# Mathematics of Signal Representations

## Math4355 - Spring 2013 - Homework

**Assignment 1, due Thursday, January 24**

- Review Sections 0.1-0.3, memorize properties of an inner
product, definitions of convergence
in square norm (Definition 0.7), pointwise and uniform
convergence (Definition 0.8).
- Do Exercises in Chapter 0, p. 34-35: 1; 3; 6; 7;
- Show that the sequence of functions given by f
_{n}(x)= x e^{-nx} converges uniformly on the interval [0,1]
to zero. Hint: Use calculus to find where the function
maximally deviates from zero.
- (Matlab project) Define a variable with the command
"x=0:0.001:1;" in Matlab. This is just a row vector with values ranging
from 0 to 1 in steps of 0.001 increment. Now plot the
functions f
_{n}(x) = x e^{-nx}
for n=2, n=5, and n=10. Refer to Matlab Help on arithmetic (element by element
product) and elementary functions (exponential) if needed. You may want to call the
resulting vectors of function values f2, f5, and f10. Plotting the
functions is simply done by "plot(x,f2);" and similarly for f5 and
f10. Export (save) the plots and print them. Attach a printout of your
plots to your homework, together with the Matlab code you used to
generate the function values.