Sampling Theory and Function Spaces

Schmeißer, Hans-Jürgen; Sickel, Winfried

doi:10.1142/9789812792686_0008

Cited by 24 publications

(21 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…By approximating a Besov function by Paley-Wiener functions and using the Classical Sampling Theorem we approach multiscale approximations of non-PaleyWiener functions. Some relevant results in the classical setting can be found in [32].…”

mentioning

confidence: 99%

Paley-Wiener Approximations and Multiscale Approximations in Sobolev and Besov Spaces on Manifolds

Pesenson

2009

J Geom Anal

View full text Add to dashboard Cite

An approximation theory by bandlimited functions (≡ Paley-Wiener functions) on Riemannian manifolds of bounded geometry is developed. Based on this theory multiscale approximations to smooth functions in Sobolev and Besov spaces on manifolds are obtained. The results have immediate applications to the filtering, denoising and approximation and compression of functions on manifolds. There exists applications to problems arising in data dimension reduction, image processing, computer graphics, visualization and learning theory.Keywords Paley-Wiener functions on Riemannian manifolds · Multiscale approximations · Schrodinger semigroup · Sobolev and Besov spaces · Sampling of non-bandlimited functions · Discrete spectral Fourier transform · Quadrature formulas on manifolds · Bernstein inequality Mathematics Subject Classification (2000) Primary 42C05 · Secondary 41A17 · 41A65 · 43A85 · 46C99

show abstract

mentioning

confidence: 99%

Paley-Wiener Approximations and Multiscale Approximations in Sobolev and Besov Spaces on Manifolds

Pesenson

2009

J Geom Anal

View full text Add to dashboard Cite

show abstract

“…The error estimate (20) with r = 1 can also be found in [17,Theorem 23], however, under the stronger condition that f ∈ L p (R) satisfies sup >0 −s m (f ; ; M(R)) p < ∞ for some 0 < s < 1/p and some m > s, i.e., f must satisfy a Lipschitz condition of order s with respect to m in order to fulfill (20) with r = 1. The proof is carried out using certain function spaces A s p,∞ .…”

Section: Theoremmentioning

confidence: 99%

“…The proof in [1] seems to be more elementary than that in [17]. It does not use any general theory and is essentially based on a Jackson-type inequality and an interpolation theorem for discrete operators.…”

Section: Theoremmentioning

confidence: 99%

See 1 more Smart Citation

Classical and approximate sampling theorems; studies in the LP(R) and the uniform norm

Butzer

Higgins²,

Stens

2005

Journal of Approximation Theory

View full text Add to dashboard Cite

The approximate sampling theorem with its associated aliasing error is due to J.L. Brown (1957). This theorem includes the classical Whittaker-Kotel'nikov-Shannon theorem as a special case. The converse is established in the present paper, that is, the classical sampling theorem for f ∈ B p w , 1 p < ∞, w > 0, implies the approximate sampling theorem. Consequently, both sampling theorems are fully equivalent in the uniform norm.Turning now to L p (R)-space, it is shown that the classical sampling theorem for f ∈ B p w , 1 < p < ∞ (here p = 1 must be excluded), implies the L p (R)-approximate sampling theorem with convergence in the L p (R)-norm, provided that f is locally Riemann integrable and belongs to a certain class p . Basic in the proof is an intricate result on the representation of the integral R |f (u)| p du as the limit of an infinite Riemann sum of |f | p for a general family of partitions of R; it is related to results of O. Shisha et al. (1973Shisha et al. ( -1978 on simply integrable functions and functions of bounded coarse variation on R. These theorems give the missing link between two groups of major equivalent theorems; this will lead to the solution of a conjecture raised a dozen years ago.

show abstract

“…the mean value w k t k+1 /w t k /w f (u)du. For references on sampling series, see [15,18,12,35,33,14,36,9,22,1,10,2,5,3,4,32]. In practice, this situation often occurs in Signal Processing, that is, more information is usually known in a neighborhood of a point than exactly at that point.…”

Section: Introductionmentioning

confidence: 99%

Approximation by means of nonlinear Kantorovich sampling type operators in Orlicz spaces

Vinti

Zampogni

2009

Journal of Approximation Theory

View full text Add to dashboard Cite

In this paper we introduce a nonlinear version of the Kantorovich sampling type series in a nonuniform setting. By means of the above series we are able to reconstruct signals (functions) which are continuous or uniformly continuous. Moreover, we study the problem of the convergence in the setting of Orlicz spaces: this allows us to treat signals which are not necessarily continuous. Our theory applies to L p -spaces, interpolation spaces, exponential spaces and many others. Several graphical examples are provided.

show abstract

Sampling Theory and Function Spaces

Cited by 24 publications

References 26 publications

Paley-Wiener Approximations and Multiscale Approximations in Sobolev and Besov Spaces on Manifolds

Paley-Wiener Approximations and Multiscale Approximations in Sobolev and Besov Spaces on Manifolds

Classical and approximate sampling theorems; studies in the LP(R) and the uniform norm

Approximation by means of nonlinear Kantorovich sampling type operators in Orlicz spaces

Contact Info

Product

Resources

About