Approximation error of the Whittaker cardinal series in terms of an averaged modulus of smoothness covering discontinuous signals

Bardaro, Carlo; Butzer, Paul L.; Stens, R. L.; Vinti, Gianluca

doi:10.1016/j.jmaa.2005.04.042

Cited by 63 publications

(59 citation statements)

References 16 publications

(20 reference statements)

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“…So we have to restrict the matter to a suitable subspace which guarantees that the series S w f is convergent for each signal function f in this space. To this end we introduced the space p in [1].…”

Section: The Space Pmentioning

confidence: 99%

“…Observe that an approximate sampling theorem, namely that the limit relation (5) holds under suitable conditions (see below), was established in [16,5,17,1]. However, it was not shown in these papers that this result does indeed follow explicitly from the classical sampling theorem.…”

Section: Introductionmentioning

confidence: 99%

“…The error entailed (see (3), (20)) is an integral part of an approximate sampling theorem; for the error in C(R)-space recall, e.g., [3,4], and in case of L p -spaces see [1].…”

Section: Introductionmentioning

confidence: 99%

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Classical and approximate sampling theorems; studies in the LP(R) and the uniform norm

Butzer

Higgins²,

Stens

2005

Journal of Approximation Theory

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The approximate sampling theorem with its associated aliasing error is due to J.L. Brown (1957). This theorem includes the classical Whittaker-Kotel'nikov-Shannon theorem as a special case. The converse is established in the present paper, that is, the classical sampling theorem for f ∈ B p w , 1 p < ∞, w > 0, implies the approximate sampling theorem. Consequently, both sampling theorems are fully equivalent in the uniform norm.Turning now to L p (R)-space, it is shown that the classical sampling theorem for f ∈ B p w , 1 < p < ∞ (here p = 1 must be excluded), implies the L p (R)-approximate sampling theorem with convergence in the L p (R)-norm, provided that f is locally Riemann integrable and belongs to a certain class p . Basic in the proof is an intricate result on the representation of the integral R |f (u)| p du as the limit of an infinite Riemann sum of |f | p for a general family of partitions of R; it is related to results of O. Shisha et al. (1973Shisha et al. ( -1978 on simply integrable functions and functions of bounded coarse variation on R. These theorems give the missing link between two groups of major equivalent theorems; this will lead to the solution of a conjecture raised a dozen years ago.

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Section: The Space Pmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Classical and approximate sampling theorems; studies in the LP(R) and the uniform norm

Butzer

Higgins²,

Stens

2005

Journal of Approximation Theory

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“…Later on, these linear approximation operators were intensively studied in e.g. [1]- [3], [7]- [10], [17], [18], [20], [21], [22], [23] (see also the references cited there).…”

Section: Introductionmentioning

confidence: 99%

Localization results for the non-truncated max-product sampling operators based on Fejér and sinc-type kernels

Coroianu

Gal

2016

Demonstratio Mathematica

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Abstract. In this paper, we obtain strong localization results and local direct results in the approximation of continuous functions by the non-truncated max-product sampling operators based on Fejér and sinc (Wittaker)-type kernels. These operators present potential applications in signal theory. IntroductionThe sinc-approximation operators were first introduced and studied in [19] This idea applied, for example, to the linear Bernstein operators B n pf qpxq " ř n k"0 p n,k pxqf pk{nq, where p n,k pxq "`n k˘x k p1´xq n´k and f is positive,

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“…[1,5,13]. The WKS sampling theorem provides an exact reconstruction formula for band limited, finite energy functions.…”

Section: Introductionmentioning

confidence: 99%

Multivariate sampling Kantorovich operators: from the theory to the Digital Image Processing algorithm

Costarelli

Vinti

2015

Proc Appl Math and Mech

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Approximation error of the Whittaker cardinal series in terms of an averaged modulus of smoothness covering discontinuous signals

Abstract: The Whittaker-Shannon-Kotel'nikov sampling theorem enables one to reconstruct signals f bandlimited to [−πW, πW ] from its sampled values f (k/W ), k ∈ Z, in terms of

Cited by 63 publications

References 16 publications

Classical and approximate sampling theorems; studies in the LP(R) and the uniform norm

Classical and approximate sampling theorems; studies in the LP(R) and the uniform norm

Localization results for the non-truncated max-product sampling operators based on Fejér and sinc-type kernels

Multivariate sampling Kantorovich operators: from the theory to the Digital Image Processing algorithm

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