Converse theorems for approximation by bernstein polynomials in Lp[0,1] (1&lt;p&lt;∞)

Ivanov, Kamen

doi:10.1007/bf01893439

Cited by 9 publications

(3 citation statements)

References 7 publications

(6 reference statements)

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“…Partial inverse theorems were then established in' [11] for the LP-error, p > 1. Concerning the Riemânn error f sup J Bk1 -/1 discussed here, an improved direct as 0.…”

Section: Hence Our Theorems Delivermentioning

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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“…Partial inverse theorems were then established in' [11] for the LP-error, p > 1. Concerning the Riemânn error f sup J Bk1 -/1 discussed here, an improved direct as 0.…”

Section: Hence Our Theorems Delivermentioning

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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“…While investigation of the rate of approximation of BJ in Lp has taken place (see, for instance, [6]), a side condition was needed, and it is the approximation of K,f that is more natural in Lp, p ~ ov. Investigation of the class of functions for which was…”

Section: Introductionmentioning

confidence: 99%

Kantorovich-Bernstein polynomials

Ditzian

Zhou²

1990

Constr. Approx

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A gap between saturation and direct-converse theorems for Kantorovich-Bernstein polynomials will be closed for a steady rate of convergence. The present theorems unify the above-mentioned results. Furthermore, it is shown that for steady rates our converse results are an improvement on both weak-type converse theorems and strong-weak-type converse theorems for the KantorovichBernstein polynomials.

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“…It is closely related to the trigonometric case; results analogous to the Jackson inequality, valid for 27r-periodic functions, However, the different moduli are not compared explicitly (which may be very difficult). In this respect Ivanov [1] remarks (without proof) that his modulus Tp is equivalent to the Ditzian-Totik modulus (*;£,. In any case, their modulus seems to be the most natural one as has been demonstrated in a masterful way in their book.…”

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confidence: 99%

Book Review: Moduli of smoothness

Butzer¹,

Wickeren²

1988

Bull. Amer. Math. Soc.

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568BOOK REVIEWS has become the first part of a two volume treatise. The second volume is intended to cover the work on the quantised systems up to the current time. Moduli of smoothness play a basic role in approximation theory, Fourier analysis and their applications. For a given function ƒ, the domain of which is a (bounded) interval Z), they essentially measure the structure or smoothness of ƒ via the rth (symmetric) difference

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Converse theorems for approximation by bernstein polynomials in Lp[0,1] (1<p<∞)

Abstract: The class of all continuous functions possessing n-" (I/p<. a <-1) order of approximation by Bemstein polynomials in Lp[0, 1] is characterized.

Cited by 9 publications

References 7 publications

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

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

Kantorovich-Bernstein polynomials

Book Review: Moduli of smoothness

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