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We shall construct a generating set of a nonfinitely generated Ga-invariant ring given in Freudenburg's counterexample by making use of an integral sequence which was constructed inductively by Freudenburg.
      
The classical Rudin-Shapiro construction produces a sequence of polynomials
      
A Generalized Sampling Theorem with the Inverse of an Arbitrary Square Summable Sequence as Sampling Points
      
In this article a generalized sampling theorem using an arbitrary sequence of sampling points is derived.
      
The mathematical concept of frames is utilized in the analysis of the properties of the sequence of sampling functions.
      
The frame operator of this sequence is expressed as a matrix-valued function multiplying a vector-valued function.
      
Using the matrix approach we prove that the sequence of sampling functions is always complete in the cases of critical sampling and oversampling.
      
A sufficient condition for the sequence of sampling functions to constitute a frame is derived.
      
When p is finite, a sequence {λn} of complex numbers will be called aframe forEp provided the inequalities hold for some positive constants A and B and all functions f inEp.
      
We say that {λn} is aninterpolating sequence forEp if the set of all scalar sequences {f (λn)}, with f εEp, coincides with ?p.
      
If in addition {λn} is a set of uniqueness forEp, that is, if the relations f(λn)=0(-∞>amp;lt;n>amp;lt;∞), with f εEp, imply that f ≡0, then we call {λn} acomplete interpolating sequence.
      
We generalize the celebrated theorem of Stein on the maximal operator of a sequence of translation invariant operators, from the scalar case to vector valued functions.
      
We show how any discrete wavelet transform or two band subband filtering with finite filters can be decomposed into a finite sequence of simple filtering steps, which we call lifting steps but that are also known as ladder structures.
      
We develop here a part of the existence theory for the inhomogeneous refinement equation where a (k) is a finite sequence and F is a compactly supported distribution on ?.
      
Given a positive definite ?1 sequence of matrices {cj}j∈S we prove that there is a positive definite matrix function f in the Wiener algebra on the bitorus such that the Fourier coefficients equal ck for k ∈ S.
      
The problem is to find a positive constant L such that for any real sequence {μn}n∈? with |μn -λn| ≤δ >amp;lt;L, is also a frame for L2[-π, π].
      
If the sequence of functions ?j, k is a wavelet frame (Riesz basis) or Gabor frame (Riesz basis), we obtain its perturbation system ψj,k which is still a frame (Riesz basis) under very mild conditions.
      
We estimate ∥f∥p from above by C∥f∥p,n and give an explicit value for C depending only on p, τ, and characteristic parameters of the sequence {tn}n∈?.
      
Let Δ denote the symmetric Laplacian on the Sierpinski gasket SG defined by Kigami [11] as a renormalized limit of graph Laplacians on the sequence of pregaskets Gm whose limit is SG.
      
We examine the asymptotics and explicitly construct a sequence of finite-length filters with the same asymptotics as the sequence of finite-length minimizers.
      
 

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