Frames in spaces with finite rate of innovation

Sun, Qiyu

doi:10.1007/s10444-006-9021-4

Cited by 52 publications

(77 citation statements)

References 56 publications

(96 reference statements)

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“…This follows from the fact that the supports of the terms in (25) are disjoint, by the definition of the sets E(j, q) and relations (20) and (24). In the sequence f q 2 , .…”

Section: Proof Of Lemma 2 2 After the Extension Fmentioning

confidence: 97%

“…Therefore, for every x ∈ (0, π) there exists j = j(x, i) such that, by (20), (24), (25), and (32), the first sum in (37) can be estimated from below as follows:…”

Section: Proof Of Lemma 2 2 After the Extension Fmentioning

confidence: 99%

“…Here, by (19), (20), (24), and (25), for estimating the first sum from below, we can take only part of the terms whose indices correspond to the modes lying from one side of x. We choose the side on which there are at least as many terms as on the opposite side; by (20) and (33), this part has at least [ By (24), every function φ j,q i vanishes at the nodes with even indices that lie in E(j, q i ).…”

Section: Proof Of Lemma 2 2 After the Extension Fmentioning

confidence: 99%

“…We choose the side on which there are at least as many terms as on the opposite side; by (20) and (33), this part has at least [ By (24), every function φ j,q i vanishes at the nodes with even indices that lie in E(j, q i ). Subtracting an upper bound for the second sum from the first sum, we use the fact that, by (35) …”

Section: Proof Of Lemma 2 2 After the Extension Fmentioning

confidence: 99%

“…The sinc-approximations are widely applied for constructing numerical methods of mathematical physics and approximation theory for functions of one or several variables [17]-[21], in the theory of quadrature formulas [6,22] and wavelet transformations [8]-[11], [23,24]. In the paper [25], an analog of the sampling theorem that employs the Hermite type interpolation was obtained.…”

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confidence: 99%

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On divergence of sinc-approximations everywhere on $(0,\pi)$

Trynin

2011

St. Petersburg Math. J.

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Abstract. Some properties of sinc-approximations of continuous functions on a segment are studied.For the first time, the sinc-approximations arose in the work of Pleynet. Later, in connection with developments in signals coding theory, E. Borel and E. T. Whittaker introduced the notions of a cardinal function and a truncated cardinal function, the restriction of which to [0, π] looks like this:(1)By the present time, the problem of sinc-approximation has been studied rather deeply for functions analytic on the real line and decaying exponentially at infinity (see, e.g.,The most complete survey of the results obtained in this direction before 1993, together with numerous important applications of sinc-approximations, can be found in [6]. An interesting historical survey of investigations on this topic is contained in [7].Also, a series of publications had their origin in what is called the sampling theorem or sometimes, the Whittaker-Kotel nikov-Shannon discretization theorem [8,9,6,11]; in those studies, various representations of entire functions were obtained in the form of series in sincs with interpolation nodes satisfying some "uniform distribution" conditions (see, e.g., [12]-[14]). Starting with the well-known paper [15] by Kramer, the relationship between the sampling theorems and the Lagrange interpolation with nodes in the SturmLiouville problem spectrum has been studied; see, e.g., [16].The sinc-approximations are widely applied for constructing numerical methods of mathematical physics and approximation theory for functions of one or several variables [17]-[21], in the theory of quadrature formulas [6,22] and wavelet transformations [8]-[11], [23,24]. In the paper [25], an analog of the sampling theorem that employs the Hermite type interpolation was obtained.The authors of [26] succeeded in proving approximability for less smooth functions. However, they were forced to modify the operator (1) somewhat. In [26], it was established that if a uniformly continuous and bounded functions f on R belongs to the Dini-Lipchitz class and satisfies the condition f (x) = O(|x| −δ ) as x → ±∞ for some 2010 Mathematics Subject Classification. Primary 42A10, 41A30.

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“…This follows from the fact that the supports of the terms in (25) are disjoint, by the definition of the sets E(j, q) and relations (20) and (24). In the sequence f q 2 , .…”

Section: Proof Of Lemma 2 2 After the Extension Fmentioning

confidence: 97%

“…Therefore, for every x ∈ (0, π) there exists j = j(x, i) such that, by (20), (24), (25), and (32), the first sum in (37) can be estimated from below as follows:…”

Section: Proof Of Lemma 2 2 After the Extension Fmentioning

confidence: 99%

Section: Proof Of Lemma 2 2 After the Extension Fmentioning

confidence: 99%

Section: Proof Of Lemma 2 2 After the Extension Fmentioning

confidence: 99%

mentioning

confidence: 99%

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On divergence of sinc-approximations everywhere on $(0,\pi)$

Trynin

2011

St. Petersburg Math. J.

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Average sampling and reconstruction in a reproducing kernel subspace of homogeneous type space

Xian

2013

Mathematische Nachrichten

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In this paper, we first introduce a reproducing kernel subspace of Lp(X,ρ,μ), where (X,ρ,μ) is a homogeneous type space. Then we consider average sampling and reconstruction of signals in the reproducing kernel subspace of Lp(X,ρ,μ),1≤p≤∞. We show that signals in the reproducing kernel subspace of Lp(X,ρ,μ) could be stably reconstructed from its average samples taken on a relatively‐separated set with small gap. Exponential convergence is established for the iterative approximation‐projection reconstruction algorithm.

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Function Spaces for Sampling Expansions

Nashed

Sun

2012

Multiscale Signal Analysis and Modeling

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Frames in spaces with finite rate of innovation

Cited by 52 publications

References 56 publications

On divergence of sinc-approximations everywhere on $(0,\pi)$

On divergence of sinc-approximations everywhere on $(0,\pi)$

Average sampling and reconstruction in a reproducing kernel subspace of homogeneous type space

Function Spaces for Sampling Expansions

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