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Proceedings of the American Mathematical Society, 2018
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2003
As we have seen, a frame {f k } k=1 ∞ in a Hilbert space H has one of the main properties of a basis: given f ∈ H, there exist coefficients {c k } k=1 ∞ ∈ l 2(ℕ) such that f = ∑ k=1 ∞ c k f k . This makes it natural to study the relationship between frames and bases. We have already seen that Riesz bases are frames.
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As we have seen, a frame {f k } k=1 ∞ in a Hilbert space H has one of the main properties of a basis: given f ∈ H, there exist coefficients {c k } k=1 ∞ ∈ l 2(ℕ) such that f = ∑ k=1 ∞ c k f k . This makes it natural to study the relationship between frames and bases. We have already seen that Riesz bases are frames.
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Macro-element Hierarchical Riesz Bases
2014We show that a nested sequence of C r macro-element spline spaces on quasi-uniform triangulations gives rise to hierarchical Riesz bases of Sobolev spaces H s (Ω ...
Oleg Davydov, Wee Ping Yeo
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Weyl-Heisenberg Riesz Bases Generated by Two Intervals
Journal of Fourier Analysis and Applications, 2012The authors consider Weyl-Heisenberg systems (also called Gabor systems) \[ (g,a,b)=\{ e^{2\pi i m bx} g(x-na) \}_{m, n \in \mathbb Z} \] with \(a=b=1\). They study two problems: I. Characterize the function \(\chi_{E}\), where \(E\) is a union of two separated intervals, so that \((\chi_{E},1,1)\) is a Riesz basis for \(L^{2}(\mathbb R)\).
He, Xing-Gang, Li, Hai-Xiong
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Steerable Wavelet Frames Based on the Riesz Transform
IEEE Transactions on Image Processing, 2010We consider an extension of the 1-D concept of analytical wavelet to n-D which is by construction compatible with rotations. This extension, called a monogenic wavelet, yields a decomposition of the wavelet coefficients into amplitude, phase, and phase direction. The monogenic wavelet is based on the hypercomplex monogenic signal which is defined using
Stefan, Held +3 more
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Edge Detection Based on Riesz Transform
2016In this paper, we present a new way of 2D feature extraction. We start by showing the direct link that exist between the Riesz Transform (RT) and the gradient and Laplacian operators. This formulation allows us to interpret the RT as a gradient of a smoothed image.
Ahror Belaid +2 more
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Perturbation Theorems for Frames and Riesz Bases
Applied Mechanics and Materials, 2013This paper gives a perturbation theorem for frames in a Hilbert space which is a generalization of a result by Ping Zhao. It is proved that the condition a linear operator is invertible can be weakened to be surjective, and a similar result also be obtained for a Riesz basis.
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Affine Riesz bases and the dual function
Sbornik: Mathematics, 2016Some parts of the abstract are as follows: ``This paper is concerned with a system of functions on the unit interval which are generated by dyadic dilations and integer translations of a given function. [\dots] Conditions, and in some particular cases, criteria for the generating function are given for the system to be Besselian, to form a Riesz basis ...
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Riesz bases of normalized reproducing kernels in Fock type spaces
Analysis and Mathematical Physics, 2021Konstantin Isaev
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Finding duality for Riesz bases of exponentials on multi-tiles
Applied and Computational Harmonic Analysis, 2021Christina Frederick +1 more
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