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The Feichtinger conjecture is considered for three special families of frames. It is shown that if a wavelet frame satisfies a certain weak regularity condition, then it can be written as the finite union of Riesz basic sequences each of which is a wavelet system. Moreover, the above is not true for general wavelet frames. It is also shown that a… (More)

- Marcin Bownik
- 2007

Using the range function approach to shift invariant spaces in L 2 (R n) we give a simple characterization of frames and Riesz families generated by shifts of a countable set of generators in terms of their behavior on subspaces of l 2 (Z n). This in turn gives a simpliied approach to the analysis of frames and Riesz families done by Gramians and dual… (More)

- Marcin Bownik
- 1997

In this paper we deal with multidimensional wavelets arising from a multiresolution analysis with an arbitrary dilation matrix A, namely we have scaling equations ' s (x) =

- Marcin Bownik, Darrin Speegle
- Journal of Approximation Theory
- 2002

- Marcin Bownik, Pierre G. Lemarié-Rieusset
- 2003

We investigate Riesz wavelets in the context of generalized multiresolution analysis (GMRA). In particular, we show that Zalik's class of Riesz wavelets obtained by an MRA is the same as the class of biorthogonal wavelets associated with an MRA.

Let A 1 and A 2 be expansive dilations, respectively, on R n and R m. Let A ≡ (A 1 , A 2) and A p (A) be the class of product Muckenhoupt weights on R n × R m for p ∈ (1, ∞]. When p ∈ (1, ∞) and w ∈ A p (A), the authors characterize the weighted Lebesgue space L p w (R n × R m) via the anisotropic Lusin-area function associated with A. When p ∈ (0, 1], w ∈… (More)

Let S be a shift-invariant subspace of L 2 (R n) defined by N generators and suppose that its length L, the minimal number of generators of S, is smaller than N. Then we show that at least one reduced family of generators can always be obtained by a linear combination of the original generators, without using translations. In fact, we prove that almost… (More)

- Marcin Bownik
- 2007

For weights in the matricial Muckenhoupt classes we investigate a number of properties analogous to properties which hold in the scalar Muckenhoupt classes. In contrast to the scalar case we exhibit for each p, 1 < p < 1, a matrix weight W 2 A p;q n S p 0 <p A p 0 ;q 0. We also give a necessary and suucient condition on W in A p;q , a \reverse inverse… (More)

- ÁRPÁD BÉNYI, MARCIN BOWNIK
- 2010

We define homogeneous classes of x-dependent anisotropic symbols ˙ S m γ,δ (A) in the framework determined by an expansive dilation A, thus extending the existing theory for diagonal dilations. We revisit anisotropic analogues of Hörmander-Mihlin multipliers introduced by Rivì ere [22] and provide direct proofs of their boundedness on Lebesgue and Hardy… (More)

- Marcin Bownik
- 2007

We give a characterization of all (quasi) aane frames in L 2 (R n) which have a (quasi) aane dual in terms of the two simple equations in the Fourier transform domain. In particular, if the dual frame is the same as the original system, i.e. it is a tight frame, we obtain the well known characterization of wavelets. Although these equations have already… (More)