The double Fourier series of functions of the generalized bounded variation class {n/ ln(n + 1)}∗BV are shown to be Pringsheim convergent everywhere. In a certain sense, this result cannot be improved. In general, functions of class Λ∗BV, defined here, have quadrant limits at every point and, for f ∈ Λ∗BV, there exist at most countable sets P and Q such that, for x / ∈ P and y / ∈ Q, f is conti...

By Theorem 2.1 the product φ1φ2 is of bounded variation and by Theorem 2.24 all three Riemann–Stieltjes integrals involved exist as the limit of Riemann–Stieltjes sums as |Γ| → 0. In particular we can find a δ1 > 0 such that ∣∣∣∣∫ b a fφ2 dφ1 −RΓ(fφ2, φ1) ∣∣∣∣ < ε , ∣∣∣∣∫ b a fφ1 dφ2 −RΓ(fφ1, φ2) ∣∣∣∣ < ε for all partitions with |Γ| < δ1. Let Γ = {xi}i=0 be any partition of [a, b] with intermed...

In this paper we prove necessary and sufficient conditions for the validity of the classical chain rule in the Sobolev space W 1,1 loc R ;R and in the space of functions of bounded variation BVloc R ;R .

Using a Riemann-Lebesgue lemma for the Fourier transform over the class of bounded variation functions that vanish at infinity, we prove the Dirichlet–Jordan theorem for functions on this class. Our proof is in the Henstock–Kurzweil integral context and is different to that of Riesz-Livingston [Amer. Math. Monthly 62 (1955), 434–437]. As consequence, we obtain the Dirichlet–Jordan theorem for f...

The present paper proposes a new condition to replace both the (O-regularly varying) quasimonotone condition and a certain type of bounded variation condition, and shows the same conclusion for the uniform convergence of certain trigonometric series still holds.

We discuss determination of jumps for functions with generalized bounded variation. The questions are motivated by A. Gelb and E. Tadmor [1], F. Móricz [5] and [6] and Q. L. Shi and X. L. Shi [7]. Corollary 1 improves the results proved in B. I. Golubov [2] and G. Kvernadze [3]. Ÿ

For a Borel-measurable function f : R→ R satisfying f(0) = 0 and

In this paper we have introduced the notion of fuzzy real-valued p-bounded variation double sequence space p F bv 2 (1 ≤ p < ∞). We have studied some important properties like completeness, convergence free, solid, symmetric. Some inclusion results are also established.

The measure approach represents a preference relation over functions by the measure of Ž . their epigraphs or hypographs . This paper proves a measure representation theorem for a class of increasing functions and shows how its proof can be modified to yield another measure representation theorem for functions of bounded variation. q 1998 Elsevier Science B.V. JEL classification: D81