b If we set e \u0394 N x \u03d5 N x E N x where E N x X n 1 \u03d5 N x n then one sees that i

# B if we set e δ n x ϕ n x e n x where e n x x n 1

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(b) If we seteΔN(x) =ϕN(x) +EN(x)whereEN(x) =X|n|≥1ϕN(x+n),then one sees that:(i)sup|x|≤1/2|E0N(x)| →0asN→ ∞. (ii)|eΔ0N(x)| ≤cN2.(iii)|eΔ0N(x)| ≤c/(N|x|3)for|x| ≤1/2.Moreover,R|x|≤1/2eΔ0N(x)dx= 0and-R|x|≤1/2xeΔ0N(x)dx1asN→ ∞.(c) The above estimates imply that iff0(x0)exists, then(f*eΔ0N)(x0+hN)f0(x0)asN→ ∞,whenever|hN| ≤C/N. Then, conclude that both the real and imaginary parts off1are nowhere differentiable, as in the proof given in Section 3, Chapter 4.(d) Leta >1,|b|<1. Also prove the Weierstrauss functionW(x) =n=1bncos(anx)is nowhere differentiable ifa|b| ≥1.34
Remark26.One can also check another characterization in [12, Section 16.H] that is differentfrom the methods of delay means.Proof.(a)(b)(c)(d)Remark27.For the differentiablity of Riemann functionn=1sinπn2xn2, see Jarnicki and Pflug[10, Chapter 13].References[1] Butzer, P. L., G. Schmeisser, and R. L. Stens: An introduction to sampling analysis. Nonuni-form sampling. Springer, Boston, MA, 2001. 17-121.[2] Chung, Soon-Yeong. ”Uniqueness in the Cauchy problem for the heat equation.” Proceedingsof the Edinburgh Mathematical Society 42.3 (1999): 455-468.[3] Daubechies, Ingrid: Ten lectures on wavelets. Vol. 61. SIAM, 1992.[4] DiBenedetto, E: Partial Differential Equations. 2nd ed., Springer, 2010.[5] Evans, L. C.: Partial Differential Equations, 2nd Edition. AMS, Providence, RI, 2010.[6] Folland, Gerald B: Fourier analysis and its applications. Vol. 4. American Mathematical Soc.,1992.[7] Folland, Gerald B: Real analysis: modern techniques and their applications. 2nd ed., JohnWiley and Sons, 1999.[8] Gilbarg, David, and Neil S. Trudinger: Elliptic partial differential equations of second order.Springer, 2001.[9] Grafakos, Loukas: Classical Fourier Analysis. 3rd ed., Springer, 2014.[10] Jarnicki, Marek, and Peter Pflug: Continuous Nowhere Differentiable Functions. SpringerMonographs in Mathematics, New York (2015).[11] John, Fritz: Partial Differential Equations. 4th ed., Springer, 1991.35
[12] Jones, Frank: Lebesgue Integration on Euclidean Space. Revised Edition. Jones & BartlettLearning, 2001.[13] Ladyzhenskaia,OlgaAleksandrovna,VsevolodAlekseevichSolonnikov,andNinaN.Ural’tseva. Linear and quasi-linear equations of parabolic type. Vol. 23. American Mathe-matical Soc., 1968.[14] Lieberman, Gary M. Second order parabolic differential equations. Revised Edition. Worldscientific, 2005.[15] Linares, Felipe, and Gustavo Ponce: Introduction to nonlinear dispersive equations. 2nd ed.,Springer, 2015.[16] Rosenbloom, P. C., and Widder, D.: A temperature function which vanishes initially. TheAmerican Mathematical Monthly 65.8 (1958): 607-609.[17] Smoller, Joel. Shock waves and reaction-diffusion equations. 2nd Edition. Vol. 258. Springer,1994.[18] Stein, Elias M., and Guido L. Weiss: Introduction to Fourier Analysis on Euclidean Spaces.Vol. 1. Princeton University Press, 1971.36