Approximation of functions by higher order Szegö kernels I. Complex variable cases

Wang, Jinxun; Qian, Tao

doi:10.1080/17476933.2014.956740

Cited by 6 publications

(9 citation statements)

References 17 publications

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“…Observe that in complex analysis the TM systems for the unit disc and upper half space can be generated by Szegö or higher order Szegö kernels through GS orthogonalization process ( [21]), heuristically, we propose the following definition.…”

Section: Takenaka-malmquist Systems In Higher Dimensionsmentioning

confidence: 99%

Orthogonalization in Clifford Hilbert modules and applications

Wang¹,

Qian²

2021

Preprint

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We prove that the Gram-Schmidt orthogonalization process can be carried out in Hilbert modules over Clifford algebras, in spite of the uninvertibility and the un-commutativity of general Clifford numbers. Then we give two crucial applications of the orthogonalization method. One is to give a constructive proof of existence of an orthonormal basis of the inner spherical monogenics of order k for each k ∈ N. The second is to formulate the Clifford Takenaka-Malmquist systems, or in other words, the Clifford rational orthogonal systems, as well as define Clifford Blaschke product functions, in both the unit ball and the half space contexts. The Clifford TM systems then are further used to establish an adaptive rational approximation theory for L 2 functions on the sphere and in R m .

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Section: Takenaka-malmquist Systems In Higher Dimensionsmentioning

confidence: 99%

Orthogonalization in Clifford Hilbert modules and applications

Wang¹,

Qian²

2021

Preprint

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“…It automatically generates a fast converging orthogonal expansion of which each entry has a meaningful instantaneous frequency. It has several variations, namely cyclic AFD, unwinding AFD, and be generalized to multi-dimensions with the Clifford and several complex variables setting with scalar-to matrix-valued signals ( [17,27,35,36,30,1,2]). In particular a variation called unwinding Blaschke expansion was studied by Coifman, Steinerberger and Peyriére making further connections with Blaschke products and outer functions ( [6,7]), being also separately developed in [18], and further developed in a recent paper on maximally unwinding AFD [28].…”

Section: Stochastic Afdsmentioning

confidence: 99%

A Sparse Representation of Random Signals

Qian

2020

Preprint

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Studies of sparse representations of deterministic signals have been well developed. Among other types there exists one called the adaptive Fourier decomposition (AFD) type for the analytic Hardy spaces. This type is recently further extended to the context of Hilbert spaces with a dictionary. Through the Hardy space decomposition of the space of L 2 -signals the AFD type algorithm gives rise to sparse representations of signals of finite energy. To deal with multivariate signals the Hilbert space context comes into play. The multivariate AFD counterpart in Hilbert spaces with a dictionary is called pre-orthogonal AFD (POAFD). In the present study we generalize AFD and POAFD to random analytic signals through formulating stochastic analytic Hardy spaces. To analyze random analytic signals we work on two models, both being called stochastic AFD, or SAFD in brief. The two models are respectively made for (i) expressible as the sum of a deterministic signal and an error term such as a white noise (SAFDI); and for (ii) being random analytic signals divided into different classes of signals obeying certain distributive law (SAFDII). In the second half of the paper we drop the analyticity assumption and generalize the SAFDI and SAFDII to what we call stochastic Hilbert spaces with a dictionary. The generalized methods are named as stochastic pre-orthogonal adaptive Fourier decompositions, SPOAFDI and SPOAFDII. Like the deterministic AFDs and POAFDs, the developed stochastic POAFD algorithms offer powerful tools to analyze random signals.

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“…In Section 3, we perform AMUCD to 2 special cases, the Hardy space

H^{2} false(double-struckD false)

and the Paley‐Wiener space

W false(\frac{π}{h} false), h > 0

. For

H^{2} false(double-struckD false)

, we show that BVC holds (see also Wang and Qian). For

W false(\frac{π}{h} false)

, we are able to show a weak BVC property.…”

Section: Introductionmentioning

confidence: 96%

Aveiro method in reproducing kernel Hilbert spaces under complete dictionary

Mai

Qian

2017

Math Methods in App Sciences

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Aveiro method is a sparse representation method in reproducing kernel Hilbert spaces, which gives orthogonal projections in linear combinations of reproducing kernels over uniqueness sets. It, however, suffers from determination of uniqueness sets in the underlying reproducing kernel Hilbert space. In fact, in general spaces, uniqueness sets are not easy to be identified, let alone the convergence speed aspect with Aveiro method. To avoid those difficulties, we propose an new Aveiro method based on a dictionary and the matching pursuit idea. What we do, in fact, are more: The new Aveiro method will be in relation to the recently proposed, the so-called pre-orthogonal greedy algorithm involving completion of a given dictionary. The new method is called Aveiro method under complete dictionary. The complete dictionary consists of all directional derivatives of the underlying reproducing kernels.We show that, under the boundary vanishing condition bring available for the classical Hardy and Paley-Wiener spaces, the complete dictionary enables an efficient expansion of any given element in the Hilbert space. The proposed method reveals new and advanced aspects in both the Aveiro method and the greedy algorithm.

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Approximation of functions by higher order Szegö kernels I. Complex variable cases

Cited by 6 publications

References 17 publications

Orthogonalization in Clifford Hilbert modules and applications

Orthogonalization in Clifford Hilbert modules and applications

A Sparse Representation of Random Signals

Aveiro method in reproducing kernel Hilbert spaces under complete dictionary

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