Adjoint classes of functions in the H1 sense

Abstract

Using the concept of the H 1 -integral, we consider a similarly defined Stieltjes integral. We prove a Riemann-Lebesgue type theorem for this integral and give examples of adjoint classes of functions.

Integration with One-Dimensional Space of Gauges

“…G? This question has already been asked [13,Problem 3.19]. From Theorems 1 and 3 we can deduce also that if G is normalized and VBG * , then each H 1 -integrand is an integrand in our new sense.…”

“…Nevertheless, the class of integrators covered by the characterization we gave in [13,Theorem 3.17], is lesser than it was expected at first, as its members have to obey some constraints at discontinuity points. In present paper, instead of modifying the characterizing condition from [13], we make an attempt to provide another Stieltjes integration, for which that condition becomes accurate for the class of integrators originally expected for H 1 -integration, namely for all VBG * -functions. Reduced to the non-Stieltjes case (i.e., with the integrator being id), it results in an integration equivalent to the H 1 one, so giving a nice (since simpler than the original) definition of that integration process.…”

“…In our considerations on H 1 -integration in its Stieltjes version [13], we sought for the widest possible (i.e., concerning also some integrators of unbounded variation) characterization theorem for integrable functions. Nevertheless, the class of integrators covered by the characterization we gave in [13,Theorem 3.17], is lesser than it was expected at first, as its members have to obey some constraints at discontinuity points.…”

“…In our considerations on H 1 -integration in its Stieltjes version [13], we sought for the widest possible (i.e., concerning also some integrators of unbounded variation) characterization theorem for integrable functions. Nevertheless, the class of integrators covered by the characterization we gave in [13,Theorem 3.17], is lesser than it was expected at first, as its members have to obey some constraints at discontinuity points. In present paper, instead of modifying the characterizing condition from [13], we make an attempt to provide another Stieltjes integration, for which that condition becomes accurate for the class of integrators originally expected for H 1 -integration, namely for all VBG * -functions.…”

“…The general line of reasoning in this section, actually follows the pattern of [13]. From now on, f, G : [a, b] → R unless otherwise stated.…”