Abstract
In this little note, we take a look at the connection between the smoothness of a function of bounded variation and its variation function. Based on a number of well-known classical results we expect a very close relationship between the smoothness of these two functions. However, we immediately see that a near at hand generalization of the classical estimates doesn’t yield correct results. Therefore, we will have to introduce a standard kind of averaging process in order to come to properly suited smoothness measures in this context. The precise averaging idea is exactly the same as the one used by Sendov and Popov to switch from the classical ω-modulus to the so-called τ-modulus.
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Lenze, B. (1999). On a Special Property of the Averaged Modulus for Functions of Bounded Variation. In: Müller, M.W., Buhmann, M.D., Mache, D.H., Felten, M. (eds) New Developments in Approximation Theory. ISNM International Series of Numerical Mathematics, vol 132. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8696-3_8
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DOI: https://doi.org/10.1007/978-3-0348-8696-3_8
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