max sums) of max-plus affine functions. It is natural to refer to the class of functions so represented as the min-max linear space (or moduloid) of max-plus hypo-convex functions. We examine this space, the associated notion of duality and min-max basis expansions. In using these methods for solution of control problems, and now games, a critical step is complexity-reduction. In particular, one needs to find reduced-complexity expansions which approximate the function as well as possible. We obtain a solution to this complexity-reduction problem in the case of min-max expansions.     

Ismagilov Type Theorems for Linear, Gel′fand and Bernstein n-Widths

Using a variational principle for s-numbers, we obtain estimates for the linear, Gel′fand. and Bernstein n-widths. A simple proof of some results concerned with the exact values of n-widths of diagonal operators is given. We also calculate the exact values at the Bernstein n-widths for the Hardy-Sobolev classes.     

Decomposition of Spaces of Distributions Induced by Hermite Expansions

Decomposition systems with rapidly decaying elements (needlets) based on Hermite functions are introduced and explored. It is proved that the Triebel-Lizorkin and Besov spaces on ℝ ^d induced by Hermite expansions can be characterized in terms of the needlet coefficients. It is also shown that the Hermite-Triebel-Lizorkin and Besov spaces are, in general, different from the respective classical spaces. The first author has been supported by NSF Grant DMS-0709046 and the second author by NSF Grant DMS-0604056.     

Sobolev Type Spaces on the Dual of the Laguerre Hypergroup

Sobolev type spaces E ^s,p (0, sR, p[1,+]) are defined on R×N by using the Fourier transform and its inverse on the Laguerre hypergroup. An analogue of H ^s (R ⁿ ), denoted by H ^s is investigated in this paper. Some properties including completeness and imbedding results for these spaces are given, Reillich-type theorem and Poincaré's inequality are proved. Also, global regularity results for certain differential operators are obtained.     

Lagrange Interpolation at Laguerre Zeros in Some Weighted Uniform Spaces

We introduce an interpolatory process essentially based on the Laguerre zeros and we prove that it is an optimal process in some weighted uniform spaces.     

Almost Diagonal Matrices and Besov-Type Spaces Based on Wavelet Expansions

This paper is concerned with problems in the context of the theoretical foundation of adaptive algorithms for the numerical treatment of operator equations. It is well-known that the analysis of such schemes naturally leads to function spaces of Besov type. But, especially when dealing with equations on non-smooth manifolds, the definition of these spaces is not straightforward. Nevertheless, motivated by applications, recently Besov-type spaces \(B^\alpha _{\Psi ,q}(L_p(\Gamma ))\) on certain two-dimensional, patchwise smooth surfaces were defined and employed successfully. In the present paper, we extend this definition (based on wavelet expansions) to a quite general class of d-dimensional manifolds and investigate some analytical properties of the resulting quasi-Banach spaces. In particular, we prove that different prominent constructions of biorthogonal wavelet systems \(\Psi \) on domains or manifolds \(\Gamma \) which admit a decomposition into smooth patches actually generate the same Besov-type function spaces \(B^\alpha _{\Psi ,q}(L_p(\Gamma ))\), provided that their univariate ingredients possess a sufficiently large order of cancellation and regularity. For this purpose, a theory of almost diagonal matrices on related sequence spaces \(b^\alpha _{p,q}(\nabla )\) of Besov type is developed.     

Fuzzy度量空间扩张型映象的不动点定理

本文给出Ｆｕｚｚｙ度量空间一些扩张型映象的不动点定理,这些结果发展和改进了普通度量空间中相应的结果。     

Fuzzy度量空间型映象的不动点定理

本文给出Ｆｕｚｚｙ度量空间一些扩张型映象的不动点定理,这些结果发展和改进了普通度量空间中相应的结果。     

Characterization of UMD Banach Spaces by Imaginary Powers of Hermite and Laguerre Operators

In this paper we characterize the Banach spaces with the UMD property by means of L ^p -boundedness properties for the imaginary powers of the Hermite and Laguerre operators. In order to do this we need to obtain pointwise representations for the Laplace transform type multipliers associated with Hermite and Laguerre operators.     

Sampling Expansions and Interpolation in Unitarily Translation Invariant Reproducing Kernel Hilbert Spaces

Sufficient conditions are established in order that, for a fixed infinite set of sampling points on the full line, a function satisfies a sampling theorem on a suitable closed subspace of a unitarily translation invariant reproducing kernel Hilbert space. A number of examples of such reproducing kernel Hilbert spaces and the corresponding sampling expansions are given. Sampling theorems for functions on the half-line are also established in RKHS using Riesz bases in subspaces of L ²(R ⁺).     

Eigenfunction Expansions on Geodesic Balls and Rank One Symmetric Spaces of Compact Type

We find sharp conditions for the pointwise convergence ofeigenfunction expansions associated with the Laplace operator and otherrotationally invariant differential operators. Specifically, we considerthis problem for expansions associated with certain radially symmetricoperators and general boundary conditions and the problem in the contextof Jacobi polynomial expansions. The latter has immediate application toFourier series on rank one symmetric spaces of compact type.