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Register a new userReconstructing real-valued functions from unsigned coefficients with respect to wavelet and other frames
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In this paper we consider the following problem of phase retrieval: Given a collection of real-valued band-limited functions ${psi_{lambda}}_{lambdain Lambda}subset L^2(mathbb{R}^d)$ that constitutes a semi-discrete frame, we ask whether any real-valued function $f in L^2(mathbb{R}^d)$ can be uniquely recovered from its unsigned convolutions ${{|f ast psi_lambda|}_{lambda in Lambda}}$. We find that under some mild assumptions on the semi-discrete frame and if $f$ has exponential decay at $infty$, it suffices to know $|f ast psi_lambda|$ on suitably fine lattices to uniquely determine $f$ (up to a global sign factor). We further establish a local stability property of our reconstruction problem. Finally, for two concrete examples of a (discrete) frame of $L^2(mathbb{R}^d)$, $d=1,2$, we show that through sufficient oversampling one obtains a frame such that any real-valued function with exponential decay can be uniquely recovered from its unsigned frame coefficients.
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We develop a natural generalization of vector-valued frame theory, we term operator-valued frame theory, using operator-algebraic methods. This extends work of the second author and D. Han which can be viewed as the multiplicity one case and extends to higher multiplicity (e.g., multiframes) their dilation approach. We prove several results for operator-valued frames concerning their parametrization, duality, disjointeness, complementarity, and composition and the relationship between the two types of similarity (left and right) of such frames. We then apply these notions to prove that the collection of multiframe generators for the action of a discrete group on a Hilbert space is norm pathwise-connected precisely when the von Neumann algebra generated by the right representation of the group has no minimal projections. The proof is obtained by parametrizing this collection by a class of partial isometries in a larger von Neumann algebra. In the multiplicity one case this class reduces to the unitary class which is path-connected in norm, but in the infinite multiplicity case this class is path connected only in the strong operator topology and the proof depends on properties of tensor product slice maps.
The like-Lebesgue integral of real-valued measurable functions (abbreviated as textit{RVM-MI})is the most complete and appropriate integration Theory. Integrals are also defined in abstract spaces since Pettis (1938). In particular, Bochner integrals received much interest with very recent researches. It is very commode to use the textit{RVM-MI} in constructing Bochner integral in Banach or in locally convex spaces. In this simple not, we prove that the Bochner integral and the textit{RVM-MI} with respect to a finite measure $m$ are the same on $mathbb{R}$. Applications of that equality may be useful in weak limits on Banach space.
We establish necessary and sufficient conditions for the stability of the finite section method for operators belonging to a certain $C^*$-algebra of operators acting on the Hilbert space $l^2_H(mathbb{Z})$ of $H$-valued sequences where $H$ is a given Hilbert space. Identifying $l^2_H(mathbb{Z})$ with the $L^2_H$-space over the unit circle, the $C^*$-algebra in question is the one which contains all singular integral operators with flip and piecewise quasicontinous $mathcal{L}(H)$-valued generating functions on the unit circle. The result is a generalization of an older result where the same problem, but without the flip operator was considered. The stability criterion is obtained via $C^*$-algebra methods and says that a sequence of finite sections is stable if and only if certain operators associated with that sequence (via $^*$-homomorphisms) are invertible.
In cite{AV99}, Antoine and Vandergheynst propose a group-theoretic approach to continuous wavelet frames on the sphere. The frame is constructed from a single so-called admissible function by applying the unitary operators associated to a representation of the Lorentz group, which is square-integrable modulo the nilpotent factor of the Iwasawa decomposition. We prove necessary and sufficient conditions for functions on the sphere, which ensure that the corresponding system is a frame. We strengthen a similar result in cite{AV99} by providing a complete and detailed proof.
For any class of operators which transform unary total functions in the set of natural numbers into functions of the same kind, we define what it means for a real function to be uniformly computable or conditionally computable with respect to this class. These two computability notions are natural generalizations of certain notions introduced in a previous paper co-authored by Andreas Weiermann and in another previous paper by the same authors, respectively. Under certain weak assumptions about the class in question, we show that conditional computability is preserved by substitution, that all conditionally computable real functions are locally uniformly computable, and that the ones with compact domains are uniformly computable. The introduced notions have some similarity with the uniform computability and its non-uniform extension considered by Katrin Tent and Martin Ziegler, however, there are also essential differences between the conditional computability and the non-uniform computability in question.
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