solve 1-10 Explain each detail. For each of the functions in the list below calculate two...
Please answer it step by step and Question 2. uniformly
converge is defined by *f=0* clear handwritten,
please, also, beware that for the x you have 2 conditions , such as
x>n and 0<=x<=n
1- For all n > 1 define fn: [0, 1] → R as follows: (i if n!x is an integer 10 otherwise Prove that fn + f pointwise where f:[0,1] → R is defined by ſo if x is irrational f(x) = 3 11 if x...
3. Recall that R([0, 1]) is the normed linear space of integrable functions, with norm 1/2 Ils le = (150)Par)". Let (fn)nen be a sequence of functions in R, defined by 1<3 fn(x) = 1 VI V 0 < (a) Prove that (fn)nen is Cauchy. (b) Prove that (fn) does not converge in R([0, 1]). (Note: If it did, then what must the limit function be? Can this candidate function be in R?)
Many thanks!!
(a) Let fn(x) max(1 - |x -n|,0) for each n 2 1. Show that {fn} is a bounded sequence in LP (R) for all p E [1, 00]. Show that fn >0 pointwise everywhere in R, i.e. fn(x) -> 0 for all x E R. Show that fn does not converge to 0 in LP (R) (b) Fix p E 1, o0). Let fn E LP(0, 1) be defined by fn(x) n1/? on [0,1/n), and fn(x)0 otherwise. Show...
2c. (10 pts) Show that f given in 2b) is intergrable and [ 1 (2) dr = 2Ě (2n-1) 2d. (10 pts) Let 0 < < be given. Show that f given in 2b) is differentiable at each 1 € (5,27 - 8). Find f' (1). Hint: Use Problem 1 and the following formula In 2 (-1)"-1 Σ 7 n=1 2. (40 pts) Let fn: R → R be given by fn (x) = sin (nx) 3 ηε Ν. n2...
Please solve the exercise 3.20 .
Thank you for your help !
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Review. Let M be a o-algebra on a set X and u be a measure on M. Furthermore, let PL(X, M) be the set of all nonnegative M-measurable functions. For f E PL(X, M), the lower unsigned Lebesgue integral is defined by f du sup dμ. O<<f geSL+(X,M) Here, SL+(X, M) stands the set of all step functions with nonnegative co- efficients. Especially, if f e Sl+(X,...
4. Suppose (fr)nen is a sequence of functions on [0, 1] such that each fn is differentiable on (0,1) and f(x) < 1 for all x € (0,1) and n e N. (a) If (fn (0))nen converges to a number A, prove that lim sup|fn(x) = 1+|A| for all x € [0, 1]. n-too : (b) Suppose that (fr) converges uniformly on [0, 1] to a function F : [0, 1] + R. Is F necessarily differentiable on (0,1)? If...
can you please solve this problem step by step, thank
you!!
1. Consider a DSB-SC signal with noise passes through a demodulator shown below. ViC0) 2(0) но но 0) The input signal plus noise is v,(t)-5,() + n'(t) where 5,0): Am(1)cos2r/rt, m(1) cos 2π/J is the message signal with f-</м , carrier frequency is f> fe , and noise n' (r) has power spectral density function G,じ)= η . The local carrier is v,(1)s 2 cosZrw. The carrier filter is...
Please explain in full detail!
For two fixed positive numbers a, b, 0 < a < b (for example, a = 1, b = 2), let a = V ab and b = - - Define aj+1 = a;b; and b;+1 = - T for any positive interger j. a+b (1) Prove that the sequence {aj} is convergent. (2) Prove that the sequence {b;} is convergent. (3) Prove that lim a;= lim bj. 1700
(B)(C)(D)(E)(G)(I)(J)
39 Write the Taylor expansion of function f order n at to given below. 1 (a) (g)2+v1+ I, n 2,ro - 0 n = 7, xo = 0 1 -2-3 (b) sin z cos(2x), (c) z In(2+3z), VI+I n= 3, To = 0 1 +e-1/ (h) 2+x n =5,xo= +o0 n= 3, ro = 1 T (i) cos (2r), n 4, xo= 6 (d) n = 7,xo = 0 COSI i) V+-VI3-, n 4, 1o =+00 (e) In(1+ arcsin(2r)),...
1. (2 pts each) The graph of some unknown function f is given below. 10 6/ 8-64-2 624 10 12 Use the graph to estimate the following quantities: (0 f (9) (g) f(4) b) lim (a) lim (e) (d) lim ( 6) (e) lim f(x) (c) lim f(x) if g(x)f(x) 6) a value of r where f is continuous but not differentiable (k) a value of r where f"(x) 0 and f"(x)>0 (1) the location of a relative maximum value...