Ref: Prof. Vershynin’s handwritten notes Lec-1: Consider the problem of numerically computing the integral of an ${ f : [0,1] ^d \to \mathbb{R} }.$ Breaking ${ [0,1] ^d }$ into (axis aligned) cubes of width ${ \epsilon },$ there are about ${ N \approx (\frac{1}{\epsilon}) ^{d} }$ many such smaller cubes. Now the integral ${ \int _{[0,1] ^d} f... Read more 07 Oct 2023 - 23 minute read
Link to lectures Instructor: Prof. Philippe Rigollet ROUGH NOTES (!) Lec-1 [Slides] Idea: Use data to get insight and perhaps make decisions. Computational view: Data is a (large) sequence of numbers that needs to be processed by a fast algorithm (approximate nearest neighbours, low dimensional embeddings, etc) Statistical view: Data comes... Read more 31 Oct 2022 - 11 minute read
Let ${ f \in \mathcal{C}[a,b] }.$ Weierstrass approximation theorem says there is a sequence of polynomials uniformly converging to ${ f }$ on ${ [a,b] }.$ That is, for every ${ \epsilon \gt 0 }$ there is a polynomial ${ P }$ with ${ \max _{x \in [a,b]} \vert f (x) - P(x) \vert }$ ${ \lt \epsilon }.$ The proof below is originally due to Lebesg... Read more 22 Jan 2022 - 7 minute read
(Ref: Stromberg’s “Classical Real Analysis”) Let ${ \alpha \in \mathbb{R} }.$ The generalised binomial theorem says ${ (1+x) ^{\alpha} }$ ${ = \sum _{0} ^{\infty} \binom{\alpha}{j} x ^j }$ for ${ \vert x \vert \lt 1 }.$ Also, radius of convergence of ${ \sum _{1} ^{\infty} \binom{\alpha}{j} x ^j }$ is ${ \infty }$ when ${ \alpha \in \mathbb{... Read more 15 Jan 2022 - 4 minute read
Let ${ F }$ be a normed space and ${ X }$ a set. For a map ${ X \overset{f}{\to} F },$ let ${ \lVert f \rVert _{\infty} }$ ${ := \sup _{ x \in X} \vert f (x) \vert }$ ${ \underline{ \in [0, \infty] } }.$ ${ \lVert \ldots \rVert _{\infty}, }$ when restricted to the space ${ \mathcal{B}(X, F) }$ of bounded functions from ${ X }$ to ${ F, }$ be... Read more 14 Jan 2022 - 15 minute read