Non-analytic Smooth Function - A Smooth Function Which Is Nowhere Real Analytic

A Smooth Function Which Is Nowhere Real Analytic

A more pathological example, of an infinitely differentiable function which is not analytic at any point can be constructed by means of a Fourier series as follows. Let A:={2n : nN } be the set of all powers of 2, and define for all xR

Since the series converge for all nN, this function is easily seen to be of class C∞, by a standard inductive application of the Weierstrass M-test, and of the theorem of limit under the sign of derivative. Moreover, for any dyadic rational multiple of π, that is x:=π p/q with pN and q ∈ A, and for all order of derivation n ∈ A, n ≥ 4 and n > q we have

where we used the fact that cos(kx)=1 for all k > q. As a consequence, at any such xR

so that the radius of convergence of the Taylor series of f at x is 0 by the Cauchy-Hadamard formula . Since the set of analyticity of a function is an open set, and since dyadic rationals are dense, we conclude that f is nowhere analytic in R.

Read more about this topic:  Non-analytic Smooth Function

Famous quotes containing the words smooth, function, real and/or analytic:

    Beautiful women seldom want to act. They are afraid of emotion and they do not try to extract anything from a character that they are portraying, because in expressing emotion they may encourage crow’s feet and laughing wrinkles. They avoid anything that will disturb their placidity of countenance, for placidity of countenance insures a smooth skin.
    Laurette Taylor (1887–1946)

    My function in life is not to be a politician in Parliament: it is to get something done.
    Bernadette Devlin (b. 1947)

    That is the real pivot of all bourgeois consciousness in all countries: fear and hate of the instinctive, intuitional, procreative body in man or woman. But of course this fear and hate had to take on a righteous appearance, so it became moral, said that the instincts, intuitions and all the activities of the procreative body were evil, and promised a reward for their suppression. That is the great clue to bourgeois psychology: the reward business.
    —D.H. (David Herbert)

    “You, that have not lived in thought but deed,
    Can have the purity of a natural force,
    But I, whose virtues are the definitions
    Of the analytic mind, can neither close
    The eye of the mind nor keep my tongue from speech.”
    William Butler Yeats (1865–1939)