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## Lower Bounds for Finite Wavelet and Gabor Systems

### Approximation Theory and Its Applications (2001-03-01) 17: 18-29 , March 01, 2001

*Given* ψ∈*L*^{2}(R) *and a finite sequence* {(*a*_{r},λ_{r})}_{r∈Γ}⫅R^{+}XR *consisting of distinct points, the corresponding wavelet system is the set of functions*
$$\left\{ {\frac{1}{{a_\gamma ^{1/2} }}\phi (\frac{x}{{a_\gamma }} - \lambda _\gamma )\gamma \varepsilon r} \right\}$$
. *We prove that for a dense set of functions* ψ∈*L*^{2}(R) *the wavelet system corresponding to any choice of* {(*a*_{r},λ_{r})}_{r∈Γ}*is linearly independent, and we derive explicite estimates for the corresponding lower* (*frame*) *bounds. In particular, this puts restrictions on the choice of a scaling function in the theory for multiresolution analysis. We also obtain estimates for the lower bound for Gabor systems*
$$\left\{ {e^{2rie_{\gamma x} } g(x - \lambda _\gamma )} \right\}\gamma \varepsilon r$$
*for functions g in a dense subset of L*^{2}(R).

## An Algebraic Method for Pole Placement in Multivariable Systems

### Approximation Theory and Its Applications (2001-06-01) 17: 64-85 , June 01, 2001

This paper considers the pole placement in multivariable systems involving known delays by using dynamic controllers subject to multirate sampling. The controller parameterizations are calculated from algebraic equations which are solved by using the Kronecker product of matrices. It is pointed out that the sampling periods can be selected in a convenient way for the solvability of such equations under rather weak conditions provided that the continuous plant is spectrally controllable. Some overview about the use of nonuniform sampling is also given in order to improve the system's performance.

## Weighted Inequalities for Certain Maximal Functions in Orlicz Spaces

### Approximation Theory and Its Applications (2001-12-01) 17: 65-76 , December 01, 2001

Let M_{g} be the maximal operator defined by
$$M_g f\left( x \right) = \sup \frac{{\int_a^b {f\left( y \right)g\left( y \right){\text{d}}y} }}{{\int_a^b {g\left( y \right){\text{d}}y} }}$$
, where g is a positive locally integrable function on R and the supremum is taken over all intervals [a,b] such that 0≤a≤x≤b/η(b−a), here η is a non-increasing function such that η (0) = 1 and
$$\mathop {{\text{lim}}}\limits_{t \to {\text{ + }}\infty } \eta \left( t \right) = 0$$
η (t) = 0. This maximal function was introduced by H. Aimar and L. L. Forzani [AF]. Let Φ be an N - function such that Φ and its complementary N - function satisfy Δ_{2}. It gives an A′_{Φ}(g) type characterization for the pairs of weights (u,v) such that the weak type inequality
$$u\left( {\left\{ {x \in {\text{R}}\left| {M_g f\left( x \right) >\lambda } \right.} \right\}} \right) \leqslant \frac{C}{{\Phi \left( \lambda \right)}}\int_{\text{R}} {\Phi \left( {\left| f \right|v} \right)} $$
holds for every f in the Orlicz space L_{Φ}(v). And, there are no (nontrivial) weights w for which (w,w) satisfies the condition A′_{Φ}(g).

## On the Uniform Convergence of the Generalized Bieberbach Polynomials in Regions with K-Quasiconformal Boundary

### Approximation Theory and Its Applications (2001-03-01) 17: 97-105 , March 01, 2001

*Let G be a finite domain in the complex plane with K-quasicon formal boundary, z*
_{0}
*be an arbitrary fixed point in G and p*>0. *Let* π(*z*) *be the conformal mapping from G onto the disk with radius r*_{0}>0 *and centered at the origin* 0, *normalized by* ϕ(*z*_{0}) = 0 *and* ϕ(*z*_{0}) = 1. *Let us set*
$$\varphi _p \left( z \right): = \int_{x_0 }^x {\left[ {\phi \left( \zeta \right)} \right]^{2/8} } d\zeta $$
, *and let* π_{n,p}(*z*) *be the generalized Bieberbach polynomial of degree n for the pair* (*G,z*_{0}) *that minimizes the integral*
$$\iint\limits_c {\left| {\varphi _p \left( z \right) - P_x^1 (z)} \right|^p d0_x }$$
*in the class*
$$\mathop \prod \limits_n $$
*of all polynomials of degree* ≤ *n**and satisfying the conditions P*_{n}(*z*_{0}) = 0 *and P*′_{n}(*z*_{0}) = 1. *In this work we prove the uniform convergence of the generalized Bieberbach polynomials* π_{n,p}(*z*) *to* ϕ_{p}(*z*) *on*
$$\bar G$$
*in case of*
$$p > 2 - \frac{{K^2 + 1}}{{2K^4 }}$$
.

## From Bounded Families of Localized Cosines to Bi-Orthogonal Riesz Bases via Shift-Invariance

### Approximation Theory and Its Applications (2001-03-01) 17: 30-47 , March 01, 2001

*The notion of bi-inner product functionals*
$$P(f,g) = \sum\limits_n{<f,f_n>
<g,g_n>}$$
*generated by two Bessel sequences* {*f*_{n}} *and* {*g*_{n}} *of functions from L*^{2}*was introduced in our earlier work as a vehicle to identify dual frames and bi-orthogonal Riesz bases of L*^{2}. *The objective was to find conditions under which P is a constant mu<iple of the inner product* <*f,g*> *of L*^{2}. *A necessary and suffici condition derived in is that P is both spatial shift-invariant and phase shift-invariant. A<hough these two shift-invariance properties are, in general, unrelated, it could happen that one is a consequence of the other for certain clases of Bessel sequences* {*f*_{n}} *and* {*g*_{n}}. *In this paper, we show that, indeed, for localized cosines with two-overlapping windoes* (*i.e., only adjacent window functions are allowed to overlap*), *spatial shift-invariance of P is already sufficient to guarantee that P is a constant mu<iple of the inner product, while phase shift-invariance is not. Hence, phase shift-invariance of P for two-overlapping localized cosine Bessel sequences is a consequence of spatial shift-invariance, but the converse is not valid. As an application, we also show that two families of localized cosines with uniformly bounded and two-overlapping windows are bi-orthogonal Riesz bases of L*^{2}, *if and only if P is spatial shift-invariant. In addition, we apply this resu< to generalize a resu< on characterization of dual localized cosine bases in our earlier work in to the mu<ivariate setting. A method for computing the dual windows is also given in this paper.*

## A Korovkin-Type Result in C k an Application to the M n Operators

### Approximation Theory and Its Applications (2001-09-01) 17: 1-13 , September 01, 2001

In this work we present a result about the approximation of the k-th derivative of a function by means of a linear operator under assumptions related to shape preserving properties. As a consequence we deduce new results about the Meyer-König and Zeller operators.

## Convergence and Rate of Approximation in BVΦ for a Class of Integral Operators

### Approximation Theory and Its Applications (2001-12-01) 17: 17-35 , December 01, 2001

We obtain estimates and convergence results with respect to ϕ-variation in spaces BV_{Φ} for a class of linear integral operators whose kernels satisfy a general homogeneity condition. Rates of approximation are also obtained. As applications, we apply our general theory to the case of Mellin convolution operators, to that one of moment operators and finally to a class of operators of fractional order.

## The Integral Formula for Calculating the Hausdorff Measure of Some Fractal Sets

### Approximation Theory and Its Applications (2001-03-01) 17: 70-75 , March 01, 2001

It is important to calculate the Hausdorff dimension and the Hausdorff mesure respect to this dimension for some fractal sets. By using the usual method of “Mass Distribution”, we can only calculate the Hausdorff dimension. In this paper, we will construct an integral formula by using lower inverse s-density and then use it to calculate the Hausdorff measures for some fractional dimensional sets.

## A New Method for the Construction of Multivariate Minimal Interpolation Polynomial

### Approximation Theory and Its Applications (2001-03-01) 17: 10-17 , March 01, 2001

The extended Hermite interpolation problem on segment points set over n-dimensional Euclidean space is considered. Based on the algorithm to compute the Gröbner basis of Ideal given by dual basis a new method to construct minimal multivariate polynomial which satisfies the interpolation conditions is given.

## Two New FCT Algorithms Based on Product System

### Approximation Theory and Its Applications (2001-09-01) 17: 33-42 , September 01, 2001

In this paper we present a product system and give a representation for consine functions with the system. Based on the formula two new algorithms are designed for computing the Discrete Cosine Transform. Both algorithms have regular recursive structure and good numerical stability and are easy to parallize.