Greedy Bases Are Best for m-Term Approximation

We study the approximation of a subset K in a Banach space X by choosing first basis B and then using n-term approximation. Into the competition for best bases we enter all unconditional bases for X. We show that if the subset K⊂X is well aligned with respect to a greedy basis B, then, in a certain sense, this basis is the best for this type of approximation. Our result improves the recent result of DeVore, Petrova and Temlyakov.      

p-Adic Haar Multiresolution Analysis and Pseudo-Differential Operators

The notion of p-adic multiresolution analysis (MRA) is introduced. We discuss a “natural” refinement equation whose solution (a refinable function) is the characteristic function of the unit disc. This equation reflects the fact that the characteristic function of the unit disc is a sum of p characteristic functions of mutually disjoint discs of radius p ⁻¹. This refinement equation generates a MRA. The case p=2 is studied in detail. Our MRA is a 2-adic analog of the real Haar MRA. But in contrast to the real setting, the refinable function generating our Haar MRA is 1-periodic, which never holds for real refinable functions. This fact implies that there exist infinity many different 2-adic orthonormal wavelet bases in ℒ²(ℚ₂) generated by the same Haar MRA. All of these new bases are described. We also constructed infinity many different multidimensional 2-adic Haar orthonormal wavelet bases for ℒ²(ℚ₂ ⁿ ) by means of the tensor product of one-dimensional MRAs. We also study connections between wavelet analysis and spectral analysis of pseudo-differential operators. A criterion for multidimensional p-adic wavelets to be eigenfunctions for a pseudo-differential operator (in the Lizorkin space) is derived. We proved also that these wavelets are eigenfunctions of the Taibleson multidimensional fractional operator. These facts create the necessary prerequisites for intensive using our wavelet bases in applications. Our results related to the pseudo-differential operators develop the investigations started in Albeverio et al. (J. Fourier Anal. Appl. 12(4):393–425, 2006).      

Bases for Cooperations in K-Theory

Gaussian polynomials are used to define bases with good multiplicative properties for the algebra K _*(K) of cooperations in K-theory and for the invariants under conjugation.     

Regular submodules of torsion modules over a discrete valuation domain

A submodule W of a p-primary module M of bounded order is known to be regular if W and M have simultaneous bases. In this paper we derive necessary and sufficient conditions for regularity of a submodule.     

Frames and bases in tensor products of Hilbert spaces and Hilbert <Emphasis Type="Italic">C</Emphasis>*-modules

In this article, we study tensor product of Hilbert C*-modules and Hilbert spaces. We show that if E is a Hilbert A-module and F is a Hilbert B-module, then tensor product of frames (orthonormal bases) for E and F produce frames (orthonormal bases) for Hilbert A ⊗ B-module E ⊗ F, and we get more results. For Hilbert spaces H and K, we study tensor product of frames of subspaces for H and K, tensor product of resolutions of the identities of H and K, and tensor product of frame representations for H and K.     

-Operator frames for a Banach space

In this paper, (p,Y)-Bessel operator sequences, operator frames and (p,Y)-Riesz bases for a Banach space X are introduced and discussed as generalizations of the usual concepts for a Hilbert space and of the g-frames. It is proved that the set of all (p,Y)-Bessel operator sequences for a Banach space X is a Banach space and isometrically isomorphic to the operator space B(X,ℓ^p(Y)). Some necessary and sufficient conditions for a sequence of operators to be a (p,Y)-Bessel operator sequence are given. Also, a characterization of an independent (p,Y)-operator frame for X is obtained. Lastly, it is shown that an independent (p,Y)-operator frame for X is just a (p,Y)-Riesz basis for X and has a unique dual (q,Y^*)-operator frame for X^*.     

A bibasic hypergeometric transformation associated with combinatorial identities of the Rogers-Ramanujan type

During the last five decades, a number of combinatorial generalizations and interpretations have occurred for the identities of the Rogers-Ramanujan type. The object of this paper is to give a most general known analytic auxiliary functional generalization which can be used to give combinatorial interpretations of generalizedq-identities of the Rogers-Ramanujan type. The derivation realise the theory of basic hypergeometric series with two unconnected bases.     

Bit-Parallel Arithmetic Implementations over Finite Fields GF(2 m ) with Reconfigurable Hardware

Galois (or finite) fields are used in a wide number of technical applications, playing an important role in several areas such as cryptographic schemes and algebraic codes, used in modern digital communication systems. Finite field arithmetic must be fast, due to the increasing performance needed by communication systems, so it might be necessary for the implementation of the modules performing arithmetic over Galois fields on semiconductor integrated circuits. Galois field multiplication is the most costly arithmetic operation and different approaches can be used. In this paper, the fundamentals of Galois fields are reviewed and multiplication of finite-field elements using three different representation bases are considered. These three multipliers have been implemented using a bit-parallel architecture over reconfigurable hardware and experimental results are presented to compare the performance of these multipliers.     

Algorithms for shortest paths and d-cycle problems

Let G be a weighted graph with n vertices and m edges. We address the d-cycle problem, i.e., the problem of finding a subgraph of minimum weight with given cyclomatic number d. Hartvigsen [Minimum path bases, J. Algorithms 15 (1993) 125–142] presented an algorithm with running time O(n²m) and O(n^2d−1m²) for the cyclomatic numbers d=1 and d2, respectively. Using a (d+1)-shortest-paths algorithm, we develop a new more efficient algorithm for the d-cycle problem with running time O(n^2d−1+n²m+n³logn).     

Ring-Shaped Functions and Wigner 6j-Symbols

We obtain an explicit expression for the ring-shaped matrix relating the ring-shaped functions corresponding to different values of an axiality parameter and find the relation between this matrix and the Wigner 6j-symbols. We investigate the motion of a quantum particle in a ring-shaped model with a zero “bare” potential and find bases factored in spherical and cylindrical coordinates for this model. We derive a formula generalizing the Rayleigh expansion for a plane wave in terms of spherical waves in a ring-shaped model. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 2, pp. 299–310, February, 2006.     

Some p-Ranks Related to Orthogonal Spaces

We determine the p-rank of the incidence matrix of hyperplanes of PG(n, p ^e) and points of a nondegenerate quadric. This yields new bounds for ovoids and the size of caps in finite orthogonal spaces. In particular, we show the nonexistence of ovoids in and . We also give slightly weaker bounds for more general finite classical polar spaces. Another application is the determination of certain explicit bases for the code of PG(2, p) using secants, or tangents and passants, of a nondegenerate conic.     

Construction of optimally conditioned cubic spline wavelets on the interval

The paper is concerned with a construction of new spline-wavelet bases on the interval. The resulting bases generate multiresolution analyses on the unit interval with the desired number of vanishing wavelet moments for primal and dual wavelets. Both primal and dual wavelets have compact support. Inner wavelets are translated and dilated versions of well-known wavelets designed by Cohen, Daubechies, and Feauveau. Our objective is to construct interval spline-wavelet bases with the condition number which is close to the condition number of the spline wavelet bases on the real line, especially in the case of the cubic spline wavelets. We show that the constructed set of functions is indeed a Riesz basis for the space L ² ([0, 1]) and for the Sobolev space H ^s ([0, 1]) for a certain range of s. Then we adapt the primal bases to the homogeneous Dirichlet boundary conditions of the first order and the dual bases to the complementary boundary conditions. Quantitative properties of the constructed bases are presented. Finally, we compare the efficiency of an adaptive wavelet scheme for several spline-wavelet bases and we show a superiority of our construction. Numerical examples are presented for the one-dimensional and two-dimensional Poisson equations where the solution has steep gradients.     

Wilson Bases and Modulation Spaces

We show that the recently discovered WILSON bases of exponential decay are unconditional bases for all modulation spaces on R, including the classical BESSEL potential spaces, the Segal algebra S_o, and the SCHWARTZ space. As a consequence we obtain new bases for spaces of entire functions. On the other hand, the WILSON bases are no unconditional bases for the ordinary L^p-spaces for p ≠ 2.     

Bases de type schauder deC[0,1] et formules de quadrature associées

Résumé On caractérise deux familles de bases deC[0,1] et l'on étudie les formules de quadrature associées. On montre en particulier que les formules de quadrature de Romberg proviennent d'une suite de bases engendrées par des polynômes.

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Bases of schauder type inC[0, 1] and associated quadrature formulas

Summary We characterize two families of bases ofC[0,1] and we study the associated quadrature formulae. In particular, we prove that the Romberg quadrature formulae come from a sequence of bases generated by polynomials.

     

On the structure of the set bases of a degenerate point

Consider an extreme point (EP)x ⁰ of a convex polyhedron defined by a set of linear inequalities. If the basic solution corresponding tox ⁰ is degenerate,x ⁰ is called a degenerate EP. Corresponding tox ⁰, there are several bases. We will characterize the set of all bases associated withx ⁰, denoted byB ⁰. The setB ⁰ can be divided into two classes, (i) boundary bases and (ii) interior bases. For eachB ⁰, there is a corresponding undirected graphG ⁰, in which there exists a tree which connects all the boundary bases. Some other properties are investigated, and open questions for further research are listed, such as the connection between the structure ofG ⁰ and cycling (e.g., in linear programs).     

A general construction of nonseparable multivariate orthonormal wavelet bases

The construction of nonseparable and compactly supported orthonormal wavelet bases of L ²(R ⁿ ); n ≥ 2, is still a challenging and an open research problem. In this paper, we provide a special method for the construction of such wavelet bases. The wavelets constructed by this method are dyadic wavelets. Also, we show that our proposed method can be adapted for an eventual construction of multidimensional orthogonal multiwavelet matrix masks, candidates for generating multidimensional multiwavelet bases.      

Base tensorielle des matrices de Hankel (ou de Toeplitz) Applications

Summary The set of all Hankel (or Toeplitz) matrices of dimensionn, is shown to possess tensorial bases: bases made ofn rank one matrices. Four families of such tensorial bases are possible. From this result, we deduce that the following computations can be performed with a number of multiplications of ordern instead of ordern ²: evaluation of the 2n+1 coefficients of the polynomial product of two polynomials of degreen, evaluation of the inverse of a lower triangular toeplitz matrix, evaluation of the quotient and of the remainder in the division of a polynomial of degree 2n by a polynomial of degreen.     

-adic refinable functions and MRA-based wavelets

The main goal of this paper is the development of the MRA theory in . We described a wide class of p-adic refinement equations generating p-adic multiresolution analyses. A method for the construction of p-adic orthogonal wavelet bases within the framework of the MRA theory is suggested. A realization of this method is illustrated by an example which gives a new 3-adic wavelet basis. Another realization leads to the p-adic Haar bases which were known before.     

Crystal Bases forUq(osp(1, 2r))

We construct ₂-graded crystal bases for the quantized universal enveloping algebra of the Lie superalgebraosp(1, 2r). We show that, like the crystal bases in the Lie algebra case, these crystal bases carry a remarkable combinatorial structure.     

Spaces with Almost Strictly Totally Positive Bases

In spline spaces there are often totally positive bases possessing a strong property called almost strictly total positivity. In this paper, it is proved that, for totally positive bases of continuous functions B, the following concepts are equivalent: (i) B is almost strictly totally positive, (ii) B satifies a Schoenberg-Whitney Theorem, (iii) The functions in B are locally linearly independent. Some classical examples of almost strictly totally positive bases are given, completing the knowledge of their properties known in the mathematical literature. Some criteria to know the existence of almost strictly totally positive bases are also derived.