Optimale Approximation linearer Funktionale auf periodischen Funktionen

Knauff, W.; Kreβ, R.

doi:10.1007/bf01436967

Cited by 15 publications

(8 citation statements)

References 11 publications

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“…Some of these results are contained in [7] and [8], but short proofs are given here for completeness. Throughout this paper we shall assume that pj4=O,j=O, 1 ..... k-1, so that 5{k is of dimension k. For f, geC2,, let…”

Section: The Orthogonal Basismentioning

confidence: 97%

“…This example is taken from [7]. The values of pj and 2.,j were computed using 3 terms in the series of (6.8) and (6.9).…”

Section: "Nkf= ~ (F Ej> Ej+ Omkfmentioning

confidence: 99%

“…Furthermore ( [7,8]), k-1 {Pj}j=O forms an orthogonal basis for ~ with respect to the inner product where 2m + 1 = k (see [1,12]), then 5~k is the space of trigonometric polynomials of degree m, and [10,13]) and trigonometric splines (see [14,15]) which will be considered in Sect. 6 and Sect.…”

Section: Offprint Requests To: Sl Leementioning

confidence: 99%

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L 2-approximation by the translates of a function and related attenuation factors

1991

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Summary. Let 5ek be the k-dimensional subspace spanned by the translates (a(.-2rtj/k), j=0, 1, ...,k-l, of a continuous, piecewise smooth, complexvalued, 2 ~-periodic function qS. For a given functionfe L z (--rt, n), its least squares approximant Szf from ~ can be expressed in terms of an orthonormal basis.If f is continuous, Skfcan be computed via its discrete analogue by fast Fourier transform. The discrete least squares approximant is used to approximate Fourier coefficients, and this complements the works of Gautschi on attenuation factors. Examples of 5~k include the space of trigonometric polynomials where ~b is the de la Valle6 Poussin kernel, algebraic polynomial splines where ~b is the periodic B-spline, and trigonometric polynomial splines where ~b is the trigonometric B-spline. (1991) : 41A 15, 42A 16, 65D07 Mathematics Subject Classification

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Section: The Orthogonal Basismentioning

confidence: 97%

“…This example is taken from [7]. The values of pj and 2.,j were computed using 3 terms in the series of (6.8) and (6.9).…”

Section: "Nkf= ~ (F Ej> Ej+ Omkfmentioning

confidence: 99%

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L 2-approximation by the translates of a function and related attenuation factors

1991

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“…The authors of [1], [4], [5] have treated the above problem in different ways for some special cases relative to either the functional L or a space setting and nodal location in the complex region of that space. In this paper, the common linear algebra background for the results of the abovementioned articles will be treated and additional results given.…”

Section: K(xy) = ~N_opm(x)p~(y) X Y E B mentioning

confidence: 99%

On the optimal approximation of bounded linear functionals in Hilbert spaces of analytic functions

Paulik¹

1976

BIT

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Abstract.Optimal numerical approximation of bounded linear funetionals by weighted sums in Hilbert spaces of functions analytic in a circle KF, in a circular annulus Krl,r~ and in an ellipse Er is investigated by Davis' method on the common algebraic background for diagonalising the normal equation matrix. The weights and error functional norms for optimal rules with nodes located angle-equidistant on the concentric circle ~Ks or on the eonfoeal ellipse ~Es and in the interval [-1,1] for an arbitrary bounded linear functional are given expllci~Iy. They are expressed in terms of a complete orthonormal system in the Hilbert space.

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“…In this context we recall the method proposed by Kress [9] and Knauff-Kress [8] to define function spaces of the considered type with the aid of area or contour integrals. As an example let E be the space of periodic functions, holomorphic in a We point to the fact that (5.12) rW(a) * í 0, iiiGZ\{0}, in contrast to the values given in [3] and [4].…”

mentioning

confidence: 99%

Interpolation on uniform meshes by the translates of one function and related attenuation factors

Locher¹

1981

Math. Comp.

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Abstract. The exact Fourier coefficients c,(PM/) are proportional to the discrete Fourier coefficients dj-"\f) if Pn is a translation invariant operator which depends only on the values of/on an equidistant mesh of width 2it/n. The proportionality factors which depend only on P" but not on / are called attenuation factors and have been calculated for several operators P" of spline type. Here we analyze first the interpolation problem which is produced by the functions

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Optimale Approximation linearer Funktionale auf periodischen Funktionen

Cited by 15 publications

References 11 publications

L 2-approximation by the translates of a function and related attenuation factors

L 2-approximation by the translates of a function and related attenuation factors

On the optimal approximation of bounded linear functionals in Hilbert spaces of analytic functions

Interpolation on uniform meshes by the translates of one function and related attenuation factors

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