First Return Approachability

Darji, Udayan B.; Evans, Michael J.; Humke, Paul D.

doi:10.1006/jmaa.1996.0160

Cited by 10 publications

(12 citation statements)

References 4 publications

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“…Since t yields the Lebesgue integral of f , it follows from Theorem 2.1 in [9] and Theorem 2 in [6] that f is first-return continuous a.e. with respect to t.…”

Section: Lemma 4 Suppose F Is Lebesgue Integrable On [A B] and The mentioning

confidence: 94%

“…Then the first-return series for f given by ∞ n=1 f r t, (a n+1 , a n ] (a n − a n+1 ) = ∞ n=1 ψ r t, (a n+1 , a n ] (a n − a n+1 ) converges via Lemma 2. Furthermore, the convergence is conditional due to (5), (6), and the fact that the Lebesgue integral of | f | on I is +∞.…”

Section: Lemma 4 Suppose F Is Lebesgue Integrable On [A B] and The mentioning

confidence: 98%

“…The next lemma will be used in constructing specific first-return series associated with a first-return integrable f . Prior to its proof we must recall another definition (see [6], e.g. ):…”

Section: Lemmamentioning

confidence: 99%

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Functions not first-return integrable

Darji¹,

Evans²

2008

Journal of Mathematical Analysis and Applications

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Benedetto Bongiorno constructed a certain class of improperly Riemann integrable functions on [0, 1] which are not first-return integrable. He asked if all improper Riemann integrable functions which are not Lebesgue integrable are not first-return integrable.Recently David Fremlin provided a clever example to show that this is not the case. It remains open as to which functions are first-return integrable. We prove two general theorems which imply the existence of a large class of improperly Riemann integrable functions which are not first-return integrable. As a corollary we obtain that there is an improperly Riemann integrable function which is C ∞ on (0, 1] yet fails to be first-return integrable.Richard O'Malley [12] borrowed the notion of first-return from dynamics to study certain types of derivatives. Subsequently, various classes of functions of interest in real analysis were characterized by first-return concepts. In particular, the class of Baire one functions [4] and the class of Baire one Darboux functions [7] were characterized by first-return techniques. In [5] we introduced the notion of first-return integrability. Our idea was to have an integration process similar to the Riemann process which integrates all the Lebesgue integrable functions. Of course, the Henstock-Kurzweil integral does this. However, we wanted an integration process where the choice of partition is completely free, as in the Riemann integral. Hence, we introduced the first-return integration process. It is clear from the definition that one can obtain many "wrong" answers for the integral. As a first step in the program, we showed [5] that for all L 1 functions one can always get the right answer by the first-return integral. The next step is to restrict the definition of the first-return integral so as to leave out the undesired consequences of the definition. Attempts were made at this in [3,8,11]. One of the conjectures was that if f : I → R is L 1 , then almost every trajectory yields the integral of f as the first-return integral, where I ≡ [0, 1]. However this is not the case: see the works of David Fremlin [10] and Jack Grahl [11]. The main goal is to define a first-return type integral which integrates a natural class of functions and gives the correct value as the integral. This problem is open.As an intermediate step to the above problem, it would be beneficial to characterize those functions which are firstreturn integrable. As mentioned earlier, we showed [5] that all Lebesgue integrable functions are first-return integrable with respect to a trajectory which gives the Lebesgue integral of function as the first-return integral. The next step is to decide if the same holds true for the Henstock-Kurzweil integral. In a clever and surprisingly simple example, Benedetto Bongiorno [1] showed that this is not the case; specifically, he showed that there is an improperly Riemann integrable function on I which is not first-return integrable. Moreover, he asked [2] if there exists a Henstock-Kurzweil integrable function f ,...

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“…Since t yields the Lebesgue integral of f , it follows from Theorem 2.1 in [9] and Theorem 2 in [6] that f is first-return continuous a.e. with respect to t.…”

Section: Lemma 4 Suppose F Is Lebesgue Integrable On [A B] and The mentioning

confidence: 94%

Section: Lemma 4 Suppose F Is Lebesgue Integrable On [A B] and The mentioning

confidence: 98%

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Functions not first-return integrable

Darji¹,

Evans²

2008

Journal of Mathematical Analysis and Applications

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“…At the end of twentieth century, a team of American mathematicians considered issues related to the theory, which can generally be called: "first return" ( [9], [10], [11]). It is worth noting that the first return continuous functions have the Darboux property.…”

Section: The Sharkovsky Propertymentioning

confidence: 99%

On rings of Darboux-like functions. From questions about the existence to discrete dynamical systems

Korczak-Kubiak¹,

Pawlak²,

Pawlak³

2013

Traditional and Present-Day Topics in Real Analysis. Dedicated to Professor Jan Stanisław Lipiński

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“…Here we wish to examine a subclass of each of these based on the following natural "universal" versions of these concepts. As was the case with universally first return continuous functions [6] and universally first return approachable functions [8], the points of quasicontinuity of the function will play a major role in understanding both UPA and SUPA.…”

Section: Introduction Definitions and Notationmentioning

confidence: 98%

Universally Polygonally Approximable Functions

Evans¹,

Humke²,

O’Malley³

2000

Journal of Applied Analysis

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It has recently been established that any Baire class one function f : [0, 1] → R can be represented as the pointwise limit of a sequence of polygonal functions whose vertices lie on the graph of f . Here we investigate the subclass of Baire class one functions having the additional property that for every dense subset D of [0, 1], the first coordinates of the vertices of the polygonal functions can be chosen from D.

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First Return Approachability

Cited by 10 publications

References 4 publications

Functions not first-return integrable

Functions not first-return integrable

On rings of Darboux-like functions. From questions about the existence to discrete dynamical systems

Universally Polygonally Approximable Functions

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