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).

learn more… | top users | synonyms (1)

1
vote
1answer
328 views

Eigenvalues and Eigenfunctions of Integral Equation

Compute the eigenvalues and eigenfunctions of $$ \varphi(x) - \lambda \int_0^1 e^{x+s} \varphi(s) ds = f(x) $$ Are there functions $f$ such that the inhomogenous equation has for every real $\lambda$ ...
1
vote
1answer
33 views

Relationship between big O notation and exponential type

Let $f: \mathbb{R} \to \mathbb{R}$, $C\in \mathbb{R}$. What, if any, is the difference between "$ f = O(e^{Cx}) $" and "$f$ is of exponential type $C$"? If they're different, is it possible to ...
1
vote
1answer
99 views

How to interchange limit and integral?

Suppose $f_{n}, f\in L^{1}(\mathbb R)$ with the properties that, $f_{n}(x)\to f(x)$ point wise for each $x\in \mathbb R;$ $\|f_{n}\|_{L^{1}(\mathbb R)} \leq \|f\|_{L^{1}(\mathbb R)}$ for every $n\...
0
votes
1answer
111 views

Fundamental group smash product

is there a way to conclude what the first fundamental group of the smash product of two path-connected spaces is? probably there should be a general way like there is for the wedge sum due to van ...
1
vote
1answer
98 views

Riesz measure associated with a subharmonic function

In page 101, corollary 4.3.3., from Armitage and Gardiner's book on potential theory, the authors prove that any subharmonic function, can be identified with a positive measure (Riesz measure). In ...
0
votes
1answer
44 views

Bounded from below or not? [closed]

If I have that a functional $J$ defined on a Hilbert space is weakly lower semi continuous and coercive is it bounded from below? Thank you
4
votes
2answers
282 views

An “obvious” statement about a nonincreasing supremum

Consider a nonnegative function $f(t,x): [0,\infty) \times [0,1] \rightarrow [0, \infty)$. Suppose we have the following property: $$ \mbox{ If } ~~~~~~~~~~f(t,y) > \frac{1}{2} \sup_{x \in [0,1]} ...
2
votes
0answers
74 views

Show a set is dense in $C(X)$

Let $X$ be a totally discontinuous compact space. Show that the algebra generated by $$\{f_F; ~f_F=\chi_F-\chi_{X/F},F \text{ is a clopen subset of }X\}$$ is dense in $C(X)$. My attempt: Suppose $\{...
4
votes
1answer
102 views

Is this another version of the gamma function?

I know that $\Gamma \left( x \right) $ is the unique function on $x \in (0, \infty)$ such that $f \left( 1 \right) =1$ $f(x+1)=xf(x)$ ${\frac {d^{2}}{d{x}^{2}}}ln(f \left( x \right))>0$ ...
1
vote
0answers
34 views

Show that $ \prod_{k=1}^n [1 + p_{nk}(e^{it} - 1)] \rightarrow e^{\lambda(e^{it} - 1)}, n \rightarrow \infty $

Suppose that $0\leq p_{nk} \leq 1, 1 \leq k \leq n$, $\max_{1 \leq k \leq n} p_{nk} \rightarrow 0, n \rightarrow \infty$ and $\sum_{k=1}^n p_{nk} \rightarrow \lambda$. Show that $$ \prod_{k=1}^n [1 ...
1
vote
0answers
43 views

can we expect a bounded approximate identity in $L^{1}(\mathbb R)$, whose Fourier transform at particular point is zero?

Let $\phi:\mathbb R \to \mathbb C$ be a function; and define $$\phi_{t}(x):=\frac{1}{t}\phi (x/t), (t>0).$$ We let, $\phi\in L^{1}(\mathbb R),$ then $\phi_{t}\in L^{1}(\mathbb R);$ in fact, we ...
1
vote
1answer
63 views

A bounded domain can be considered as a compact manifold?

A bounded domain $\Omega$ with smooth boundary $\Gamma$ can be considered as a compact connect Riemannian manifold?
4
votes
2answers
109 views

Intuition about Taking an Integral

My hope is to personally develop some further intuition for taking an integral (measuring the area under a curve). Consider a normal distribution and I need the area under the curve from $a$ to $b$. I ...
0
votes
1answer
41 views

Show that Fourier series arising in solution of differential eqn. converges uniformly

Let $f \in L_2(0,\pi)$ have the Fourier expansion $f(x) = \sum_{n=2}^{\infty} f_n\sin(nx)$. Compute (formally) the boundardy value problem $$ u''(x) + u(x) = f(x) \qquad \mbox{ for } 0 < x < \...
0
votes
2answers
54 views

Total boundness of Lipschitz densities

In the article Almost Sure Testability of Classes of Densities by Devroye and Lugosi in 1999. They claim in Example 10 (page 9) that Lipschitz densities on [0,1] with Lipschitz constant bounded by ...
0
votes
1answer
115 views

How to interchange sum and integral?

We fix the point $\xi_{0}\in \mathbb R.$ Choose sequence $\{f_{n}\}_{n\in \mathbb N}\subset L^{1}(\mathbb R)$ with the following property : (1) $\|f_{n}\|_{L^{1}(\mathbb R)} \leq 1, $ for $n\in \...
0
votes
1answer
55 views

Prove two solutions of differential equation are the same

In a recent work I had to solve the following differential equation: $$ r x''(r)+r x'(r)^2+x'(r)-\frac{4}{r}=0~~. $$ To do so I used two methods and I got, using each, two solutions with different ...
3
votes
3answers
685 views

Is this proof of the fundamental theorem of calculus correct?

A student friend of mine recently gave me a proof of the fundamental theorem of calculus which does not correspond to any I can find in the textbooks. It starts by considering an increasing continuous ...
1
vote
1answer
68 views

Fredholm index for 1-d Schroedinger operator

if I look at a Schroedinger-operator $-\frac{d^2}{dx^2} +V$ on a compact intervall $[a,b] \subset \mathbb{R}$ and I take boundary conditions that this operator is self-adjoint (for example periodic ...
1
vote
1answer
68 views

How to find an everywhere discontinuous real function with $F((a+b)/2)<(F(a)+F(b))/2$?

In here I posted a non-constructive everywhere discontinuous real function with $$F((a+b)/2)=(F(a)+F(b))/2$$ based on the using of Hamel basis. And Conifold answered there that there is no explicit ...
2
votes
4answers
456 views

Analysis question limit sin function as n goes to infinity [closed]

can you help me with the following: $\lim_{n \rightarrow \infty} \sin^{2} \pi \sqrt{n^2 + n}$ Thanks a lot!
1
vote
1answer
30 views

Tangent line to a curve statement

I am having problems understanding some parts of the proof of some statement related to tangent line to a curve. I'll copy the exact statement and proof and then my doubts. Statement If $\mathcal C$ ...
2
votes
1answer
130 views

Does differentiability imply absolute continuity? [duplicate]

Suppose $f:[a,b] \rightarrow \mathbb{R}$ is a function which is (i) differentiable at all $x \in (a,b)$ (ii) the right-derivative at $x=a$ exists and the left-derivative at $x=b$ exists. Does it ...
2
votes
0answers
54 views

Discontinuous linear operator on $\ell^{2}$

Let $e_{n} = (0, 0, \ldots, 0, 1, 0, \ldots)$ where $1$ is in the $n$th position. Then $\{e_{n}\}$ is an orthonormal basis for the Hilbert space $\ell^{2}(\mathbb{N})$. Does there exists a linear ...
2
votes
2answers
156 views

Asymptotic Behaviour Of A Bizarre Function

It is relatively easy to show that the asymptotic behaviour of $f(x)$, where $$ f(x)= \left[\frac{x}{2}\right] + \left[\frac{x}{4}\right] + \left[\frac{x}{8}\right] + \left[\frac{x}{16}\right] + \...
1
vote
0answers
124 views

Banach Fixed Point Theorem. Measurable version.

The Banach fixed point theorem has the following statement THEOREM ( Banach contraction principle). Let $(Y,d)$ be a complete metric space and $F:Y\to Y$ be contractive . Then $F$ has a uniqe ...
3
votes
1answer
105 views

find a $B_{n,j}$ such that $|A_{n,j}-L_j| \leq B_{n,j}$ $\forall n,j$ and $\sum_{j=0}^{\infty}B_{n,j}$ converges

We have $A_{n,j}= 3(-1)^j2^{n-j+1}\frac{(2(n-j)-4)!}{(n-j)!(n-j-2)!}\binom{j+2}{2}\frac{n^\frac{5}{2}}{8^n}$ and $L_j=(-\frac{1}{8})^j\binom{j+2}{2}\frac{3}{8\sqrt{\pi}}$ So I know $\lim_{n \to \...
2
votes
1answer
103 views

Properties of the Double Layer Potential

Consider the double layer potential $$ W_{\nu}(x) = \int_{\partial\Omega} \nu(y) \frac{\partial}{\partial n_y}\left( \frac{1}{|x - y|} \right) d \sigma_y $$ for a bounded region $\Omega \in \mathbb R^...
1
vote
0answers
327 views

Absolutely continuous function admits weak derivative

How to prove that an Absolutely continuous function admits weak derivative? Absolutely continuous function: Let $(X, d)$ be a metric space and let I be an interval in the real line R. A function $f: ...
4
votes
2answers
101 views

Riemann integral enigma

I tried to solve this problem from Souza Silva - Berkeley Problems In Mathematics: In the Solutions part, I founded next solution for this problem: I do not understand the last statement, so why $|...
4
votes
1answer
187 views

An explicit construction for an everywhere discontinuous real function with $F((a+b)/2)\leq(F(a)+F(b))/2$?

I would like to know an explicit method on constructing an everywhere discontinuous real function $F$ with the property: $$F((a+b)/2)\leq(F(a)+F(b))/2.$$ There is a non-constructive example (with the ...
-2
votes
1answer
849 views

Best/Worst Test Cases

For the two functions, would their worst/best Big O cases change compared to each others? If so, how would they change. I believe that when dealing with Big O, order is the only thing that matters, so ...
2
votes
2answers
713 views

Modulo Big O Problem

I know this may be really basic, but I am unsure of the complexity of this procedure in Python: def modten(n): return n%10 edit: It is done with Python. That ...
2
votes
1answer
87 views

Group action and set define via their quotient topology open/closed equivalence relations

If we have a topological space $X$ and a subset $A \subset X$, we can define $X \backslash A$. My question is: Is it true that this equivalence relation is closed iff $A$ is closed as a subset of $X$ ...
0
votes
1answer
476 views

Fundamental polygon

So, I have seen fundamental polygons quite a few times now and I was always wondering what they are actually good for. Let's take the sphere. It's fundamental polygon can be seen here image. Does ...
3
votes
1answer
64 views

Does the series $\sum_{i=0}^{\infty}\exp\{{-(m!)!}\}(D^m\phi)(0)$ converge for every $\phi \in C^\infty$?

Does the series $$\sum_{m=0}^{\infty}\exp\{{-(m!)!}\}(D^m\phi)(0)$$ converge for every $\phi \in C^\infty$? For analytic function $\phi$, we can show that the series converges by using Caushy-...
1
vote
1answer
33 views

$|\int_{\mathbb R} e^{-t^{2}} e^{-(t/\lambda -x)^{2}} e^{-2\pi i w\cdot t}| dt \leq G(x,w), G\in L^{1} ? $

Put $\lambda >0,$ and we define, $$F_{\lambda}(x, w)= \int_{\mathbb R} e^{-t^{2}} e^{-(t/\lambda -x)^{2}} e^{-2\pi i w\cdot t} dt;(x,w) \in \mathbb R^{2}$$ we note that, $F_{\lambda} \in L^{1}(\...
2
votes
1answer
232 views

Does a function that is twice weakly differentiable have a version that is classically differentiable?

I have been wondering about the idea of functions that are weakly differentiable. My intuition tells me that the weak derivative allows one to differentiate functions that either have a removable ...
-1
votes
2answers
83 views

a question about sequence and series. prove $ \lim_{n \to \infty}( n\ln n)a_{n}=0$? [closed]

Suppose $a_{n}>0$. $na_{n}$ is monotonic, and it approaches 0 as n approaches infinity. $\sum_{n=1}^{\infty} a_{n}$ is convergent. please prove $$ \lim_{n \to \infty} n\ln(n)\,a_{n}=0$$ I totally ...
2
votes
1answer
66 views

an inequality derived from conformal automorphisms of unit disk

Let $f$ be a holomorphic function on $D(0,1)$ such that $|f(z)|<1$ for all $z\in D(0,1)$. I have obtained $$ \frac{|f(0)|-|z|}{1+|f(0)||z|}\leq |f(z)|\leq \frac{|f(0)|+|z|}{1-|f(0)||z|}. $$ Is it ...
1
vote
1answer
165 views

Constructing a set that contains at most one point on vertical and horizontal

I'm not sure how to answer this question: Construct a set $A$, which is a subset of $[0, 1] \times [0, 1]$, such that $A$ contains at most one point on the horizontal and vertical lines, but ...
4
votes
7answers
169 views

Convergence in a metric space

Is it possible to define a metric on $\mathbb R$ such that $(1,0,1,0,...)$ converges on $(\mathbb R, d)$? I believe it is impossible. But how to show analytically? Any hint would be appreciated.
2
votes
1answer
49 views

Minimum of a potential function

I'm looking for extremes (minimum) of $$V = \frac{\alpha}{|\vec{r}_1-\vec{r}_2|} + \beta (\vec{r}_1 + \vec{r}_2)\cdot \vec{e}_z$$ where $\vec{r}_i = R(\cos\phi_i\sin\theta_i,\sin\phi_i\sin\theta_i,\...
0
votes
1answer
95 views

Find $\lim_{n \rightarrow \infty}$$\int_0^n(1 + \frac{-x}{n})^n\cos(\frac{x}{\sqrt{n}})e^{x/2}dx$

Find $\lim_{n \rightarrow \infty}$$\int_0^n(1 + \frac{-x}{n})^n\cos(\frac{x}{\sqrt{n}})e^{x/2}dx$ I want to use dominated convergence theorem obviously. However, not sure how to dominate it. ...
2
votes
1answer
95 views

Differentiation under the integral if and only if we have an $L^1$ dominator

Let $f(x)\in L^2(\mathbb{R})$ and define $$g(t) = \int_\mathbb{R} f^2(x)\exp(-tx^2)dx$$ for $t\geq0$. We want to show that $g(t)$ is continuously differentiable if and only if $xf(x)\in L^2(...
0
votes
1answer
81 views

Proving or disproving continuity of a function

Consider a function $f:\mathbb{R}^{n}\times \mathbb{R}^{+} \rightarrow \mathbb{R}$, with the property that for a fixed vector $a:=(a_1,a_2,\cdots,a_n) \in \mathbb{R}^{n}$, there exist a finite ...
1
vote
1answer
62 views

The functions $\{f_n(x) = n\}$ are analytic and each miss the points $-2, -3$. But, they are not a normal family. So what am I missing. Thanks.

Here is a theorem of Montel: Let $\mathcal{F}$ be a family of analytic functions defined on a domain $\Omega$ . If there are two fixed complex numbers $a$ and $b$ that are omitted from the range of ...
11
votes
0answers
424 views

norm of a singular integral operator

My question is from Harmonic Analysis, about the study of singular kernels (in the Calderon Zygmund sense.) Suppose that a kernel $K$ is a singular kernel, extending to a bounded operator on $L^p$, ...
1
vote
0answers
50 views

Prove that $f\ast g$ is defined a.e., integrable, and such that $∥f\ast g∥_1 ≤ ∥f∥_1 · ∥g∥_1$

Let $f,g : \mathbb{R} → \mathbb{R}$ be $L_1$-functions. Set $h(x) = \int_\mathbb{R}f (x − y)g(y) \, dm(y).$ Prove that $h(x)$ is defined a.e., $h ∈ L_1(\mathbb{R})$ and $∥h∥_1 ≤ ∥f∥_1 · ∥g∥_1.$ So I ...
1
vote
1answer
150 views

Let $f(z) = \sum_{n = 0}^\infty a_nz^n$ be the Taylor series around $0$. Prove that lim $a_n/a_{n+1} = z_0.$ [duplicate]

Let $f(z) = \sum_{n = 0}^\infty a_nz^n$ be the Taylor series around $0$ of a function which is analytic in $\mathbb{C}$ \ ${z_0}$, $z_0\neq 0$ and has only a simple pole at $z_0.$ Prove that $lim_{n \...