A quantitative condensation of singularities on arbitrary sets

Dickmeis, W.; Nessel, R. J.

doi:10.1016/0021-9045(85)90114-5

Cited by 13 publications

(17 citation statements)

References 4 publications

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“…In [18] sharpness of such error bounds is shown on the basis of quantitative extensions of the uniform boundedness principle developed by Dickmeis, Nessel and van Wickeren (cf. [9,10]). We follow this approach and modify it to fit for band-limited functions.…”

Section: Sharpnessmentioning

confidence: 99%

A Sharp Error Estimate for Numerical Fourier Fransform of Band-Limited Functions Based on Windowed Samples

Goebbels¹

2013

Z. Anal. Anwend.

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W. Dickmeis and R.J. Nessel published their first version of a quantitative extension of the classical uniform boundedness principle in [J. Approx. Theory 31 (1981), 161-174]. It is a general approach to finding counterexamples that prove sharpness of error estimates. So far applications of this principle include error bounds for approximation processes, cubature rules, ordinary and partial differential equations, and reconstruction from samples. Here we discuss the error of discrete approximations of the Fourier transform based on windowed samples for band-limited functions. The results can be applied to the Hann-and Blackmann-Harris-window but also to window-functions that enable higher orders of convergence. We describe a class of such windows.

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Section: Sharpnessmentioning

confidence: 99%

A Sharp Error Estimate for Numerical Fourier Fransform of Band-Limited Functions Based on Windowed Samples

Goebbels¹

2013

Z. Anal. Anwend.

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“…The central assumption in [1] was that the operators are sublinear, i.e., fulfill (4.2) and (4.3). Further, in [1] the sequences of operators were analyzed for fixed t. In [10,11,18] conditions were discussed that allow results for more general sets T ⊂ R. For convergence almost everywhere a new approach was developed in [19] that extends the theorem of Banach and Steinhaus. All papers [10,11,18,19] have in common that they need the sublinearity of the involved operators.…”

Section: The Threshold Operator and Basic Propertiesmentioning

confidence: 99%

“…Moreover, there exists a f (2) ∈ PW 2 π such that f (2) (k) = 0 only for finitely many k ∈ Z and f (1) 10) and, according to the assumption ∞ k=−∞ |h T (t − k)| = ∞, at least one of the sums on the right-hand side of (5.10) must be infinity. Without loss of generality, we assume that…”

Section: Behavior For Fixed Tmentioning

confidence: 99%

On the Behavior of the Threshold Operator for Bandlimited Functions

Boche

Monich

2013

J Fourier Anal Appl

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One interesting question is how the good local approximation behavior of the Shannon sampling series for the Paley-Wiener space PW 1 π is affected if the samples are disturbed by the non-linear threshold operator. This operator, which is important in many applications, sets all samples whose absolute value is smaller than some threshold to zero. In this paper we analyze a generalization of this problem, in which not the Shannon sampling series is disturbed by the threshold operator but a more general system approximation process, were a stable linear time-invariant system is involved. We completely characterize the stable linear time-invariant systems that, for some functions in PW 1 π , lead to a diverging approximation process as the threshold is decreased to zero. Further, we show that if there exists one such function then the set of functions for which divergence occurs is in fact a residual set. We study the pointwise behavior as well as the behavior of the L ∞-norm of the approximation process. It is known that oversampling does not lead to stable approximation processes in the presence of thresholding. An interesting open problem is the characterization of the systems that can be stably approximated with oversampling. Math Subject Classifications. 94A20, 94A12.

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“…Uh :!9 M min {1, a(6)/q}-(6.> 0,. n E N), In the present context X will be the Banach space R with norm IIIIB US = r2 (f, (5) with q',, = ,u,, and a(ô) = 62, and R,,f = f sup ITkI -/1 . The sharpness of the estimate (10) is then established by kn -…”

Section: T(f+ G)^s Tf+tg T(af) I Al T!mentioning

confidence: 99%

“…Proof: To construct the elements h,,, following [5], let H be four times continuously Ix -x.,, > d,, for j =f= k, one has h,,,(x) = 0,'thus in view of (23)…”

Section: / Knmentioning

confidence: 99%

A Quantitative Bohman–Korovkin Theorem and its Sharpness for Riemann Integrable Functions

Mevissen¹,

Nessel²,

Wickeren³

1988

Z. Anal. Anwend.

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Ausgehend von einer Arbeit von Pôlya (1933) uber die Konvergenz von Quadraturverfahren haben wir kürzlich einen Folgenkonvergenzbegriff im Raum der Riemann-integrierbaren Funktionen eingefuhr, unter dem sich dieser klassische Raum als Vervollstandigung der ste. tigen Funktionen darstellt. In der sich dann anschliel3enden Diskussion mäglicherAuswirkungen auf di'Approximationstheorie.wurde unter anderem der Satzvon BohmanKorovkin (mit und ohne Ordnung) von stetigen auf R.iemann-integrierbare Funktionen ubertragen. Die dabei erzielten Schranken wurden uber den ersten t-Modul ausgedruckt. Ein Ziel dieser Arbeit ist es, entsprechende Abschätzungen gegen den zwéiten T-Modul herzuleiten. Die Schärfe dieser Resultate wid dann mittels-eines quantitativen Beschränktheitsprinzips gezeigt. Schliel3lich werden die aligemeinen Ergebnisse im Zusammenhang mit Bernstein-Polynomen und linearcn, interpolierenden Splines getestet.-14cxojx.q OT oH61 paGoTar floiiva (1933) 0 C X OJ 1IMOCTB MeTOOB RBaApaTyp Mlii suejiu IIeaBHo noHnTIle CX0HM0CTH nocJlelxouaTeJlbnocTell B EIOCT}1CTBC IlHTerpnpyeMblx no PuslaHy)yIIxun5, npu KoTopoM DTO inaccwiecioe npocTpaucTBo oHa3a.noch90110J1HeH11eM Henpepain-HliIX yHH1uti. B xoje noc.uejyiou.1et1)1.HcHycclsH 0 B03M0HIIbIX nocnecTBHHx na Teoplilo annpoKdnMausm nepeHoclulacb, Me-. , y npo'iut, TeopeMa BoMaiIa-1-copon141,11-1a (C nopnxoi H 6e3 Hero) C IlenpepblBHhlx HaIIHTerpnpyeMh1e no PHMaHy yuiiuf1. ,LocTnrHyTbIe npis 3TOM rpaiinti norpewHocTH nwpaarnic 'lepea nepBa!n T-MoyJ1b. Oiia no uejieft OTotl pa60Tbl-BhIBOHTb cooTBeTcTBylouIIe OICHKI1 oTfroduTejmIIo BTOporO r-M0Jy.un. HOTOM nocpecTBoM ICoJlIl'iecTBeIIHoro HJUfIIHfla orpaHll'leIIHocTII n0Fa3bIBaeTcJ 'ITO 0TH pe3yii-Tam! HBJImOTCR TO4HLIMH. UaHoHeu, o6une peayJiami flOBeHI0TCH B CBHCH c noimHo-MaMH BepHwTeftHa 11 J1141-lefIlhlMH uhITepnoJlIlpyIoaulMn duJ1alHaMu. Inspired by work of POlya (193 on the convergence of quadrature formulas, we previously introduced a concept of sequential convergence in the space of-Riemann integrable functions under which this classical space is in fact the completion of Continuous functions. Discussing its consequences to approximation theory, among others we already extended the (qualitative as well as quantitative) Bohnian-Korovkin theorem from continuous to Riemann integrable functions. The error bounds obtained were given in terms of the first r-modul. One purpose of this paper is to derive corresponding results, involving the second r-modul. As a consequence of our previous quantitative uniform boundedness principle-it is then shown that these estimates are indeed sharp. The general results obtained are illustrated in connection with Brnstein polynomials and linear interpolating splines.

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A quantitative condensation of singularities on arbitrary sets

Cited by 13 publications

References 4 publications

A Sharp Error Estimate for Numerical Fourier Fransform of Band-Limited Functions Based on Windowed Samples

A Sharp Error Estimate for Numerical Fourier Fransform of Band-Limited Functions Based on Windowed Samples

On the Behavior of the Threshold Operator for Bandlimited Functions

A Quantitative Bohman–Korovkin Theorem and its Sharpness for Riemann Integrable Functions

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