A Theory on Non-Constant Frequency Decompositions and Applications

Chen, Qiuhui; Qian, Tao; Tan, Lei

doi:10.1007/978-3-030-40120-7_1

Cited by 15 publications

(31 citation statements)

References 93 publications

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“…Under some mild condition we have the same convergent rate O(n −1/2 ) as K-L expansion ( [43,44,15]).…”

Section: Poafd and Spoafd Algorithmsmentioning

confidence: 83%

“…The Bwownian bridge example shows that the K-L expansion converging rate for general random fields should not be better than M √ n , because, in fact, for the Brownian bridge case M = 1 π 2 ([28]). In a Hilbert space with a dictionary with boundary vanishing condition (BVC) the convergence rate for the AFD type expansions of f is dominated by M √ n , where M is any bound of ∞ k=1 |c k | for f = ∞ k=1 c k e a k ( [15,60]). This shows that SAFD is at least as fast as the K-L method in approximating a random signal.…”

Section: Discussionmentioning

confidence: 99%

“…We note that the solution spaces h 2 (B 1 ) and H 2 heat (R 1+n + )) are RKHSs ( [57]), briefly denoted as H 2 in the sequel. The idea of POAFD algorithm was initiated in [51] and further formulated in [43] and [44] (also see [15]). The work [46] and [47] carry out the POAFD algorithm to the weighted Bergman spaces in the unit disc, and the work [45] carries out it to the h 2 (B 1 ) and H 2 heat (R 1+n + ) spaces which are all for deterministic signals.…”

Section: Dirichlet Problem With Stochastic Boundary Value and Cauchy ...mentioning

confidence: 99%

“…(see [44,15]). Iteratively applying the above process to G, one obtains a sequence of such maximally selected {q k } ∞ k=1 , and eventually has…”

Section: Poafd and Spoafd Algorithmsmentioning

confidence: 99%

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Sparse Representations of Solutions to a class of Random Boundary Value Problems

Yang¹,

Chen²,

Chen³

et al. 2021

Preprint

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In the present study, we consider sparse representations of solutions to Dirichlet and heat equation problems with random boundary or initial conditions. To analyze the random signals, two types of sparse representations are developed, namely stochastic pre-orthogonal adaptive Fourier decomposition 1 and 2 (SPOAFD1 and SPOAFD2). Due to adaptive parameter selecting of SPOAFDs at each step, we obtain analytical sparse solutions of the SPDE problems with fast convergence.

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“…Under some mild condition we have the same convergent rate O(n −1/2 ) as K-L expansion ( [43,44,15]).…”

Section: Poafd and Spoafd Algorithmsmentioning

confidence: 83%

Section: Discussionmentioning

confidence: 99%

Section: Dirichlet Problem With Stochastic Boundary Value and Cauchy ...mentioning

confidence: 99%

“…(see [44,15]). Iteratively applying the above process to G, one obtains a sequence of such maximally selected {q k } ∞ k=1 , and eventually has…”

Section: Poafd and Spoafd Algorithmsmentioning

confidence: 99%

See 2 more Smart Citations

Sparse Representations of Solutions to a class of Random Boundary Value Problems

Yang¹,

Chen²,

Chen³

et al. 2021

Preprint

Self Cite

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“…The lemma may be extended to the cases where a 1 , • • • , a m may have multiplicities. In the extended cases the concept of a multiple reproducing kernel is involved ( [1]).…”

Section: Scheme I: Phase Retrieval Based On 1-intensity Measurements Andmentioning

confidence: 99%

Analytic Phase Retrieval Based on Intensity Measurements

Qian

Deng

et al. 2021

Acta Math Sci

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This paper concerns the reconstruction of a function f in the Hardy space of the unit disc D by using a sample value f (a) and certain n-intensity measurements | f, Ea 1 •••an |, where a1, • • • , an ∈ D, and Ea 1 •••an is the n-th term of the Gram-Schmidt orthogonalization of the Szegö kernels ka 1 , • • • , ka n , or their multiple forms. Three schemes are presented.The first two schemes each directly obtain all the function values f (z). In the first one we use Nevanlinna's inner and outer function factorization which merely requires the 1-intensity measurements equivalent to know the modulus |f (z)|. In the second scheme we do not use deep complex analysis, but require some 2-and 3-intensity measurements. The third scheme, as an application of AFD, gives sparse representation of f (z) converging quickly in the energy sense, depending on consecutively selected maximal n-intensity measurements | f, Ea 1 •••an |.

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Sparse representations of random signals

Qian

2021

Math Methods in App Sciences

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Sparse (fast) representations of deterministic signals have been well studied. Among other types, there exists one called adaptive Fourier decomposition (AFD) for functions in analytic Hardy spaces. Through the Hardy space decomposition of the L 2 -space, the AFD algorithm also gives rise to sparse representations of signals of finite energy. To deal with multivariate signals, the general Hilbert space context comes into play. The multivariate counterpart of AFD in general Hilbert spaces with a dictionary has been named preorthogonal AFD (POAFD). In the present study, we generalize AFD and POAFD to random analytic signals through formulating stochastic analytic Hardy spaces and stochastic Hilbert 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) those expressible as the sum of a deterministic signal and an error term (SAFDI) and for (ii) those from different sources obeying certain distributive law (SAFDII). In the later part of the paper, we drop off 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 POAFDs, SPOAFDI and SPOAFDII. Like AFDs and POAFDs for deterministic signals, the developed stochastic POAFD algorithms offer powerful tools to approximate and thus to analyze random signals.

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A Theory on Non-Constant Frequency Decompositions and Applications

Cited by 15 publications

References 93 publications

Sparse Representations of Solutions to a class of Random Boundary Value Problems

Sparse Representations of Solutions to a class of Random Boundary Value Problems

Analytic Phase Retrieval Based on Intensity Measurements

Sparse representations of random signals

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