please comment if more explanation is needed
a)I have done this in Matlab i am attaching the code and the generated figure
code:
n=linspace(0,19,20); x=cos(pi*n/5); X=fft(x); stem(abs(X));xlabel('k');ylabel('X[k]');
Figure:
b)
= 5a. (10 pts) Let fr : [0, 1] → R, fn(x) ce-nzº, for m = = 1, 2, 3, .... Check if the sequence (fn) is uniformly convergent. In the case (fr) is uniformly convergent find its limit. Justify your answer. Hint: First show that the pointwise limit of (fr) is f = 0, i.e., f (x) = 0, for all x € [0, 1]. Then show that 1 \Sn (r) – 5 (w) SS, (cm) - Vžne 1...
Please write the answer clearly, Thak you so much! Question 4 Consider a p-variate sample with size T1.,n For some CE RxP and a E R9, consider the linear transformation Furthermore, for some D E R* and bER, consider the linear transformation For any j -1.....q and k-1.....r, denote by sy,z the sample covariance between (yjl-1 and (-k1 Define the matrix Y,Z Show that Y,Z where Sx is the sample covariance matrix of ...fn Question 4 Consider a p-variate sample...
The sequence r[n] = cos (Fn). oo < n < oo was obtained by sampling the continuous-time signal ra(t) = cos (Rot), -oo < t < oo at a sampling rate of 1000 samples/sec, what are two possible values of Ω0 that could have resulted in the sequence zn]?
Problem 2 1. Let fn(ar) n As the metric take p(x, y) = |x - y. Does lim, fn(x) exist for all E R? If it exists, is the convergence uniform. Justify 2. Consider fn(x) = x2m, x E [0, 1]. Is it true that lim (lim fn(= lim( lim fn(x)) noo x-1 Justify.
5. For any real number L > 0, consider the set of functions fx(x) = cos ("I") and In(x) = sin (^) se hos e mais a positive in where n is a positive integer. Show that these functions are orthonormal in the sense that (a) 1 L È Lsu(w) m(e)dx = {if m=n. fn (2) fm(x) dx = {. if m En if m =n -L 1 L il fn(x)9m(x)dx = 0 (c) il 9.(X)gm()dx = {{ if m=n...
3. For each n E N let fn : (1, 0) -+ R be given by f/(x) = Find the function f : (1, 0) - R to which {fn} converges pointwise. Prove that the convergence is not uniform 3. For each n E N let fn : (1, 0) -+ R be given by f/(x) = Find the function f : (1, 0) - R to which {fn} converges pointwise. Prove that the convergence is not uniform
For n ∈ N let fn : [0, 1] → R be given by fn(x) = sin((1 + n)x) / ( 1 + n ) ^(1/2) . Prove that {fn} is equicontinuous on [0, 1].
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...
For each n E N, define a function fn A - R. Suppose that each function fn is uniformly continuous. Moreover, suppose there is a function f : A R such that for all є 0, there exists a N, and for all x E A, we have lÍs(x)-f(x)|く for all n > N. Then f is uniformly continuous. Note: We could say that the "sequence of functions" f "converges to the function" f. These are not defined terms for...
Consider the sequence of functions fn : [0,1| R where each fn is defined to be the unique piecewise linear function with domain [0, 1] whose graph passes through the points (0,0) (, n), (j,0), and (1,0) (a) Sketch the graphs of fi, f2, and f3. (b) Computefn(x) dx. (Hint: Compute the area under the graph of any fn) (c) Find a function f : [0, 1] -> R such that fn -* f pointwise, i.e. the pointwise limit of...