Complexity of implementation of Boolean functions by real formulas

Upper complexity estimates are proved for implementation of Boolean functions by formulas in bases consisting of a finite number of continuous real functions and a continuum of constants. For some bases upper complexity estimates coincide with lower ones.     

N–Term Approximation in Anisotropic Function Spaces

In L₂(0, 1)²) infinitely many different biorthogonal wavelet bases may be introduced by taking tensor products of one–dimensional biorthogonal wavelet bases on the interval (0, 1). Most well–known are the standard tensor product bases and the hyperbolic bases. In [23, 24] further biorthogonal wavelet bases are introduced, which provide wavelet characterizations for functions in anisotropic Besov spaces. Here we address the following question: Which of those biorthogonal tensor product wavelet bases is the most appropriate one for approximating nonlinearly functions from anisotropic Besov spaces? It turns out, that the hyperbolic bases lead to nonlinear algorithms which converge as fast as the corresponding schemes with respect to specific anisotropy adapted bases.     

Error Estimates and Reproduction of Polynomials for Biorthogonal Local Trigonometric Bases

A general approach for biorthogonal local trigonometric bases in the two-overlapping setting was given by Chui and Shi. In this paper, we give error estimates for the approximation with such basis functions. In particular, it is shown that for a partition of the real axis into small intervals one obtains better approximation order if polynomials are reproduced locally. Furthermore, smooth trigonometric bases are constructed, which reproduce constants resp. linear functions by only one resp. a small number of basis functions for each interval.     

Wavelets of Wilson Type with Arbitrary Shapes

Motivated by the Gaussian bases of Coifman and Meyer and the need of bases with arbitrary shapes which may have to be different at different locations, we derive complete characterizations of window functions and their duals for localization of all appropriate sines and cosines that give rise to biorthogonal Schauder bases, Riesz bases, and frames. In addition, when the window functions are simply integer translates of a single window function, we give an explicit formulation of its dual that generates the biorthogonal basis, regardless of the shape and support of the window function. Besides the Coifman–Meyer Gaussian bases, several other examples of wavelets of Wilson type are given.     

Band-limited refinable functions for wavelets and framelets

Extending band-limited constructions of orthonormal refinable functions, a special class of periodic functions is used to generate a family of band-limited refinable functions. Characterizations of Riesz bases and frames formed by integer shifts of these refinable functions are obtained. Such families of refinable functions are employed to construct band-limited biorthogonal wavelet bases and biframes with desirable time-frequency localization.     

Read-once functions with hard-to-test projections

An effect of an increase in minimum test length for functions under constant substitutions of constants instead of variables in a checking test problem for read-once functions is described. A family of bases is described, and sequences of functions that are read-once in these bases and have projections whose testing requires more vectors than these functions themselves are constructed.     

Linear Approximation and Reproduction of Polynomials by Wilson Bases

Wilson bases are constituted by trigonometric functions multiplied by translates of a window function with good time frequency localization. In this article we investigate the approximation of functions from Sobolev spaces by partial sums of the Wilson basis expansion. In particular, we show that the approximation can be improved if polynomials are reproduced. We give examples of Wilson bases, which reproduce linear functions with the lowest-frequency term only.     

Algebra of Calderón-Zygmund Operators Associated to Para-Accretive Functions

By use of special wavelet bases associated to accretive or pseudo-accretive functions, it was proved that all Calderón-Zygmund operators satisfying certain conditions form an algebra. In this article, a similar result is proved for more general para-accretive functions. Since wavelet bases are not available for this general setting, the new idea used here is to apply the discrete Calderón-type reproducing formula associated to para-accretive functions developed in [14]. This new method can be applied to many other problems, where wavelet bases are not available.     

Optimized Local Trigonometric Bases with Nonuniform Partitions

The authors provide optimized local trigonometric bases with nonuniform partitions which efficiently compress trigonometric functions. Numerical examples demonstrate that in many cases the proposed bases provide better compression than the optimized bases with uniform partitions obtained by Matviyenko.     

О ДИФФЕРЕНцИРОВАНИИ ИНтЕгРАлОВ От ФУНкцИ И Иж сИММЕтРИЧНых пРОс тРАНстВ ДИФФЕРЕНцИАльНыМИ Б АжИсАМИ

We consider new methods of constructing differential bases. Symmetric spaces with essentially different fundamental functions at zero can be defined by means of differential bases. Even the Lorentz and Marcinkiewicz or the Lebesgue and Marcinkiewicz spaces can be defined by means of differential bases.     

Radial basis approximation and its application to biharmonic equation

Order of approximating functions and their derivatives by radial bases on arbitrarily scattered data is derived. And then radial bases are used to construct solutions of biharmonic equations that approximate potential integrals for the exact solutions with the order of approximation derived.     

On the Problem of the Flow of an Ideal Gas around Bodies

We construct biorthogonal bases of spaces of an n-separate multiresolution analysis and wavelets for n scaling functions. Fast algorithms are presented for finding the coefficients of expansions of functions in such bases.     

The weighted dual functions for Wang-Bézier type generalized Ball bases and their applications

By introducing the inner-product matrix of two vector functions and using conversion matrix, explicit formulas for the dual basis functions of Wang-Bézier type generalized Ball bases (WBGB) with respect to the Jacobi weight function are given. The dual basis functions with and without boundary constraints are also considered. As a result, the paper includes the weighted dual basis functions of Bernstein basis, Wang-Ball basis and some intermediate bases. Dual functionals for WBGB and the least square approximation polynomials are also obtained.     

Algebras of Functionally Invariant Solutions of the Three-Dimensional Laplace Equation

In commutative associative third-rank algebras with principal identity over a complex field, we select bases such that hypercomplex monogenic functions constructed in these bases have components satisfying the three-dimensional Laplace equation. The notion of monogeneity for these functions is similar to the notion of monogeneity in the complex plane.     

Positive Refinable Operators

Recently, linear positive operators of Bernstein–Schoenberg type, relative to B-splines bases, have been considered. The properties of these operators are derived mainly from the total positivity of normalized B-spline bases. In this paper we shall construct a generalization of the operator considered in [15] by means of normalized totally positive bases generated by a particular class of totally positive scaling functions. Next, we shall study its approximation properties. Our results can be established also for more general sequences of normalized totally positive bases.     

Classical symmetric functions in superspace

We present the basic elements of a generalization of symmetric function theory involving functions of commuting and anticommuting (Grassmannian) variables. These new functions, called symmetric functions in superspace, are invariant under the diagonal action of the symmetric group on the sets of commuting and anticommuting variables. In this work, we present the superspace extension of the classical bases, namely, the monomial symmetric functions, the elementary symmetric functions, the completely symmetric functions, and the power sums. Various basic results, such as the generating functions for the multiplicative bases, Cauchy formulas, involution operations as well as the combinatorial scalar product are also generalized.     

Optimized Local Trigonometric Bases

This paper generalizes Malvar–Coifman–Meyer (MCM) wavelets by extending the choice of bell functions. We dispense with the orthonormality of MCM wavelets to produce a family of smooth local trigonometric bases that efficiently compress trigonometric functions. Any such basis is, in general, not orthogonal, but any element of the dual basis differs from the corresponding element of the original basis only by the shape of the bell. Furthermore, in our scheme the bell functions are bounded by 1 and the dual bell functions are bounded by (2^1/2+ 1)/2 ≈1.2. These bounds ensure the numerical stability of the forward and the inverse transformations in these bases. Numerical examples demonstrate that in many cases the proposed bases provide substantially better (up to a factor of two) compression than the standard MCM wavelets.     

Structure of Regular Solutions of the Three-Body Schrödinger Equation near the Pair Impact Point

We investigate the six-dimensional Schrödinger equation for a three-body system with central pair interactions of a more general form than Coulomb interactions. Regular general and special physical solutions of this equation are represented by infinite asymptotic series in integer powers of the distance between two particles and in the sought functions of the other three-body coordinates. Constructing such functions in angular bases composed of spherical and bispherical harmonics or symmetrized Wigner D-functions is reduced to solving simple recursive algebraic equations. For projections of physical solutions on the angular bases functions, we derive boundary conditions at the pair impact point.     

Orthonormal polynomial wavelets on the interval

We use special functions and orthonormal wavelet bases on the real line to construct wavelet-like bases. With these wavelets we can construct polynomial bases on the interval; moreover, we can use them for the numerical resolution of degenerate elliptic operators.

     

Maximal functions with respect to differential bases measuring mean oscillation

In this paper we study maximal sharp functions associated with arbitrary differential bases. The definition of these functions goes back to the papers by F. John (1965), and by C. Fefferman and E. M. Stein (1972), where the classical bases consisting of cubic intervals were considered. We obtain conditions imposed on the basis, under which inequalities, known earlier in the case of a basis of cubes, are valid for the considered maximal functions. The main results are formulated in terms of nonincreasing rearrangements. In the capacity of applications, we obtain estimates of the rearrangements of subadditive operators acting in BMO. In particular, the estimate for the Hilbert transform, obtained earlier by C. Bennett and K. Rudnick, follows.