Number of lattice points in the hyperbolic cross

Добровольский, Николай Михайлович; Roshchenya, A. L.

doi:10.1007/bf02317776

Cited by 6 publications

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“…For a proof of this in a more general setting we refer to [8]. A simple inductive argument appears in [15], which we now repeat here, since similar methods will be used in the sequel:…”

Section: Hyperbolic Cross Index Setsmentioning

confidence: 99%

Multivariate modified Fourier series and application to boundary value problems

Adcock

2010

Numer. Math.

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In this paper we analyse the approximation-theoretic properties of modified Fourier series in Cartesian product domains with coefficients from both full and hyperbolic cross index sets. We show that the number of expansion coefficients may be reduced significantly whilst retaining comparable error estimates. In doing so we extend the univariate results of Iserles, Nørsett and S. Olver. We then demonstrate that these series can be used in the spectral-Galerkin approximation of second order Neumann boundary value problems, which offers some advantages over standard Chebyshev or Legendre polynomial discretizations.

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“…For a proof of this in a more general setting we refer to [8]. A simple inductive argument appears in [15], which we now repeat here, since similar methods will be used in the sequel:…”

Section: Hyperbolic Cross Index Setsmentioning

confidence: 99%

Multivariate modified Fourier series and application to boundary value problems

Adcock

2010

Numer. Math.

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“…Proof We forego the longer (and more general) proof in (Dobrovol'skii & Roshchenya 1998) for an elementary inductive argument. Clearly, κ 1 (t) = t and this is consistent with (3.1).…”

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confidence: 99%

From high oscillation to rapid approximation IV: accelerating convergence

Huybrechs

Iserles

Nørsett

2010

IMA Journal of Numerical Analysis

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Modified Fourier expansion is a powerful means for the approximation of non-periodic smooth functions in a univariate or multivariate setting. In the current paper we consider further enhancement of this approach by two techniques familiar from conventional Fourier analysis: polynomial subtraction and the hyperbolic cross. We demonstrate that, judiciously subtracting simpler functions dependent on linear combinations of derivatives along boundaries, it is possible to accelerate convergence a great deal and this procedure is considerably more efficient than in the case of conventional Fourier expansion. Moreover, examining the pattern of decay of coefficients in a multivariate setting, we demonstrate that most of them can be disregarded without any ill effect on quality of approximation.

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