A dimension invariance property of finite frames of translates and Gabor frames is discussed. Under appropriate support conditions among the frame and dual frame generating functions, we show that a pair of dual frames evaluated in a given space remains a valid dual set if they are naturally embedded in the underlying space of almost arbitrarily enlarged dimension. Consequently, the evaluation ...

We construct discrete analogs of multidimensional singular integral operators and study their invertibility. Moreover, we give a comparison between continual case. the theory periodic Riemann problem also, because it is needed for studying invertibility so-called paired equations. For more general case pseudo-differential operators, solvability equations in Sobolev–Slobodetskii spaces. Some res...

We study the efficient numerical solution of infinite matrix equations Au = f for a matrix A in the Jaffard algebra. These matrices appear naturally via frame discretizations in many applications such as Gabor analysis, sampling theory, and quasi-diagonalization of pseudo-differential operators in the weighted Sjöstrand class. The proposed algorithm has two main features: firstly, it converges ...

Ionic liquid 1,3-disulfonic acid imidazolium trifluoroacetate ([Dsim][TFA]) has been exploited as a highly efficient catalyst for the one-pot pseudo five-component reaction of phenylhydrazine (2 eq.) with ethyl acetoacetate (2 eq.) and arylaldehydes (1 eq.) in ethanol (reflux conditions). In this reaction, 4,4′‐(arylmethylene)‐bis(3‐methyl‐1‐phenyl‐1H‐ pyrazol‐5‐ol)s have been obtained in high ...

In this note we will show that the so called Sobolev dual is the minimizer over all linear reconstructions using dual frames for stable r order Σ∆ quantization schemes under the so called White Noise Hypothesis (WNH) design criteria. We compute some Sobolev duals for common frames and apply them to audio clips to test their performance against canonical duals and another alternate dual correspo...

Approximately dual frames as a generalization of duality notion in Hilbert spaces have applications Gabor systems, wavelets, coorbit theory, and sensor modeling. In recent years, the computing associated deviations canonical alternate from original ones has been considered by functional analysts. this paper, we consider quantitative measurement approximately ones. More precisely, it is proved t...

In this article, we discuss some recent developments on pairs of dual wavelet frames in Sobolev spaces constructed by the Oblique Extension Principle (OEP) from refinable function vectors. Our investigation sheds a new light on understanding the OEP method for constructing MRA dual wavelet frames.

Starting from any two compactly supported d-refinable function vectors in ( L2(R) )r with multiplicity r and dilation factor d, we show that it is always possible to construct 2rd wavelet functions with compact support such that they generate a pair of dual d-wavelet frames in L2(R) and they achieve the best possible orders of vanishing moments. When all the components of the two real-valued d-...

It is well known that the category KHaus of compact Hausdorff spaces is dually equivalent to the category KRFrm of compact regular frames. By de Vries duality, KHaus is also dually equivalent to the category DeV of de Vries algebras, and so DeV is equivalent to KRFrm, where the latter equivalence can be described constructively through Booleanization. Our purpose here is to lift this circle of ...

Abstract. Dual Gramian analysis is one of the fundamental tools developed in a series of papers [37, 40, 38, 39, 42] for studying frames. Using dual Gramian analysis, the frame operator can be represented as a family of matrices composed of the Fourier transforms of the generators of (generalized) shiftinvariant systems, which allows us to characterize most properties of frames and tight frames...