Uniform partitions of frames of exponentials into Riesz sequences

The Feichtinger conjecture, if true, would have as a corollary that for each set E⊂[0,1] and Λ⊂Z, there is a partition Λ₁,…,ΛN of Z such that for each 1?i?N, is a Riesz sequence. In this paper, sufficient conditions on sets E⊂[0,1] and Λ⊂R are given so that can be uniformly partitioned into Riesz sequences.     

ON THE BOX DIMENSION FOR A CLASS OF NONAFFINE FRACTAL INTERPOLATION FUNCTIONS

We present lower and upper bounds for the box dimension of the graphs of certain nonaffine fractal interpolation functions by generalizing the results that hold for the affine case.     

Bessel sequences,wavelet frames,duals and extensions

From the perspectives of duality and extensions, Gabor frames and wavelet frames have contrasting behaviour. Our chief concern here is about duality. Canonical duals of wavelet frames may not be wavelet frames, whereas canonical duals of Gabor frames are Gabor frames. Keeping these in view, we give several constructions of wavelet frames with wavelet canonical duals. For this, a simple characterisation of Bessel sequences and a general commutativity result are given, the former also leading naturally to some extension results.     

Variation and Minkowski dimension of fractal interpolation surface

The fractal interpolation surface on the rectangular domain is discussed in this paper. We study the properties of the oscillation and the variation of bivariate continuous functions. Then we discuss the special properties of bivariate fractal interpolation function, and estimate the value of its variation. Using the relation between the Minkowski dimension of the graph of continuous function and its variation, we obtain the exact value of the Minkowski dimension of the fractal interpolation surface.     

一类分形插值曲面的可微性

在矩形域上,构造一类迭代函数系,并由此生成一类分形曲面,分别给出该分形曲面几乎处处可微和不可微点的条件,得到相应的结论.     

Basic definition and properties of Bessel multipliers

This paper introduces the concept of Bessel multipliers. These operators are defined by a fixed multiplication pattern, which is inserted between the analysis and synthesis operators. The proposed concept unifies the approach used for Gabor multipliers for arbitrary analysis/synthesis systems, which form Bessel sequences, like wavelet or irregular Gabor frames. The basic properties of this class of operators are investigated. In particular the implications of summability properties of the symbol for the membership of the corresponding operators in certain operator classes are specified. As a special case the multipliers for Riesz bases are examined and it is shown that multipliers in this case can be easily composed and inverted. Finally the continuous dependence of a Bessel multiplier on the parameters (i.e., the involved sequences and the symbol in use) is verified, using a special measure of similarity of sequences.     

Riesz s-equilibrium measures on d-dimensional fractal sets as s approaches d

Let A be a compact set in Rp of Hausdorff dimension d. For s∈(0,d), the Riesz s-equilibrium measure μs,A is the unique Borel probability measure with support in A that minimizes the double integral over the Riesz s-kernel |x−y|^−s over all such probability measures. In this paper we show that if A is a strictly self-similar d-fractal, then μs,A converges in the weak-star topology to normalized d-dimensional Hausdorff measure restricted to A as s approaches d from below.     

Sub-Riemannian vs. Euclidean dimension comparison and fractal geometry on Carnot groups

We solve Gromov's dimension comparison problem for Hausdorff and box counting dimension on Carnot groups equipped with a Carnot-Carathéodory metric and an adapted Euclidean metric. The proofs use sharp covering theorems relating optimal mutual coverings of Euclidean and Carnot-Carathéodory balls, and elements of sub-Riemannian fractal geometry associated to horizontal self-similar iterated function systems on Carnot groups. Inspired by Falconer's work on almost sure dimensions of Euclidean self-affine fractals we show that Carnot-Carathéodory self-similar fractals are almost surely horizontal. As a consequence we obtain explicit dimension formulae for invariant sets of Euclidean iterated function systems of polynomial type. Jet space Carnot groups provide a rich source of examples.     

Beurling dimension and self-similar measures

In this paper the authors study the Beurling dimension of Bessel sets and frame spectra of some self-similar measures on ${\mathbb{R}}^d$ and obtain their exact upper bound of the dimensions, which is the same given by Dutkay et al. (2011) [8]. The upper bound is attained in usual cases and some examples are given to explain our theory.     

Vector-valued fractal functions: Fractal dimension and fractional calculus

There are many research available on the study of a real-valued fractal interpolation function and fractal dimension of its graph. In this paper, our main focus is to study the dimensional results for a vector-valued fractal interpolation function and its Riemann–Liouville fractional integral. Here, we give some results which ensure that dimensional results for vector-valued functions are quite different from real-valued functions. We determine interesting bounds for the Hausdorff dimension of the graph of a vector-valued fractal interpolation function. We also obtain bounds for the Hausdorff dimension of the associated invariant measure supported on the graph of a vector-valued fractal interpolation function. Next, we discuss more efficient upper bound for the Hausdorff dimension of measure in terms of probability vector and contraction ratios. Furthermore, we determine some dimensional results for the graph of the Riemann–Liouville fractional integral of a vector-valued fractal interpolation function.     

Orthogonal exponentials on the generalized three-dimensional Sierpinski gasket

The self-affine measure μM,D corresponding to the expanding integer matrix

     

The cardinality of certain -orthogonal exponentials

The self-affine measures μ_M,D corresponding to the case (i) M=pI₃, D={0,e₁,e₂,e₃} in the space and the case (ii) M=pI₂, D={0,e₁,e₂,e₁+e₂} in the plane are non-spectral, where p>1 is odd, I_n is the n×n identity matrix, and e₁,…,e_n are the standard basis of unit column vectors in . One of the non-spectral problem on μ_M,D is to estimate the number of orthogonal exponentials in L²(μ_M,D) and to find them. In the present paper we show that, in both cases (i) and (ii), there are at most 4 mutually orthogonal exponentials in L²(μ_M,D) each, and the number 4 is the best.     

Hilbert空间中的Bessel列构成的 $C^*$-代数 $B_H(I)$

Let H be a separable Hilbert space, B H(I), B(H) and K(H) the sets of all Bessel sequences {f i}i∈I in H, bounded linear operators on H and compact operators on H, respectively. Two kinds of multiplications and involutions are introduced in light of two isometric linear isomorphisms αH : B H(I) → B(?2), β : B H(I) → B(H), respectively, so that B H(I) becomes a unital C*-algebra under each kind of multiplication and involution. It is proved that the two C*-algebras(B H(I), ?, ?) and(B H(I), ·, *) are *-isomorphic. It is also proved that the set F H(I) of all frames for H is a unital multiplicative semi-group and the set R H(I) of all Riesz bases for H is a self-adjoint multiplicative group, as well as the set K H(I) := β-1(K(H)) is the unique proper closed self-adjoint ideal of the C*-algebra B H(I).     

Finite extensions of generalized Bessel sequences to generalized frames

The objective of this paper is to investigate the question of modifying a given generalized Bessel sequence to yield a generalized frame or a tight generalized frame by finite extension. Some necessary and sufficient conditions for the finite extensions of generalized Bessel sequences to generalized frames or tight generalized frames are provided, and every result is illustrated by the corresponding example.     

Closed fractal interpolation surfaces

Based on the construction of bivariate fractal interpolation surfaces, we introduce closed spherical fractal interpolation surfaces. The interpolation takes place in spherical coordinates and with the transformation to Cartesian coordinates a closed surface arises. We give conditions for this construction to be valid and state some useful relations about the Hausdorff and the Box counting dimension of the closed surface.     

Parameter identification of 1D fractal interpolation functions using bounding volumes

Fractal interpolation functions are very useful in capturing data that exhibit an irregular (non-smooth) structure. Two new methods to identify the vertical scaling factors of such functions are presented. In particular, they minimize the area of the symmetric difference between the bounding volumes of the data points and their transformed images. Comparative results with existing methods are given that establish the proposed ones as attractive alternatives. In general, they outperform existing methods for both low and high compression ratios. Moreover, lower and upper bounds for the vertical scaling factors that are computed by the first method are presented.     

Generalized Bessel and Riesz Potentials on Metric Measure Spaces

We introduce generalized Bessel and Riesz potentials on metric measure spaces and the corresponding potential spaces. Estimates of the Bessel and Riesz kernels are given which reflect the intrinsic structure of the spaces. Finally, we state the relationship between Bessel (or Riesz) operators and subordinate semigroups.      

Banach函数空间上的Bessel(Riesz)位势及其应用(I)理论

本文首先引入Ｂｅｓｅｌ（Ｒｉｅｓｚ）位势Ｋ¨ｏｔｈｅ函数空间Ｘｓ（Ｘｓ）的概念，然后讨论一类算子在Ｌｅｂｅｓｇｕｅ－位势Ｋ¨ｏｔｈｅ函数空间Ｌｑ（－Ｔ，Ｔ；Ｘｓ）上的对偶估计．由此我们得到半群ｅｘｐ（ｉｔ（－Δ）ｍ／２）和算子Ａ：＝∫ｔ０ｅｘｐ（ｉ（ｔ－τ）（－Δ）ｍ／２）·ｄτ在Ｌｅｂｅｓｇｕｅ－Ｂｅｓｏｖ空间Ｌｑ－Ｔ，Ｔ；·Ｂｓｐ，２中的一些时间－－空间Ｌｐ－Ｌｐ′估计．本文的系列文将给出这些估计的应用     

A new idea to construct the fractal interpolation function

The aim of this paper is to present a new idea to construct the nonlinear fractal interpolation function, in which we exploit the Matkowski and the Rakotch fixed point theorems. Our technique is different from the methods presented in the previous literatures.