The regulated primitive integral

Abstract

A function on the real line is called regulated if it has a left limit and a right limit at each point. If f is a Schwartz distribution on the real line such that f = F ′ (distributional or weak derivative) for a regulated function F then the regulated primitive integral of f is (a,b) with similar definitions for other types of intervals. The space of integrable distributions is a Banach space and Banach lattice under the Alexiewicz norm. It contains the spaces of Lebesgue and Henstock-Kurzweil integrable functions as continuous embeddings. It is the completion of the space of signed Radon measures in the Alexiewicz norm. Functions of bounded variation form the dual space and the space of multipliers. The integrable distributions are a module over the functions of bounded variation. Properties such as integration by parts, change of variables, Hölder inequality, Taylor's theorem and convergence theorems are proved. 2000 subject classification: 26A39, 46E15, 46F05, 46G12∞ −∞ f φ defines a distribution. The differentiation formula T ′ , φ = − T, φ ′ ensures that all distributions have derivatives of all orders which are themselves distributions. This is known as the distributional derivative or weak derivative. The formula follows by mimicking integration by parts in the case of T f where f ∈ C 1 . We will usually denote distributional derivatives by F ′ and pointwise derivatives by F ′ (t). For T ∈ D ′ and t ∈ R the translation τ t is defined byMost of the results on distributions we use can be found in [11].It will be shown in Theorem 4 below that primitives are unique and that the spaces A R and B R are isometrically isomorphic, the integral constituting a

The one-dimensional heat equation in the Alexiewicz norm

“…For example, the Dirac measure is the distributional derivative of the Heaviside step function H = χ (0,∞] . The regulated primitive integral is discussed in [24]. While many of the properties proven in Theorem 3.1 continue to hold, a key difference between the two integrals is that, for the regulated primitive integral, translation is not continuous in the Alexiewicz norm.…”

Integrals and Banach spaces for finite order distributions

“…And, {a} f = F (a+) − F (a−). This integral was described in detail in [28]. The multipliers are the functions of bounded variation.…”

“…This agrees with the action of δ as a tempered distribution, for which g must be in the Schwartz space S. Notice that changing the value of H 0 (0) does not affect the value of . See [28] for more examples of integration in A 1 r . Using (9), an example in A n r is…”

“…In [26] a change of variables formula was proved in A 1 c when ψ ∈ C(R), i.e., no monotonicity or pointwise differentiability is assumed. In [28] a change of variables formula was proved in A 1 r when ψ was piecewise monotonic. For A n r we have the simple case of composition with a linear function.…”

“…Then A n r is the set of distributions that are the nth distributional derivative of a function in B r . The case n = 1 was studied in [28]. The space A 1 r contains A 1 c as well as all finite signed Borel measures.…”