Analysis

The infinite impulse response is a property of a filter of digital signal processing. The filter response of the impulse signal does not end infinitely. The filter can be represented in either the time domain or frequency domain. Time-domain filter modifies time-domain input signal to time-domain output signal. The frequency domain lets us understand or to design the effect of the filter. The discrete Fourier transformation ${\Sigma }_{n=0}^{N-1}x\left(n\right)e^{-i2\pi nm/N}$ transforms time domain to frequency domain in the finite impulse response.

Cauchy’s integral formula defines analytic function evaluation with path integral with denominator translation at evaluation point ($\frac{1}{z-a}$). $f\left(a\right)=\frac{1}{2\pi i}{\oint }_{\gamma }\frac{f\left(z\right)}{z-a}dz$ $f^{\left(n\right)}\left(a\right)=\frac{n!}{2\pi i}{\oint }_{\gamma }\frac{f\left(z\right)}{{(z-a)}^{n+1}}dz$ Cauchy’s integral formula is a limit of path. ${lim}_{r\to 0}\gamma :|z-z_0|=r$ Taylor series evaluated a analytic function by approximation at an open disc $D(z_0,r)$. $f\left(x\right)={\sum }_{n=0}^{\infty }{a}_n(x-b{)}^n$ $\frac{f^{\left(n\right)}\left(b\right)}{n!}=a_n$

The presence of primitive function is a strong condition that makes a function is analytic in a disc $D(a,R)$. The meaning is the presence of primitive function is confusing at first to me. If a function is integrable, then integration value and a primitive function can be determined. But in complex analysis this is not the case. In real analysis, the integral interval $[a,b]$ is unique, but in complex analysis the integral interval should be determined by line path $\Gamma =g\left(x\right)$.

A norm of linear transformation $\Lambda :X\to Y$ is defined by $\parallel \Lambda \parallel =sup\{\parallel \Lambda (x)\parallel :x\in X,\parallel x\parallel \le 1\}$. We can give a norm to a space or a set. A norm determines the size of a vector in the function space. The way of measure the size of a vector gives important properties of the space like boundness, completeness or orthogonality.

It is the main subject of analysis that finding conditions making sequential mathematical objects like a set, sequence, series to be convergent. Induction changes $S=N$ to $s_1\in S$ and if $s_n\in S$ then $s_{n+1}\in S$. The natural number has a property of endless addable with one. But, induction can prove only natural number $N$ not infinity $\infty$. $$Induction{s}_1\in Sif{s}_n\in Sthen{s}_{n+1}\in SThenS=N$$ The limit is the way $N$ goes to $\infty$.

Differential equations describe the change of state. The change relates to the state. The solutions of the differential equations are the status equations. The initial conditions set the time $t$ and status $y$. The boundary conditions are the value of boundary $y_0$ and $y_1$. $\frac{dy}{dt}$ $=ay+q\left(t\right)$ starting from $y\left(0\right)$ at $t=0$. inital conditions $t=0$ and $y=1$ $q\left(t\right)$ is a input and $y\left(t\right)$ is a response.

$$f\left(x\right)=\sum _{k=0}^{\infty }{c}_k{x}^k=c_0+c_{1}x+c_2{x}^2+\cdots .$$ This is an approximation that is a function of h and derivatives of $f\left(x\right)$ are elements of parameters. $f(x\pm h)=f\left(x\right)\pm h{f}^{\prime }\left(x\right)+\frac{h^2}{2}{f}^{\prime \prime }\left(x\right)\pm \frac{h^3}{6}{f}^{\prime \prime \prime }\left(x\right)+O\left(h^4\right)$ Let’s think about $\sin \left(x\right)$. $$f\left(x\right)=\sin \left(x\right)f\left(0\right)=0,f^{\prime }\left(x\right)=\cos \left(x\right)f^{\prime }\left(0\right)=1,f^{\prime \prime }\left(x\right)=-\sin \left(x\right)f^{\prime \prime }\left(0\right)=0$$ Thus, \[\begin{align*} \sin(x) &= 0 + \frac{1}{1!

Here I summarize some tools for proof of the Riesz representation theorem. They are the limit of inequality of sequence and $ϵ$. The Rudin’s proof of the Riesz representation theorem construct measure $\mu$ and measurable set $M$, then prove the $\mu$ and $M$ have properties. Countable additivity (not subadditivity) is an important property. The strategy of proving equality (additivity) is bidirectional inequality. Limit of inequality of sequence gives us a tool that finite inequality makes infinite inequality.

This is a note of real and complex analysis chapter 2. Chapter 2 is about measures. The measure already defined in chapter 1. In chapter 2, every linear functionals, not combination, of a continuous function space on compact set ($C$) ($\Lambda f$) represents the integration of the function ($\int fdu$) (Riesz representation theorem). Let X be a locally compact Hausdorf space, and let $\Lambda$ be a positive linear functional on $C_c\left(X\right)$.

This is a note for Rudin’s real and complex analysis chapter 1. The key concepts are $\sigma$-algebra, measure ($\mu$) zero, and linear combination. The three concepts bring me abstract integration. The $\sigma$-algebra makes that countable sum and measure of complement (subtract measure) can be possible. Measure zero completes the system. linear combination integrates a measurable function. After a measure space established, Lebesgue’s monotone convergence theorem, Fatou’s lemma, and Lebesgue’s dominant convergence theorem follow.