Lecture Notes

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Hello! My name is Muireann,  //pronounced like Marin County with a soft ending//, I'm an exchange student from Ireland! I grew up and live in Cork and study in the University there (UCC). I'm a double major in Mathematical Sciences and Physics.

I began university thinking I'd only like Algebra and Applied Maths but have found interest in subjects like analysis. I took Complex Analysis M185 last semester. I'm really liking the approach to this class so far it is very different to anything I've taken before, so I am really looking forward to it.

My other interests are music and sport. I play piano and saxophone, I teach piano at home and currently trying to teach myself the trumpet. I also ran the half marathon across the golden gate last November! 

**Study Methods**  // \\
**//Revision Questions//**I find the best way to learn analysis is starting with definitions. So, every analysis class I make revision questions from every book. This helps  me to form proofs ect later on. Maybe some of you will find them useful too. I will link them here: //
Lebesgue Outer Measure (Pugh Chp6 and Tao Chp 7) //

{{ :math105-s22:s:mheaney:m105_fourier_series_revision_questions.pdf |}}

**//Tool Box//** As I go through revising for exams and for problem sets. I try to put together a toolbox. Just definitions and theorems we have covered without the proofs. I'll add that  here when it is complete


__Class Notes__
//Class Notes will be uploaded in a pdf format//

__Homework Problems__
//Homework will be uploaded here in a pdf format//

__Interesting Links / Problems__

//Here I will post any useful links or resources I find myself using. As well as any interesting problems I have that a class mate could shed some light on.//


//**HW4 Lebesgue Integral summary**//
Up until now we had only been solving Reimann Integral. The Lebesgue integral is in some sense a generalization of the Riemann integral.  This was only possible with 'Reimann Integrable' functions, i.e not all functions could be integrated. A classic example is f(x) = 1, x is a rational number and zero otherwise on the interval [0,1]. 

//The steps for Lebesgue Integral //
1. subdivide the range of function into infinitely many intervals
2. construct a simple function by taking a function whose values are those finitely many numbers
3. Take limit of these simple functions, when more points are added in the range of original functions. 


//**HW6 Summary of results for Lebesgue Measure**//

**Outer Measure**
1. From Pugh's approach, he defines the outer measure of a set using Intervals, Rectangles, and boxes. 
Lebesgue outer measure of a set $A \subset \R$ is\\

$m^{*}A$ = $inf${ $\Sigma_{k}$ $\vert$ $I_{k}$ | : {$I_{k}$ is a covering of A by open intervals} \\

The important theorem for outer measure is proving its properties: \\
//a) The outer measure of the empty set is 0, $m^{*}\emptyset$ = 0// \\
//b) If A $\subset$ B then $m^{*}A$ $\leq$ $m^{*}B$// \\
//c) if A = $\cup$ $A_{n}$ then $m^{*}A$ $\leq$ $\Sigma$ $m^{*}A_{n}$ \\

Another definition we continued to use throughout Lebesgue Theory was ** If $Z \subset \R^{n}$ has outer measure zero then it is a zero set** \\
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2. The second topic we learnt about was **Measurability** \\
First defining **(Lebesgue) measurable**  \\
A set $E \subset \R$ is Lebesgue measurable if the division $E|E^{c}$ of $\R$ is so "clean" that for each "test set" $X \subset \R$ we have \\ \\
$m^{*}X$ = $m^{*}$( X $\cap$ E) + $m^{*}$(X $\cap$ $E^{c}$) \\
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The fact an empty set is measurable and that if a set is measurable then so is its compliment was needed for futher chapters as we will see...

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In this section we were introduced to $\sigma$ - algebra, which is a collection of sets that includes the empty set, is closed under complement and is closed under countable union. \\
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3. **Mesomorphism** \\
// measure space, differences between mesemorphism, meseomorphism, and mesisometry //
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**4. Regularity** \\
**Theorem 11** // Lebesgue measure is **regular** in the sense that each measurable set E can be sandwiched between an $F_{\sigma}-set$ and a $G_{\delta}-set$ , F $\subset$ E $\subset$ G , such that G\F is a zero set. Conversely, if there is such an F $\subset$ E $\subset$ G, E is measurable. //
Affine motions..
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**__Final Essay- Fast Fourier Transforms (FFT)__**
I took a perspective based on my own background and how I visualize and use FT and FFT's in general
{{ :math105-s22:s:mheaney:final_essay_what_is_fast_fourier_transform.pdf |}}
Lecture Notes

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