The <i>p<sup>n</sup></i>-integral

Bullen, P. S.

doi:10.1017/s1446788700010065

Cited by 13 publications

(21 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Also our integrals are shown to be equivalent to the r-} D -and rintegrals in Bullen [2] and Bullen and Mukhopadhyay [3] . It is thus evident that a function / is r-integrable if and only if it is C, -P -integrable and a result that the extension of the (L ) integral actually has.…”

Section: If the Condition (Ii) Is Replaced By The Condition That F Ismentioning

confidence: 76%

See 1 more Smart Citation

Riemann derivatives and general integrals

Sarkar

Das

1987

Bull. Austral. Math. Soc.

View full text Add to dashboard Cite

Sargent and later Bullen and Mukhopadhyay obtained a definition of absolutely continuous functions, functions, that is related to kth Peano derivatives. The generalised notions of ACkG*, [ACkG*], ACkG* above, etcetera functions led Bullen and Mukhopadhyay to define certain general integrals of the kth order.The present work is concerned with a further simplification of the definitions of such functions by the use of divided differences but still retaining similar fundamental properties. These concepts lead to the introduction of Denjoy and Ridder type integrals which are shown to be equivalent to a Perron type integral that corresponds to kth Riemann* derivatives. All three of these integrals are shown to be equivalent to the three integrals of Bullen and Mukhopadhyay.

show abstract

Section: If the Condition (Ii) Is Replaced By The Condition That F Ismentioning

confidence: 76%

“…Peano derivative and one sided upper (respectively lower) kth Riemann* derivative at x are equal. Thus it follows that the above integral is same as the P^-integral, for n = k , defined in Bullen [2].…”

Section: If Ismentioning

confidence: 84%

Riemann derivatives and general integrals

Sarkar

Das

1987

Bull. Austral. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…[22] gives necessary use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788700027087 conditions for integrals of the type using derivatives like (3,4), and [22], page 114, (27), (28), leads to the following details. Given a < b and each w, v in a =£ u < v < b let N s (u, v, t)(u < t < v) be monotone increasing in t with…”

Section: Generalized Perron and Tv-variational Integrationmentioning

confidence: 99%

A problem in two-dimensional integration

Henstock

1983

J Aust Math Soc A

View full text Add to dashboard Cite

If two functions of a real variable are integrable over two intervals, say of t, T, respectively, then the product of the two functions should be integrable over the rectangular product of the two intervals of t and T. For the Lebesgue integral, definable using non-negative functions alone, the proof is easy. For non-absolute integrals such as the Perron, Cesaro-Perron, and Marcinkiewicz-Zygmund integrals we have difficulties since the functions cannot be assumed non-negative. But the present paper gives a proof. [a, b], [a, /?] of the real line, respectively, then f(t) that is, (1) / f(t) 0.[26], page 63, Theorem 7, is more general, but the assumption of bounded variation shows that we are again dealing with absolute integrals. The problem ought to be easy even when the integrals are not absolute, but the changing of sign of / and

show abstract

“…These two integrals are in fact equivalent provided that the space uL is closed under uniform convergences. For these results, see Remark 3, Theorems (6,1) and (7,1).…”

mentioning

confidence: 96%

“…The C" ^-integral has been further investigated by many authors. See, for example, [2], [3], [7], [9], [13], [17], [19]- [24], [39]- [41], [44]. It should be pointed out that a defect in [12] has been corrected by Verblunsky in [43], and another defect in [39] also has been corrected by Verblunsky in [44].…”

mentioning

confidence: 99%

Generalizations of Cesàro continuous functions and integrals of Perron type

Lee¹

1981

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. The linear space of all the Cesaro continuous functions of any order is extended by introducing pointwisely Cesàro continuous functions and exact generalized Peano derivatives. Then six generalized integrals of Perron type are defined and studied. They are based on three recent monotonicity theorems and each depends on an abstract upper semilinear space of certain functions. Some of the integrals are more general than all the integrals in the Cesàro-Perron scale provided that the abstract semilinear space is taken to be the linear space of all the pointwisely Cesàro continuous functions or all the exact generalized Peano derivatives. That such a concrete general integral is possible follows from the fact proved here that each exact generalized Peano derivative is in Baire class one and has the Darboux property. Relations between the pointwisely Cesàro continuous functions or the exact generalized Peano derivatives and functions defined by means of the values of certain Schwartz's distributions at "points" are also established.

show abstract

The pⁿ-integral

Cited by 13 publications

References 8 publications

Riemann derivatives and general integrals

Riemann derivatives and general integrals

A problem in two-dimensional integration

Generalizations of Cesàro continuous functions and integrals of Perron type

Contact Info

Product

Resources

About

The pn-integral

Cited by 13 publications

References 8 publications

Riemann derivatives and general integrals

Riemann derivatives and general integrals

A problem in two-dimensional integration

Generalizations of Cesàro continuous functions and integrals of Perron type

Contact Info

Product

Resources

About

The pⁿ-integral