Determination of bounds similar to the Lebesgue constants

Dikshit, H. P.; Kumar, Anil

doi:10.2140/pjm.1981.97.339

Cited by 3 publications

(3 citation statements)

References 4 publications

(7 reference statements)

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“…As observed by us in [10], the conditions {Rn} G B and {N,p) is regular, used by Hille and Tamarkin in [19] for the proof of Theorem 10 follow from {V^ } G 5, since …”

Section: An Application Of Splines For Lebesgue-type Constantsmentioning

confidence: 86%

“…We briefly sketch the interesting analysis evolved in the proof given in [10]. For positive integers r, we consider disjoint intervals/^ = ((2r+ l/3)7r/n, (2r + 4/9)7r/n) c (2/n, 1), so that t e Ir implies that cos(47r/9) < cos(n -k)t, since (n -k)t G ((2r + l/3)7r-1, (2r+4/9)TT).…”

Section: An Application Of Splines For Lebesgue-type Constantsmentioning

confidence: 99%

“…nPn = -Y. HkPk) = Pn-i -S K^Pk-l)-0 i=lDenoting the {N,p) transformation of {kcoskt} by NP{t;2) we proved the following in[10].Theorem 11. Letpn > 0, and {V^} e B then KX< [\mt;2)\dt<K,Sl;…”

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confidence: 92%

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Geometric Modelling and Shape Preserving Properties of Wachspress Functions

Dikshit¹

2009

AIP Conference Proceedings

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Wachspress rational basis functions have been studied with the following two objectives. We first introduce barycentric coordinates for polygons in terms of Wachspress rational functions and also give a more general definition in terms of volumes. We use these to develop quadrilateral and pentagonal patches for geometric modelling to construct surfaces of desired shape and smoothness. We also compare this method with the well known Bezier-Bemstein methods. Next we define rational complex planar splines using complex analogues of real Wachspress functions and study their shape preserving properties like quasiregularity and quaziconformality of the spline interpolants. Finally, we present a beautiful application of quintic splines in obtainig a sharp result on bounds for Lebesgue type constant for the Norlund transformation of Fourier series.where K-is chosen so that ffj.(a') = 1. These basis functions posses following interesting properties, which make them especially useful for geometric modelling applications [6].V^{aJ) = 5j-•; where 5^ • is 1 for ; = J and 0 otherwise; FJ-is linear along each edge of Q;(2) 20

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“…As observed by us in [10], the conditions {Rn} G B and {N,p) is regular, used by Hille and Tamarkin in [19] for the proof of Theorem 10 follow from {V^ } G 5, since …”

Section: An Application Of Splines For Lebesgue-type Constantsmentioning

confidence: 86%

Section: An Application Of Splines For Lebesgue-type Constantsmentioning

confidence: 99%

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Geometric Modelling and Shape Preserving Properties of Wachspress Functions

Dikshit¹

2009

AIP Conference Proceedings

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Matrix summability of Fourier series based on inclusion theorems

Rhoades

1986

Math. Proc. Camb. Phil. Soc.

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Letdenote the Fourier series expansion of a function. Féver's celebrated theorem states that the series is summable (C, 1) at each point of continuity off, where (C, 1) denotes the Cesàro method of summability of order 1. Riesz extended this result to (C, α) for each α > 0. Since then many authors have established sufficient conditions on various methods of summability to guarantee similar results. Over the years the pattern has been to strive for weaker conditions on the matrix, and to replace the condition of continuity on the function by a less stringent one. Theorems have been proved not only for the summability off, but the summability of the derived series, and other series, related tof. Beginning with the work of Hille and Tamarkin[13], many of the theorems have been extended to absolute summability.

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Absolute total-effectiveness of a total effective (𝑁,𝑝_{𝑛}) method

Kumar¹

1982

Proc. Amer. Math. Soc.

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Determination of bounds similar to the Lebesgue constants

Cited by 3 publications

References 4 publications

Geometric Modelling and Shape Preserving Properties of Wachspress Functions

Geometric Modelling and Shape Preserving Properties of Wachspress Functions

Matrix summability of Fourier series based on inclusion theorems

Absolute total-effectiveness of a total effective (𝑁,𝑝_{𝑛}) method

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