Infinite Riemann Sums, the Simple Integral, and the Dominated Integral

Lewis, James L.; Osgood, Charles F.; Shisha, Oved

doi:10.1007/978-3-0348-5563-1_25

Cited by 4 publications

(3 citation statements)

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“…In this respect it was shown in [15] that a function g ∈ R loc satisfies condition (iii) of our Lemma 5 if and only if it is simply integrable over R. Simple integrability in turn implies in the terminology of Lemma 6(b) that (cf. [12,Theorem 2])…”

Section: Lemma 6 Let 1 P < ∞ (A) P Is a Proper Linear Subspace Of Lmentioning

confidence: 99%

“…the interchange of the limit and the integral being justified by Lebesgue's dominated convergence theorem, noting that in view of (12),…”

Section: The Approximate Sampling Theorem In the Uniform Normmentioning

confidence: 99%

“…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.…”

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confidence: 98%

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

Butzer

Higgins²,

Stens

2005

Journal of Approximation Theory

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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.

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Section: Lemma 6 Let 1 P < ∞ (A) P Is a Proper Linear Subspace Of Lmentioning

confidence: 99%

“…the interchange of the limit and the integral being justified by Lebesgue's dominated convergence theorem, noting that in view of (12),…”

Section: The Approximate Sampling Theorem In the Uniform Normmentioning

confidence: 99%

mentioning

confidence: 98%

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