By Marek Jarnicki, Peter Pflug

ISBN-10: 3319126695

ISBN-13: 9783319126692

ISBN-10: 3319126709

ISBN-13: 9783319126708

Presents a scientific reference for learn into, and classes on, the topic of constant nowhere differentiable functions
Anthologizes over a hundred and fifty years of development with regards to non-stop nowhere differentiable functions
Consolidates an array of verified effects right into a unmarried, self-contained quantity together with specified proofs

This publication covers the development, research, and conception of constant nowhere differentiable features, comprehensively and accessibly. After illuminating the importance of the topic via an outline of its historical past, the reader is brought to the delicate toolkit of principles and tips used to check the categorical non-stop nowhere differentiable capabilities of Weierstrass, Takagi–van der Waerden, Bolzano, and others. sleek instruments of practical research, degree thought, and Fourier research are utilized to check the primary nature of continuing nowhere differentiable features, in addition to linear buildings in the (nonlinear) area of continuing nowhere differentiable services. To around out the presentation, complicated strategies from numerous parts of arithmetic are introduced jointly to offer a cutting-edge research of Riemann’s non-stop, and purportedly nowhere differentiable, function.

For the reader’s profit, claims requiring elaboration, and open difficulties, are in actual fact indicated. An appendix comfortably presents history fabric from research and quantity conception, and complete indices of symbols, difficulties, and figures improve the book’s application as a reference paintings. scholars and researchers of research will price this certain ebook as a self-contained advisor to the topic and its methods.

Topics
Real Functions
Fourier Analysis
Functional research

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Example text

Hence −∞ = min{D+ f (x), D− f (x)} < max{D+ f (x), D− f (x)} = +∞. 9 Behrend’s Method This whole section is based on [Beh49]. Let f := C a,b . 1. 1) (cf. 3). Then f ∈ ND∞ (R). It is clear that the result gets better as we increase the constant K. Proof . Let A := 1 1−a . Note that |Δf (x, x + h)| ≤ 2A |h| . We will prove that ∀x∈R, C>0 ∃h=hx,C , h =hx,C ∈R∗ : Δf (x, x + h) > C, Δf (x, x + h ) < −C. 2) Observe that |hx,C |, |hx,C | ≤ 2A C . In particular, h(x, C), h (x, C) −→ 0 when C −→ +∞. 2) implies that f ∈ ND∞ (R).

9. We have cos(πx) ≥ cos(πε) =: δ > 0, x ∈ Ip := [2p − ε, 2p + ε], p ∈ Z. Take an m ∈ Z and let J0 := Im . Observe that every interval of length ≥ 2 + 2ε contains an interval Ip for some p. Suppose that Jn := b1n Ipn (for some pn ). The length of Jn equals 1 2ε 2 + 2ε 2ε 2ε = n+1 . = n+1 b ≥ n+1 1 + bn b b ε b 1 Thus there exists a pn+1 such that Jn+1 := bn+1 Ipn+1 ⊂ Jn ⊂ · · · ⊂ J1 ⊂ J0 . Let 1 δ {x2m }. Then f ( 2 x2m ) ≥ 1−a and |2m − x2m | ≤ ε. ∞ n=0 Jn = 42 3 Weierstrass-Type Functions I δ Analogously, one gets a point x2m+1 such that f ( 12 x2m+1 ) ≤ − 1−a and |2m + 1 − x2m+1 | ≤ 1 ε.

For n ≥ m, we have p n−m (αm ± 1)) = −(−1)αm , cosp (2πbn x± m ) = cos (πb cosp (2πbn x) = cosp (πbn−m (hm + αm )) = (−1)αm cosp (πbn−m hm ). 28 3 Weierstrass-Type Functions I Hence ∞ an Qm,± = 2 n=m −(−1)αm (1 + cosp (πbn−m hm )) (±1 − hm )b−m ∞ = ∓(−1)αm (ab)m 2 an n=0 where Tm,± ≥ a0 1 + cosp (πbn hm ) = ∓(−1)αm (ab)m 2Tm,± , 1 ∓ hm 1 + cosp (πhm ) 2 ≥ . 1) where Um,± ≥ 1, |Vm,± | ≤ 1. The condition ab > 1 + 32 π implies that − sgn Δf (x, x+ m ) = − sgn Δf (x, xm ), |Δf (x, x± m )| −→ +∞. m→+∞ Hence, either (D+ f (x) = +∞ and D− f (x) = −∞) or (D− f (x) = +∞ and D+ f (x) = −∞).

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Continuous Nowhere Differentiable Functions: The Monsters of Analysis by Marek Jarnicki, Peter Pflug


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