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## 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.

## The Equivalence Theory of Native Spaces

### Approximation Theory and Its Applications (2001-03-01) 17: 76-96 , March 01, 2001

In this paper some equivalence definitions are given for native spaces which were introduced by Madych and Nelson and have become influential in the theory of radial basis functions. The abstract elements in native spaces are interpreted. Moreover, Weinrich and Iske's theories are unified.

## 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).

## 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.

## Some Rough Operators on Product Spaces

### Approximation Theory and Its Applications (2001-03-01) 17: 48-69 , March 01, 2001

In this survey report, we shall mainly summarize some recent progress, interesting problems and typical methods used in the theory related to rough Marcinkiewicz integrals and rough singular integrals on product spaces. In addition, we give new proofs for some known results.

## 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.

## A Super-Halley Type Approximation in Banach Spaces

### Approximation Theory and Its Applications (2001-09-01) 17: 14-25 , September 01, 2001

The Super-Halley method is one of the best known third-order iteration for solving nonlnear equations. A Newton-like method which is an approximation of this method is studied. Our approach yields a fourth R-order iterative process which is more efficient than its classical predecessor. We establish a Newton-Kantorovich-type convergence theorem using a new system of recurrence relations, and give an explicit expression for the a priori error bound of the iteration.

## On Nonlinear Coapproximation in Banach Spaces

### Approximation Theory and Its Applications (2001-06-01) 17: 54-63 , June 01, 2001

The problem of nonlinear coapproximation in Banach spaces is considered, and characterization results and strong coapproximation results are presented.

## 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).