Wavelet Multipliers Research Articles

Multipliers, Phases and Connectivity of MRA Wavelets in L 2(ℝ2)

Let A be any 2×2 real expansive matrix. For any A-dilation wavelet ψ, let \(\widehat{\psi}\) be its Fourier transform. A measurable function f is called an A-dilation wavelet multiplier if the inverse Fourier transform of \((f\widehat{\psi})\) is an A-dilation wavelet for any A-dilation wavelet ψ. In this paper, we give a complete characterization of all A-dilation wavelet multipliers under the condition that A is a 2×2 matrix with integer entries and |{det }(A)|=2. Using this result, we are able to characterize the phases of A-dilation wavelets and prove that the set of all A-dilation MRA wavelets is path-connected under the L2(ℝ2) norm topology for any such matrix A.

HILBERT–SCHMIDT OPERATORS AND FRAMES — CLASSIFICATION, BEST APPROXIMATION BY MULTIPLIERS AND ALGORITHMS

In this paper we deal with the theory of Hilbert–Schmidt operators, when the usual choice of orthonormal basis, on the associated Hilbert spaces, is replaced by frames. We More precisely, we provide a necessary and sufficient condition for an operator to be Hilbert–Schmidt, based on its action on the elements of a frame (i.e. an operator T is [Formula: see text] if and only if the sum of the squared norms of T applied on the elements of the frame is finite). Also, we construct Bessel sequences, frames and Riesz bases of [Formula: see text] operators using tensor products of the same sequences in the associated Hilbert spaces. We state how the [Formula: see text] inner product of an arbitrary operator and a rank one operator can be calculated in an efficient way; and we use this result to provide a numerically efficient algorithm to find the best approximation, in the Hilbert–Schmidt sense, of an arbitrary matrix, by a so-called frame multiplier (i.e. an operator which act diagonally on the frame analysis coefficients). Finally, we give some simple examples using Gabor and wavelet frames, introducing in this way wavelet multipliers.

Wavelet multipliers on 𝐿^{𝑝}(ℝⁿ)

We give results on the boundedness and compactness of wavelet multipliers on L p ( R n ) , 1 ≤ p ≤ ∞ L^p(\mathbb {R}^n),\,1\leq p\leq \infty .

Nonstationary Gaussian processes in wavelet domain: Synthesis, estimation, and significance testing

We propose an equivalence class of nonstationary Gaussian stochastic processes defined in the wavelet domain. These processes are characterized by means of wavelet multipliers and exhibit well-defined time-dependent spectral properties. They allow one to generate realizations of any wavelet spectrum. Based on this framework, we study the estimation of continuous wavelet spectra, i.e., we calculate variance and bias of arbitrary estimated continuous wavelet spectra. Finally, we develop an areawise significance test for continuous wavelet spectra to overcome the difficulties of multiple testing; it uses basic properties of continuous wavelet transform to decide whether a pointwise significant result is a real feature of the process or indistinguishable from typical stochastic fluctuations. This test is compared to the conventional one in terms of sensitivity and specificity. A software package for continuous wavelet spectral analysis and synthesis is presented.

Formula omitted]-Boundedness of localization operators associated to Bessel left regular representations

Motivated from Wong [M.W. Wong, L p -Boundedness of localization operators associated to left regular representations, Proc. Amer. Math. Soc. 130 (2002) 2911–2929], an L μ p -boundedness of localization operators associated to Bessel's left regular representations of locally compact Hausdorff group is obtained and an application of Bessel wavelet multipliers is investigated by using Zemanian theory of Hankel transformation.

$L^{p}$-boundedness of wavelet multipliers

We give conditions on the symbol σ in L ∞ (R n ) to ensure that the corresponding wavelet multiplier is a bounded linear operator on L 2 (R n ) and on L p (R n ), 4/3 < p < 4.

TRACE CLASS NORM INEQUALITIES FOR WAVELET MULTIPLIERS

In this paper, the authors give sharp estimates on the norms in the trace class of wavelet multipliers Pσ,ϕ:L2(Rn)→L2(Rn) in terms of the symbols σ and the admissible wavelets ϕ. 2000 Mathematics Subject Classification 47G10.

On a class of bidimensional nonseparable wavelet multipliers

Let E= 1 1 1 −1 or 0 2 1 0 . A measurable function v is called an E- wavelet multiplier if (v ψ ̂ ) ̆ is an E-wavelet whenever ψ is an E-wavelet. In this paper, some characterizations of E-wavelet multiplier are obtained. As an application of these techniques, we prove that the set of E-wavelets is arcwise connected.

Wavelet multipliers and signals

AbstractThe Schatten-von Neumann property of a pseudo-differential operator is established by showing that the pseudo-differential operator is a multiplier defined by means of an admissible wavelet associated to a unitary representation of the additive group Rn on the C*-algebra of all bounded linear operators from L2(Rn) into L2(Rn). A bounded linear operator on L2(R) arising in the Landau, Pollak and Slepian model in signal analysis is shown to be a wavelet multiplier studied in this paper.