Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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6
votes
1answer
55 views

Limit of measurable functions is measurable?

Suppose $(\Omega, \cal F)$ is a measurable space and $(X, \mathcal B_X)$ is a topological space with its Borel sigma algebra. If $f_n: \Omega \to X$ is a sequence of $(\cal F , B$$_X)$-measurable ...
1
vote
1answer
32 views

Particular $f \in C_c^1$

is there a way to construct a function $f \in C_c^1( B(0,R))$ such that $f|_{B(0,\frac{R}{2})}=1, \quad f|_{B(0,R)^C}=0$ and $0\le f\le 1$ everywhere such that $|Df(x)| \le \frac{4}{R}$ for almost all ...
2
votes
1answer
37 views

Prove that $Df(p)=f(p)T$ where $T(q)=\int_{0}^1q$

Let $E=\mathcal{C}[0,1]$ provide with $\|\cdot\|_\infty$ norm. Let $f:E\to \mathbb{R}$, given by $f(p)=e^{\int_0^1 p}$. I need to prove that $f$ is differentiable. My approach: Let ...
2
votes
0answers
52 views

A few questions on the later chapters in Principles of Mathematical Analysis by Walter Rudin (3rd Edition)

I am currently reading Principles of Mathematical Analysis by Walter Rudin (3rd Edition). I am enjoying the book and it's terseness, which isn't an issue for me. What I do have a problem with is that ...
0
votes
2answers
22 views

If the iteration $x^{k+1}=x^k-t_kH_k^{-1}\nabla f(x^k)$ converges superlinearly to a stationary point $x^*\ne x^k$, then $t_k\to 1$

Let $f\in C^2(\mathbb{R}^n)$ $(H_k)_{k\in\mathbb{N}_0}\subseteq\text{GL}_n(\mathbb{R})$ $x^0\in\mathbb{R}^n$ and $$x^{k+1}:=x^k+t_k d^k\;\;\;\text{for }k\in\mathbb{N}_0\tag{1}$$ with ...
-1
votes
0answers
22 views

Continuously differentiable operator

if i consider the operator $A$ defined on $H^1_0$ by $$Au(t)=\int_0^1 G(t,s) f(s,u(s))ds$$ where $$ G(t,s)=\begin{cases} t(1-s),~~t\leq s\\s(1-t),~~s\leq t\end{cases}$$ What is the expretion of $A'u$ ...
3
votes
1answer
32 views

Estimate for a weak solution to a PDE

Let $f \in L^2(B_R(0))$ and let $u \in W^{1,2}(B_R(0))$ be a weak solution of the equation $$Lu = - \sum_{i,j=1}^{n} D_i(a_{ij}D_ju)+ \sum_{i=1}^{n} b_i D_i u + cu =f.$$ There are constants $0 \le ...
0
votes
1answer
32 views

Finding Laurent series of a function $f(z)=\frac{3z^2-6z+1}{(z-1)(3z-1)}$ [on hold]

How do i transform this function into Laurent series $$f(z)=\frac{3z^2-6z+1}{(z-1)(3z-1)}$$ where $ \frac{1}{3} < |z| < 1 $.
1
vote
2answers
78 views

Showing that $f$ is $C^\infty$

Question: Let $f: U \to \mathbb R$ be a continuous function, with $U \subset \mathbb R^2$ open, such that $$(x^2 +y^4)f(x,y) + f(x,y)^3 = 1,\, \,\, \forall (x,y) \in U$$ Show that $f$ is of ...
0
votes
0answers
16 views

combine analysis and artificial intelligence

I'm sorry if I ask this question at the wrong place, but I don't know a better one. I am a Master's student and I am really interested in analysis, but I also want to get into AI. Does anyone know a ...
0
votes
1answer
25 views

Prob. 8, Sec. 4.2 in Kreyszig's functional analysis book: Nonnegativity of a subadditive functional outside a sphere implies nonnegativity

If a subadditive functional $p$ defined on a normed space $X$ is non-negative outside a sphere $\{ \ x \in X \ \colon \ \Vert x \Vert = r \ \}$, then how to show that $p$ is non-negative for all $x ...
0
votes
1answer
23 views

Prob. 3, Sec. 4.2 in Erwin Kreyszig's Functional Analysis: How to show that $\lim\sup$ is sublinear?

Let's consider the real space $\ell^\infty$ of all bounded sequences of real numbers. Let $p \colon \ell^\infty \to \mathbb{R}$ be defined by $$p(x) \colon= \lim\sup_{n \to \infty} \xi_n \ \mbox{ for ...
0
votes
0answers
34 views

Is the field of real algebraic numbers a complete field?

Let $\mathbb{R}_{alg}$ be the field of real algebraic numbers. Is there exist a metric $|\cdot|$ for which $(\mathbb{R}_{alg}, |\cdot|)$ is a complete field (i.e. any Cauchy sequence converges in ...
4
votes
2answers
66 views

Product rule for Hessian matrix

Let $f: \mathbb{R}^n \to \mathbb{R}$ and $g: \mathbb{R}^n \to \mathbb{R}$. Is there a general formula for the Hessian matrix of their product? That is, what is $H(f(x) g(x))$, where $H(f(x)) = ...
1
vote
1answer
18 views

Error bound of midpoint rules with unbounded second derivative

It is well known that error bound of midpoint rule for function $f[a,b]$ is given by $$ E\leq K\frac{(b-a)^3}{24 n^2} $$ where $|f(x)''\leq K|$ and $n$ is the number of time steps. if second ...
1
vote
2answers
62 views

find $\lim_n\frac{a^\frac1n}{n+1}+\frac{a^\frac 2n}{n+\frac 12}+…+\frac{a^\frac nn}{n+\frac 1n}$ using Riemann integral

Here is the question: prove that $S_n=\frac{a^\frac1n}{n+1}+\frac{a^\frac 2n}{n+\frac 12}+...+\frac{a^\frac nn}{n+\frac 1n}$ is convergent for $a>0$ then find its limit. My attempt: If we accept ...
2
votes
3answers
277 views

Non linear Differential Equation

Let $\Omega:=\{(x_1,x_2) \subset \mathbb{R}^2 | x_2>0\}$. I want to solve the differential equation $$\begin{pmatrix} \dot{x_1} \\\dot{x_2} \end{pmatrix}=\begin{pmatrix}x_2^2-x_1^2 ...
1
vote
0answers
25 views

Approximating functions such that the left- and right limit exists everywhere

Every continuous function $f : \mathbb R \to \mathbb R$ could be uniformly approximated by step functions. For a proof consider an interval $[a,b]$, then $f$ is bounded on this compact interval, i.e. ...
3
votes
1answer
90 views

How to show that the normed space $(\ell^2, \|\cdot\|_2)$ is complete

Show that the normed space $(\ell^2, \|\cdot\|_2)$ is complete. I am not sure where to even start with this question. I'm quite sure it involves Cauchy and also Bolzano-Weierstrass but I'm not sure ...
2
votes
1answer
423 views

Approximating integrals with step functions

For $f \colon [1,2] \to \mathbb{R}$ , $f(x) = 1/x$, Choose a sequence of step functions $\phi_n$ approximating $f$ with partition $P_n := [r/n : n < r < 2n]$ to show that $ 1/(n+1) + \cdots + ...
3
votes
1answer
99 views

Iteration with bounded converging sequence as input

Consider the sequence of vectors $( x_k )_{k=0}^{\infty}$ such that $x_k \in X \subset \mathbb{R}^n$ for all $k$, where $X$ is compact, and $x_k \rightarrow \bar{x} \in X$. Consider a vector $y_0 \in ...
1
vote
1answer
28 views

Prob. 10, Sec. 3.10 in Kreyszig's functional analysis book: Every isometric linear operator on a finite-dimensional inner product space is unitary? [duplicate]

Let $X$ be an inner product space such that $\dim X < \infty$, and let $T \colon X \to X$ be an isometric linear operator. Since $\dim X < \infty$, $X$ is complete and thus a Hilbert space; ...
1
vote
0answers
11 views

Boundedness of binomial coefficient multi-index [hold-on]

Let be $\alpha,\beta\in \Bbb{N}^n$ multi-index, such that $|\alpha|=m+1$ e $\beta <\alpha$, with $m\in\Bbb{N}$ fixed. Is it possible to bound $\large\large{\alpha \choose \beta}$ by something that ...
3
votes
3answers
342 views

Introduction to Pseudodifferential operators

I'm interested in elementary introduction to pseduodifferential operators and its application to hyperbolic PDE's. I know measure theory, Fourier analysis and some elementary(linear) hyperbolic PDE's ...
0
votes
1answer
48 views

Prove that the function $\xi\in R \mapsto {e^{i\cdot \xi\cdot λ}-1\over i\cdot \xi}-λ$ is $C^{\infty}$

Prove that the following function is $C^\infty$ in the point $\xi=0$: $$f:\Bbb R\to\Bbb C:\xi\mapsto {e^{i\cdot\xi\cdot λ}-1\over i\cdot\xi}-λ$$ Any ideas how to prove this? I am trying to ...
1
vote
3answers
142 views

Show the following set is connected

For any $x \in \Bbb R^n$ how do I show that the set $B_x := \{{kx\mid k \in \Bbb R}$} is connected. It should also be concluded that $\Bbb R^n$ is connected. I was thinking of starting by assuming ...
0
votes
3answers
85 views

Locally Lipschitz and Gâteaux Derivative if and only if Frechet Derivative

Consider $f$ locally Lipschitz. So $f$ is Gâteaux Derivative if and only if $f$ is Frechet Derivative. PS.: the converse is trivial.
3
votes
0answers
65 views

A derivation of the Euler-Maclaurin formula?

The generating function for the Bernoulli numbers $B_n$ is $$\frac{x}{e^x-1}=\sum_{n=0}^\infty\frac{B_n}{n!}x^n$$ The sum of an infinite geometric series is $$\frac{1}{1-x}=\sum_{k=0}^\infty x^k$$ ...
2
votes
0answers
29 views

non-analytic smooth function

Given the function $$f(z) = \sum_{k \in \mathbb{Z}} \exp\left(-\sqrt{|k|}\right) z^{k}$$ Defined on the unit circle $S^1$. That $f$ is smooth is not hard to see. Why is there not a $\Phi$ on an open ...
2
votes
1answer
23 views

What are Carnot groups?

I'm trying to learn the Pansu differentiability theorem and I need to know what Carnot groups are. Can someone please explain what Carnot groups are? An introductory reference would be greatly ...
0
votes
1answer
31 views

Prove that the following function is $C^\infty$ [duplicate]

Prove that the following function is $C^\infty$ (and in the point $ξ=0$) : $$f:\Bbb R\to\Bbb C:\xi\mapsto {e^{i\cdot\xi\cdot λ}-1\over i\cdot\xi}-λ$$ for whichever $$λ>0$$ I am trying to find a ...
-1
votes
0answers
19 views

If $\{x_n\}$ and $\{y_n\}$ are two bounded sequences then prove that $\liminf x_n -\liminf y_n \leq \liminf(x_n - y_n)$ [on hold]

Please help me in finding the proof of this theorem.I am in a fix If $(x_n)_{n\in\mathbb N}$ and $(y_n)_{n\in\mathbb N}$ are two bounded sequences then prove that $$\liminf_{n\to\infty}\,x_n - ...
1
vote
1answer
66 views
+50

Show that if $\int_0^x f(y)dy \sim Ax^\alpha$ then $f(x)\sim \alpha Ax^{\alpha -1}$

Let $f$ be a real, continuous function defined on $[0,\infty)$ such that $xf(x)$ is increasing for all sufficiently large values of $x$. Show that if $$\int_0^x f(y)dy \sim Ax^\alpha \; (x\to ...
0
votes
3answers
93 views

Prove that the following function is $C^{\infty}$ [duplicate]

Prove that the following function: $$r:x \mapsto \begin{cases} e^{-{1\over (1-x^2)}}, & \text{if $|x|<1$} \\ 0, & \text{if $|x| \ge 1$} \end{cases}$$ is $C^{\infty}$ I found this problem ...
2
votes
1answer
29 views

If a function is enclosed by lower and upper sums, does its limit w.r.t. partitions equals the integral

Let $a < b$ and denote by $\mathcal P[a,b]$ the set of all finite partitions of the compact interval $[a,b]$, i.e. all sets of the form $P = \{ a = x_0 < x_1 < \ldots < x_n = b \}$. ...
3
votes
2answers
76 views

Boundedness of the norm of the Riemann curvature tensor

Let $(M,g)$ be a Riemannian manifold and let $R(X,Y)Z$ be its $(3,1)$ Riemann curvature tensor given by $$R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z$$ Let the input vectors $X,Y,Z$ ...
2
votes
0answers
20 views

Exponential of powers of the derivative operator

A translation operator The Taylor series of a function $f$ is $$f(x)=\sum_{n=0}^\infty\frac{(\partial_x^nf)(a)}{n!}(x-a)^n$$ where $\partial_x$ is the derivative operator. Expanding about $x+b$: ...
6
votes
2answers
181 views

Show that $\lim_{n\to\infty}n\int_0^1f(x)g(x^n)dx=f(1)\int_0^1\frac{g(x)}{x}dx$

Let $g:[0,1]\mapsto\mathbb{R}$ be a continuous function, and $\lim_{x\to0^+}g(x)/x$ exists and is finite. Prove that $\forall f:[0,1]\mapsto\mathbb{R}$, ...
4
votes
3answers
117 views

How to prove $\lim \limits_{n\to\infty}(n+1)\int_{0}^{1}x^nf(x)dx=f(1)$ [on hold]

I need help to prove this in real analysis. I think it uses IMVT, but not sure how to do it. Let $f(x)$ be a real valued continuous function on $[0,1]$. Show that $$ \lim ...
3
votes
1answer
441 views

Derivation of poisson kernel for disk of radius $R$ from unit disk

Is there a way to derive poisson kernel for disk of radius $R$ from unit disk?
1
vote
1answer
31 views

Why periodic functions form a dense subset in $ C[a,b]$ with $L^2$ norm?

Let's consider the linear space $C[a,b]$ but with $L^2$ norm $$ \|f\|=(\int_a^b |f(t)|^2dt)^{\frac{1}{2}} $$ How to prove that the subspace $$ V=\{f\in C[a,b]: f(a)=f(b)\} $$ is dense in this normed ...
1
vote
1answer
32 views

Prove the concavity of the transformation from a concave function to another

Let's say we have $f_1$ and $f_2$, both strictly increasing and strictly concave on $[0,+\infty)$. $f_1(0)=f_2(0)=0$ and the difference $f_1-f_2$ is strictly positive and strictly increasing. That is, ...
3
votes
2answers
109 views

Show that $\frac{(x^2 + y^2 )}{4} \leq e^{x+y-2}$

Show that \begin{equation} \frac{x^2 + y^2}{4} \leq e^{x+y-2} \end{equation} is true for $x,y \geq 0$. As far, I have prove that \begin{equation} x^2 + y^2 \leq e^{x}e^{y}\leq e^{x+y} ...
1
vote
0answers
37 views

Prob. 9, Sec. 3.10 in Kreyszig's functional analysis book: The image of ann isometric non-unitary operator on a Hilbert space

Let $H$ be a Hilbert space, let $T \colon H \to H$ be a linear operator such that $T$ is isometric but not unitary. Then how to show that the image $T[H]$ is a proper closed subspace of $H$? My ...
1
vote
0answers
20 views

Let $A,B:V\to V$ positive definite operators in complex linear space with inner product $V$, $dimV<\infty$

Let $$A,B:V\to V$$ positive definite operators in complex linear space with inner product $$V$$, $$dimV<\infty$$ Show that $$log det(A\cdot B^{-1})=-\int_{0}^\infty tr(e^{-t\cdot A}-e^{-t\cdot ...
2
votes
1answer
37 views

Slopes of curves from complex derivative [on hold]

Show that the slopes of the level curves$$u(x,y)=\text{constant} \ \ \text{and} \ \ v(x,y)=\text{constant}$$ are respectively given by $$\cot(\arg(f'(z))) \ \ \text{and} \ \ -\tan(\arg(f'(z)))$$ If ...
2
votes
0answers
48 views

Can we find an example?

I have this theorem, and i want to know if i can find an expression of the operator $A$ which satisfy this theorem:
0
votes
0answers
23 views

How do I formulate a specific formula for a sequence?

I have three arrays, for instance s = [1:2], j = [1:20] and b = [1:8], and I am trying to build a single row. The problem that I actually have is that I need to find a formula f(s,j,b) such that ...
7
votes
1answer
387 views

Prove that $\mathbb{R}^k$ is separable

I'd like to show that $\mathbb{R}^k$ is separable. (A metric space is called separable if it contains a countable dense subset.) Here's what I have and I'd like to confirm with everyone to see if ...
2
votes
2answers
50 views

Prove that a classical solution of $-\langle\nabla,A\nabla u\rangle=f$ is also a weak one

Let $\Omega\subseteq\mathbb{R}^n$ a domain $f\in L^2(\Omega)$ $A:\Omega\to\mathbb{R}^{n\times n}$ be Borel-measurable and $A(x)$ be symmetric, for all $x\in\Omega$ $u\in C^2(\Omega)$ with $A\nabla ...