On the differentiability class of the admissible square roots of regular nonnegative functions

Bony, Jean-Michel; Colombini, Ferruccio; Pernazza, Ludovico

doi:10.1007/978-0-8176-4521-2_4

Cited by 7 publications

(8 citation statements)

References 5 publications

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“…The condition f ∈ F β requires a certain notion of "flatness" on the derivatives of f that is different from that considered in [1][2][3]. In particular, it contains functions not covered by their results for which Theorem 1 nonetheless applies [see (2.2) for a simple example].…”

Section: Theorem 1 Formentioning

confidence: 99%

“…One can exploit extra regularity by assuming that f and its derivatives vanish at all local minima (e.g., in [1][2][3] for admissible square roots). However, we take a different approach permitting functions to take small nonzero values.…”

Section: Definition and Comparison With Other Flatness Constraintsmentioning

confidence: 99%

“…The key property for additional regularity of roots is flatness, in the sense that whenever f is small then so are its derivatives, and in a series of recent papers, Bony et al [1][2][3] consider flatness conditions for admissible square roots. Recall that g is an admissible square root of f if f = g 2 (allowing g to switch signs).…”

Section: Introductionmentioning

confidence: 99%

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A regularity class for the roots of nonnegative functions

Ray

Schmidt-Hieber

2017

Annali di Matematica

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We investigate the regularity of the positive roots of a nonnegative function of one-variable. A modified Hölder space F β is introduced such that if f ∈ F β then f α ∈ C αβ . This provides sufficient conditions to overcome the usual limitation in the square root case (α = 1/2) for Hölder functions that f 1/2 need be no more than C 1 in general. We also derive bounds on the wavelet coefficients of f α , which provide a finer understanding of its local regularity.

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Section: Theorem 1 Formentioning

confidence: 99%

Section: Definition and Comparison With Other Flatness Constraintsmentioning

confidence: 99%

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A regularity class for the roots of nonnegative functions

Ray

Schmidt-Hieber

2017

Annali di Matematica

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“…Note that there always is a continuous parameterization of the roots in this case, e.g., by ordering them increasingly. Variations on this fundamental result (and its proof) appeared in [29], [51], [1], [25], [5], [6], [50], [7], [13], [32].…”

Section: 2mentioning

confidence: 99%

Selections of bounded variation for roots of smooth polynomials

Parusiński

Rainer

2020

Sel. Math. New Ser.

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We prove that the roots of a smooth monic polynomial with complex-valued coefficients defined on a bounded Lipschitz domain Ω in R m admit a parameterization by functions of bounded variation uniformly with respect to the coefficients. This result is best possible in the sense that discontinuities of the roots are in general unavoidable due to monodromy. We show that the discontinuity set can be chosen to be a finite union of smooth hypersurfaces. On its complement the parameterization of the roots is of optimal Sobolev class W 1,p for all 1 ≤ p < n n−1 , where n is the degree of the polynomial. All discontinuities are jump discontinuities. For all this we require the coefficients to be of class C k−1,1 (Ω), where k is a positive integer depending only on n and m. The order of differentiability k is not optimal. However, in the case of radicals, i.e., for the solutions of the equation Z r = f , where f is a complex-valued function and r ∈ R >0 , we obtain optimal uniform bounds.

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“…Broglia and the authors proved in [3] that this result is sharp in the sense that it is not possible to have in general a greater regularity for g. They also showed that if f is of class C 4 and vanishes at all its (local) minimum points, one can always find g of class C 2 and that the result is sharp. Later, in [4] it was proved that for f of class C 6 vanishing at all its minimum points one can find g of class C 2 and three times differentiable at every point.…”

Section: Introductionmentioning

confidence: 99%

On square roots of class Cm of nonnegative functions of one variable

Bony

Colombini²,

Pernazza³

2010

Annali Scuola Normale Superiore - Classe Di Scienze

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We investigate the regularity of functions g such that g 2 = f , where f is a given nonnegative function of one variable. Assuming that f is of class C 2m (m > 1) and vanishes together with its derivatives up to order 2m − 4 at all its local minimum points, one can find a g of class C m . Under the same assumption on the minimum points, if f is of class C 2m+2 then g can be chosen such that it admits a derivative of order m + 1 everywhere. Counterexamples show that these results are sharp.

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On the differentiability class of the admissible square roots of regular nonnegative functions

Cited by 7 publications

References 5 publications

A regularity class for the roots of nonnegative functions

A regularity class for the roots of nonnegative functions

Selections of bounded variation for roots of smooth polynomials

On square roots of class Cm of nonnegative functions of one variable

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