Riemann Sums: A Simple Illustration

2007 ◽  
Keyword(s):  
Riemann Sums

Globally Small Riemann Sums and the Henstock Integral

1990 ◽  
Vol 16 (2) ◽  
pp. 537 ◽  
Author(s):  
Shi-Pan ◽  
Peng-Yee
Keyword(s):  
Henstock Integral ◽  
Riemann Sums

Definite Integrals and Riemann Sums

1939 ◽  
Vol 46 (9) ◽  
pp. 538
Author(s):  
J. A. Shohat
Keyword(s):  
Riemann Sums ◽  
Definite Integrals

Summability Methods Defined by Riemann Sums

1953 ◽  
Vol 5 ◽  
pp. 289-296
Author(s):  
J. D. Hill
Keyword(s):  
Riemann Sums ◽  
Summability Methods ◽  
Image Position

Let f(x) be real valued, bounded and, integrable in the sense of Riemann on the interval X = (0 < x < 1), with the value of its integral over X equal to one. For brevity we call such a function admissible.The symbol Xnk will always denote the interval an arbitrarily chosen point of Xnk and δ any specified set of intermediate points

Relationship Between a Vector Differential and the Corresponding Scalar Differential in Physics

2019 ◽  
Vol 01 (03) ◽  
pp. 1950011
Author(s):  
Carl E. Mungan
Keyword(s):  
Linear Momentum ◽  
Introductory Physics ◽  
Riemann Sums ◽  
Starting Point ◽  
Vector Quantity ◽  
Mass Ratios ◽  
Small Change ◽  
The Relationship

In physics, a differential is an infinitesimal change in or amount of some quantity. For example, [Formula: see text] is a small change in linear momentum, and [Formula: see text] is a small amount of mass. Ratios of differentials become derivatives, while Riemann sums of differentials become integrals. Given some vector quantity X, what is the relationship between [Formula: see text] and [Formula: see text] according to the standard conventions of introductory physics? Surprisingly, there are two distinct answers, depending on exactly what quantity X happens to be. The distinction is illustrated here with specific examples. After discussing this ambiguity in some detail, some recommendations to physics instructors and textbook authors are preferred. Although not everyone will agree with these conclusions and suggestions, this article provides a starting point for further deliberations.

scholarly journals On some random Riemann-sums

1956 ◽  
Vol 8 (3) ◽  
pp. 245-257
Author(s):  
Shigeru Takahashi
Keyword(s):  
Riemann Sums

scholarly journals On well-bounded operators of class Г

1983 ◽  
Vol 24 (1) ◽  
pp. 1-5
Author(s):  
Adnan A. S. Jibril
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Bounded Operators ◽  
Strong Limit ◽  
Riemann Sums ◽  
Image Position

Let T be a linear operator acting in a Banach space X. It has been shown by Smart [5] and Ringrose [3] that, if X is reflexive, then T is well-bounded if and only if it may be expressed in the formwhere {E(λ)} is a suitable family of projections in X and the integral exists as the strong limit of Riemann sums.

scholarly journals On approximating Lebesgue integrals by Riemann sums

1991 ◽  
Vol 33 (2) ◽  
pp. 129-134
Author(s):  
Szilárd GY. Révész ◽  
Imre Z. Ruzsa
Keyword(s):  
Characteristic Function ◽  
Lebesgue Measure ◽  
Real Function ◽  
Riemann Sums ◽  
Function Classes ◽  
Open Set ◽  
Measurable Set ◽  
Image Position ◽  
Good Sequences

If f is a real function, periodic with period 1, we defineIn the whole paper we write ∫ for , mE for the Lebesgue measure of E ∩ [0,1], where E ⊂ ℝ is any measurable set of period 1, and we also use XE for the characteristic function of the set E. Consistent with this, the meaning of ℒp is ℒp [0, 1]. For all real xwe haveif f is Riemann-integrable on [0, 1]. However,∫ f exists for all f ∈ ℒ1 and one would wish to extend the validity of (2). As easy examples show, (cf. [3], [7]), (2) does not hold for f ∈ ℒp in general if p < 2. Moreover, Rudin [4] showed that (2) may fail for all x even for the characteristic function of an open set, and so, to get a reasonable extension, it is natural to weaken (2) towhere S ⊂ ℕ is some “good” increasing subsequence of ℕ. Naturally, for different function classes ℱ ⊂ ℒ1 we get different meanings of being good. That is, we introduce the class of ℱ-good sequences as

The strong sweeping out property for lacunary sequences, Riemann sums, convolution powers, and related matters

1996 ◽  
Vol 16 (2) ◽  
pp. 207-253 ◽  
Author(s):  
Mustafa Akcoglu ◽  
Alexandra Bellow ◽  
Roger L. Jones ◽  
Viktor Losert ◽  
Karin Reinhold-Larsson ◽  
...  
Keyword(s):  
Fourier Transform ◽  
Characteristic Function ◽  
Ergodic Averages ◽  
Riemann Sums ◽  
Lacunary Sequences ◽  
Linearly Independent ◽  
Convolution Powers ◽  
The Fourier Transform

AbstractIn this paper we establish conditions on a sequence of operators which imply divergence. In fact, we give conditions which imply that we can find a set B of measure as close to zero as we like, but such that the operators applied to the characteristic function of this set have a lim sup equal to 1 and a lim inf equal to 0 a.e. (strong sweeping out). The results include the fact that ergodic averages along lacunary sequences, certain convolution powers, and the Riemann sums considered by Rudin are all strong sweeping out. One of the criteria for strong sweeping out involves a condition on the Fourier transform of the sequence of measures, which is often easily checked. The second criterion for strong sweeping out involves showing that a sequence of numbers satisfies a property similar to the conclusion of Kronecker's lemma on sequences linearly independent over the rationals.

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