m -approximate Taylor polynomial
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© Springer-Verlag GmbH Germany, part of Springer Nature 2019
Silvano Delladio
m-approximate Taylor polynomial Received: 9 April 2019 / Accepted: 8 November 2019 Abstract. We define and investigate the class τmN (x) of functions provided with the notion of m-approximate N th degree Taylor polynomial at x ∈ Rn . In particular we compare τmN (x) with the class t k, p (x) of functions having the derivative of order k in the L p sense at x (Ziemer in Weakly differentiable functions. GTM, Springer, Berlin, 1989, Section 3.5). We apply this machinery to prove several properties of Sobolev functions, such as a new sectional superdensity property.
1. Introduction In this manuscript we will take a further step along the path on the applications of the graded notion of “m-density point” that we have undertaken in papers [6–8] and then continued through several subsequent papers [9–15]. Recall that (compare [8]) a point x ∈ Rn is said to be a m-density point of E ⊂ Rn , with m ∈ [n, +∞), if Ln (Br (x)\E) = o(r m )
(as r → 0+)
where Ln and Br (x) denote, respectively, the Lebesgue outer measure on Rn and the open ball in Rn of radius r centered at x. According to this definition, the larger m, the higher the concentration of E at x (and the more x will resemble an interior point, in the sense of Euclidean topology). Observe that, in particular, x is a n-density point of E if and only if x is a point of Lebesgue density of E, that is Ln (Br (x) ∩ E)/Ln (Br (x)) → 1 (as r → 0+). Moreover, as one expects, the notion of m-density point extends easily to the context of C 1 submanifolds of an Euclidean space, compare [15]. The set of m-density points of E is denoted by E (m) . As examples of application of m-density points, we also mention [19, Section 6.D] (about superdensity topologies and Lusin–Menchoff property) and [20] (where higher order densities of exceptional sets vanishing at almost all are employed to prove second order rectifiability of integral varifolds of locally bounded first variation). In this context we have recently provided in [16] the definition of m-approximate limit at x ∈ Rn of a function f : A ⊂ Rn → R, which for m = n reduces to S. Delladio (B): Department of Mathematics, University of Trento, Via Sommarive 14, Povo, 38123 Trento, Italy e-mail: [email protected] Mathematics Subject Classification: 28Axx · 26B05 · 54C08 · 26A45 · 46E35
https://doi.org/10.1007/s00229-019-01167-0
S. Delladio
ordinary approximate limit (e.g., see [17, Section 1.7.2]). The following characterization clarifies such a notion: A number l ∈ R is the m-approximate limit at x of f if and only if x ∈ [ f −1 (I )](m) for every open interval I containing l. Based on this definition, as expected, it was natural to give (always in [16]) the pointwise notions of m-approximate continuity and m-approximate differentiability of a function. Here is a summary of the main results obtained in [16]: • A function f : A ⊂ Rn → R is m-approximately continuous (resp. differentiable) at x ∈ A if and only if there exists a set E ⊂ A suc
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