On chebyshev quadrature for a special class of weight functions

Förster, Klaus-Jürgen

doi:10.1007/bf01933712

Cited by 6 publications

(8 citation statements)

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“…The second statement of the corollary follows immediately by Corollary 3. , 1), does not admit T -q for odd n since the maximum degree of exactness is of order In n, a result which has been shown in [3] by a completely different approach. Thus the question arises whether there is a distribution function at all admitting T-q on a set of several intervals which is symmetric with respect to zero and does not contain the point zero.…”

Section: Admits (M(nk) Nk Da) T--q For Each Ken and That The Nodes mentioning

confidence: 88%

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Gauss-Tchebycheff quadrature formulas

Peherstorfer

1990

Numer. Math.

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Summary. It is well known that the Tchebycheff weight function (l--x2) -1/2 is the only weight function (up to a linear transformation) for which the n point Gauss quadrature formula has equal weights for all neN. In this paper we describe explicitly all weight functions which have the property that the nk-point Gauss quadrature formula has equal weights for all keN, where (nk), nx < n2 <..., is an arbitrary subsequence of N. Furthermore results on the possibility of Tchebycheff quadrature on several intervals are given.

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Section: Admits (M(nk) Nk Da) T--q For Each Ken and That The Nodes mentioning

confidence: 88%

“…[1][2][3][4][5][6][7][8][9][10][11][12][13]). Furthermore let us note that the constant given in [1][2][3][4][5][6][7][8] is not so precise than that one given in the next theorem. we get with the help of (3.6) and ( …”

Section: Admits (M(nk) Nk Da) T--q For Each Ken and That The Nodes mentioning

confidence: 99%

Gauss-Tchebycheff quadrature formulas

Peherstorfer

1990

Numer. Math.

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“…Our proof of this theorem uses general theorems of Peherstorfer [21] and Förster and Ostermeyer [10] which, when taken together, show that d n (σ ) may rise at most logarithmically in n for odd n whenever σ is a symmetric measure having 0 outside its support. We remark also that the phenomenon that d n (σ ) may have very different orders of magnitude for odd and even n was first discovered, in a particular case, by Förster [9].…”

Section: Lower Boundsmentioning

confidence: 94%

“…The bounds (10) and (11) follow directly from the bounds (8) and (9) and the definitions of d n (σ 0 ), n 0 σ 0 (k) and n σ 0 (k). The bound (8) follows from Bernstein's Theorem 1.2 on replicating the Chebyshev-type quadrature given by the upper bound of that theorem to each of the two intervals in the support of σ 0 .…”

Section: Lower Bounds For the Number Of Nodesmentioning

confidence: 99%

Simple universal bounds for Chebyshev-type quadratures

Peled

2010

Journal of Approximation Theory

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A Chebyshev-type quadrature for a probability measure σ is a distribution which is uniform on n points and has the same first k moments as σ . We give an upper bound for the minimal n required to achieve a given degree k, for σ supported on an interval. In contrast to previous results of this type, our bound uses only simple properties of σ and is applicable in wide generality. We also obtain a lower bound for the required number of nodes which only uses estimates on the moments of σ . Examples illustrating the sharpness of our bounds are given. As a corollary of our results, we obtain an apparently new result on the Gaussian quadrature.In addition, we suggest another approach to bounding the minimal number of nodes required in a Chebyshev-type quadrature, utilizing a random choice of the nodes, and propose the challenge of analyzing its performance. A preliminary result in this direction is proved for the uniform measure on the cube. Finally, we apply our bounds to the construction of point sets on the sphere and cylinder which form local approximate Chebyshev-type quadratures. These results were needed recently in the context of understanding how well a Poisson process can approximate certain continuous distributions. The paper concludes with a list of open questions.

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“…For odd n := 2m + 1, however, one has for all Chebyshev-type rules Qrm+ 1 the inequality (6) deg(Q~m+ 1 ) < const -In m (see [3]), which indicates that, in this case, the cost of obtaining minimal variance increases exponentially. On the other hand, by the Theorem, minimal variance can be obtained asymptotically, even with the condition of maximal degree deg(Q,) = 2n-1.…”

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confidence: 95%