Kyle Roth's Math 320 blog

The most important things I learned in this course were big-Oh notation, the Master Theorem, the various algorithm styles available, and probability theory. To review for the test I'll make a list of important theorems and definitions to memorize, and practice proving simple facts about each of them. I'll also go over past tests and make sure I could get 100% on them. In class tomorrow I'd like to go over an example of Monte Carlo integration. I feel like in this class I've learned how to apply the mathematics I've learned to real-world problems, more than in any other class I've taken. I feel that way because I'm most interested in using computers to solve problems, and the principles we've talked about this semester have been foundational to my understanding of how to do that.

I see the graphs of the Daubechies db2 scaling function, but I really don't understand how to use it since there's no definition that I can easily look at. We'll need to talk about that in class in order for me to understand it. Also, the proof for 8.7.5(iv) was long and I couldn't follow the algebra very well. The idea that you can use any version of the wavelets you want to reproduce the function you're looking at is very powerful. It's interesting to see such a complicated function like the Daubechies function be used. I'd like to know more about how to choose a function that works well for different types of data.

I'm having a hard time understanding how to build a basis out of wavelet vectors. It's not immediately obvious to me how to know what phi functions belong to each V. Honestly, this section was hard because I didn't initially understand Haar wavelets from the last section It's interesting to see that the FWT doesn't take as long as the FFT for a given input. When we practice with FWTs in the lab, I feel like I'll get better understanding of the concept then. I can see this being useful for all sorts of curve analyses when they need to be approximated but what you want to see is general trends.

I like understanding how it's possible to create different representations of the same function depending on what aspects of the curve we're interested in. I would be interested in doing a coding lab where we use these Haar wavelets to solve problems, so I can get a feel for how each type of decomposition is used in applications. The most difficult part of the reading was understanding the proof that sons and daughters are complements that make up the space V_j. I don't yet have a good sense for how to use the definitions of these different relatives (sons, daughters, father, mother, etc.).

Now I know what antialiasing is in computer graphics! It makes so much sense now. It's cool to see how something as simple as the Nyquist value can tell you so much about the parameters of a given problem. The methods for antialiasing are intuitive as well, kind of in the same way as least-square solutions were in Volume 1. It would be nice to get some intuition for why the Nyquist rate is twice the Nyquist value, and why the function becomes uniquely determined at that point and not before. Also, I'd like to run through a full example in class of how we can sample from a function, find the DFS, and perform ant-aliasing. That would help me understand the process a little better than before.