3. Prove that for all f e L(T), the following estimate holds sup If(n)l lILL nEz
An important fact we have proved is that the family (enr)nez is orthonormal in L (T,C) and complete, in the sense that the Fourier series of f converges to f in the L2-norm. In this exercise, we consider another family possessing these same properties. On [-1, 1], define dn Ln)-1) 0, 1,2, Then Lv is a polynomial of degree n which is called the n-th Legendre polynomial. (a) Show that if f is indefinitely differentiable on [-1,1], thern In particular,...
3. (a) Given n e N, prove that sup{.22 : 0<x<1} = 1 and inf{.22n: 0<x<1} = 0. (b) Find the supremum of the set S = {Sn: ,ne N}. Give a proof.
Prove the reverse Fatou lemma: If (f)Ro is a sequence in L'(a, bl; R) and if there exists some nonnegative g E L (a, b]; R) with fk K g a.e. for all k E N, then innsupf.[a.bfk/[a,blim sup fk.
{ <N> : L(M) contains a string starting with a). Rice's theorem can be F 20, L used to prove that LD. T L(M2) >. Rice's theorem can be used to prove T F 21. L that L D. <M,, M2> L(M,) 22. L-( <M,M> : L(M) = L(M2) }, and R is a mapping reduction function from H to L. It is possible that R retur a TM. T F ns <M#>, where M # is the string encoding...
(7) In this problem let X denote the vector space C(0, 1) with the sup norm. (a) Given f e X, define d(f) = f2. : X → X is differentiable, and Prove that φ find φ'(f). (b) Given f e X, define 9(f) = J0 [f(t)]2dt. Prove that Ψ : X → R is differentiable. and find Ψ(f).
(7) In this problem let X denote the vector space C(0, 1) with the sup norm. (a) Given f e X,...
Prove the Binomial Theorem, that is Exercises 173 (vi) x+y y for all n e N C) Recall that for all 0rS L is divisible by 8 when n is an odd natural number vii))Show that 2 (vin) Prove Leibniz's Theorem for repeated differentiation of a product: If ande are functions of x, then prove that d (uv) d + +Mat0 for all n e N, where u, and d'a d/v and dy da respectively denote (You will need to...
6.) Use induction to prove that the following holds for each n 2
N; make
sure to state your induction hypothesis carefully:
6
(74n + 5):
6.) Use induction to prove that the following holds for each n E N; make sure to state your induction hypothesis carefully: 6|(74 + 5). 4n
(5) Let f: [0, 1 R. We say that f is Hölder continuous of order a e (0,1) if \f(x) -- f(y)| . , y sup [0, 1] with 2 # 1£l\c° sup is finite. We define Co ((0, 1]) f: [0, 1] -R: f is Hölder continuous of order a}. = (a) For f,gE C ([0, 1]) define da(f,g) = ||f-9||c«. Prove that da is a well-defined metric Ca((0, 1) (b) Prove that (C ([0, 1]), da) is complete...
Let f(n) = 5n^2. Prove that f(n) = O(n^3). Let f(n) = 7n^2. Prove that f(n) = Ω(n). Let f(n) = 3n. Prove that f(n) =ꙍ (√n). Let f(n) = 3n+2. Prove that f(n) = Θ (n). Let k > 0 and c > 0 be any positive constants. Prove that (n + k)c = O(nc). Prove that lg(n!) = O(n lg n). Let g(n) = log10(n). Prove that g(n) = Θ(lg n). (hint: ???? ? = ???? ?)???? ?...
Question #4
Consider the linear map, Prove that L^n x goes to 0 for all x
in R^2. prove that if x does not lie on the y axis then the orbit
of x tends to 0 tangentially to the x -axis.
4. Consider the linear map 0 L(x) = X. Prove that L"X → 0 for all x E R2. Prove that, if x does not lie on the y-axis, then the orbit of x tends to 0 tangentially...