0
votes
1answer
24 views

Measurability of Modulus

Context: This problem came up while reading an essay on Bochner integrability. Let $\Omega$ be a measure space and $E$ a Banach space. Consider two plain functions $f:\Omega\to E$ and $g:\Omega\to ...
1
vote
0answers
51 views

How to prove the uniqueness of a specific root?

Let us define: $$F(x):=\int_0^Tf(t)\cos(x\,t)dt-\frac{\sin(T_0\,x)}{T_0\,x}$$ where: 1). $0<T_0<T \in\mathbb{R}^+$ are both positive real constants, and 2). $0\leqslant f(x)\in C^{\infty}$ ...
0
votes
2answers
49 views

Extending Positive Functionals: Linearity

How does regularity provide linearity? Given the full Banach space of bounded functions over a suitable set: $$\mathcal{B}:=\{f:\Omega\to \mathbb{C}:\|f\|_\Omega<\infty\}$$ and a linear subspace ...
0
votes
0answers
19 views

integration formula

help me please all these functions are regular. How we can found this formulation $$ \displaystyle\int_{\Omega} (f(u)-f(k)) \nabla p(g(u)-g(k)) \xi dx = - \displaystyle\int_{\Omega} H(u,k) \nabla \xi ...
0
votes
2answers
46 views

Bochner: Absolute Integrability

For a Bochner measurable function it holds: $$f\text{ Bochner integrable}\iff\|f\|\text{ Bochner integrable}$$ for any positive measure $\lambda\geq 0$. The one direction is relatively simple when ...
5
votes
1answer
86 views

What is an example of a function that is measurable but not strongly measurable?

Let $(\Omega, \Sigma)$ be a measurable space and $X$ a Banach space. Let $f: \Omega \rightarrow X$. $f$ is called measurable if every the preimage of every Borel set in $X$ is an element of ...
2
votes
1answer
47 views

Positive Linear Transformations: What good for?

Positivity is a concept appearing quite frequently in the study of algebras and its related spectral theory. Positive elements naturally give rise to an ordering and therefore allows to construct ...
0
votes
0answers
40 views

Darboux Integrable Functions vs. Uniform Closure of Simple Functions

Is there a Darboux integrable function $f:[0,1]\to E$ with values in a Banach space $E$ that is not the uniform limit of simple functions $s=\sum_\alpha\chi_{A_\alpha}$ taken over finite sums with ...
0
votes
0answers
35 views

Equation derived by integration by parts?

I am reading the paper "Coupling of Wiener processes by using copulas" by P.Jaworski and M.Krzywda and as I am reading the proofs, one derivation I can't quite understand. Let $L^{-}$ and $L^{+}$ be ...
1
vote
1answer
18 views

Reference on Riesz representation theorem for $L^p(0,T,X)$ spaces.

Brezis Functional Analysis book proves the following Riesz representation theorems for usual $L^p(\Omega)$ spaces: In what book can we find an analogous of these theorems for $L^p(0,T,X)$ spaces? ...
6
votes
1answer
135 views

Separability of a set with norm $\thickapprox$ $L^1$ +$L^{\infty}$

Let $(M, \mathcal{A}, \mu)$ a complete separable probability space. Recall that complete means that any subset of a measurable set with zero measure is measure (and has zero measure) and separable ...
2
votes
2answers
43 views

Banach Space: Open Unit Ball Totally Bounded?

Just to be sure: In an infinite dimensional Banach space the open unit ball cannot be totally bounded, right? The context is that I need this in order to find a lack in here...
1
vote
1answer
62 views

estimation of gradient

$$(\mathcal{P}_{\varepsilon}) : \left\{\begin{array}{ll} \displaystyle -div\left(A(x)\nabla u_\varepsilon(x)\right)= \dfrac{a(x)}{|u_\varepsilon(x)|+\varepsilon} &\mbox{ in }\Omega \\\\ ...
17
votes
1answer
432 views

Reinventing The Wheel - Part 2: The Lebesgue Integral

Disclaimer After struggling for some time to find an appropriate definition for the notion of integration I came across another attempt for which I would need your help deciding to what extend this ...
4
votes
0answers
23 views

Distributions and primitives

I was wondering: if distributions are seen as a generalization of functions that "removes obstructions" to the operation of derivation, is there a generalization of functions that would remove any ...
1
vote
0answers
24 views

Helffer-Sjöstrand-Formula: Idea behind?

I have to present the Helffer-Sjöstrand-Formula. Now I'm wondering: Why does it include a factor $\chi(y\langle x\rangle^{-1})$ for some bump function $\chi$ and the chinese symbol ...
1
vote
1answer
47 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 $\lambda$ ...
0
votes
0answers
26 views

Show that different eigenfunctions of integral kernel are orthogonal

Consider the integral operator $K \varphi := \int_0^1 k(x,s) \varphi(s) ds$ with a continuous and symmetric kernel $k : [0,1]^2 \to \mathbb R$ which has at least two different eigenvalues $\lambda_1$ ...
1
vote
1answer
36 views

Finite Measure Space: Integral Closure = Bochner Integral

I can't sleep for so long time as the integral gives me headaches. I was looking for so many approaches. Now another one. Hope this works... Let $\Omega$ be a finite measure space and $E$ a Banach ...
2
votes
1answer
46 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 ...
0
votes
1answer
29 views

Integral of a composition of piecewise linear function with polynomial

Fix a number $k > 0$ and let $$T(x) = \begin{cases} k &: x \geq k\\ x &: |x| < k\\ -k &: x \leq -k \end{cases}. $$ Define $S(s) = \int_0^s T(|x|^{m-1}x)\;dx.$ I want to show that ...
2
votes
1answer
47 views

Are these two expression square integrable?

I have two expressions (let's call them functions $f,g$) on $[0,1]$, where I want to find out whether they are square-integrable or better: for which $m \in \mathbb{Z}$ they are square-integrable ( ...
0
votes
2answers
62 views

How can the double intergal expression be reduced to the single intergal expression

Consider the following expression where $x(s)$ and $y(s)$ are continuous as is necessary on the closed interval [a,b]. (This is a functional analysis question -- see below for details.) $$x(s) = ...
2
votes
1answer
51 views

Defining the integral on an arbitrary metric space

I am trying to prove a version of Mercer's Theorem for an arbitrary compact metric space; that is, I do not wish to restrict myself to the space of real-valued continuous functions $C[a,b]$. I ...
4
votes
2answers
48 views

p-norm of a function

Let $f\in L^1(\mu)\cap L^\infty(\mu)$. I have proved for any $1<p<\infty$, $f\in L^p(\mu)$, $w(p)=||f||_p$ is continuous w.r.t. $p$, and $\lim_{p\to \infty}||f||_p=||f||_\infty$. Is $w(p)$ ...
5
votes
2answers
220 views

Prove that $f$ is constant on $[a,b]$

$\displaystyle \int_{a}^{b} f^2(x) \, \mathrm{d}x$ = $\displaystyle \int_{a}^{b} f^4(x) \, \mathrm{d}x$ = $\displaystyle \int_{a}^{b} f^3(x) \, \mathrm{d}x$ And $f$ is continious on $[a,b]$ and ...
-2
votes
2answers
87 views

Group of Unitaries: Strong Continuity

Let $\mathcal{L}^2(\mathbb{R})$ be the the Hilbert space of square integrable functions, shortly $\mathcal{L}^2$. Consider the group of unitaries: $$U:\mathbb{R}\to ...
3
votes
1answer
52 views

Solution of a functional integral

I am trying to show the following integral has the following result $$-\int \nabla^2\psi \text{d} \psi^*=|\nabla\psi|^2$$ Going backwards I write ...
0
votes
1answer
27 views

a non-decreasing sequence of functions with bounded L^p-norm is a Cauchy sequence in L^p space

Let $\{f_k\}_{k=1}^\infty$ be a sequence in $L^p(\mathbb{R})$ for $1\leq p<\infty$. Suppose $f_1\leq f_2\leq\cdots$ and $\sup ||f_k||_p<\infty$. Prove that $\{f_k\}_{k=1}^\infty$ converges in ...
3
votes
2answers
53 views

What is $T^nf(t)$? (Question on integrals)

I am supposed to prove the following: For the operator $T$ defined by $$Tf(t)=\int_0^t(t-s)f(s)\,ds,\quad f\in C[0,1]$$ Show that $$T^nf(t)=\int_0^t\frac{(t-s)^{2n-1}}{(2n-1)!}f(s)\, ds$$ I ...
1
vote
1answer
35 views

bounded sequence in $L^p(\mathbb{R}^n)$ that converges a.e.

Let $1<p<\infty$. Let $\{f_k\}$ be a sequence in $L^p(\mathbb{R}^n)$. Suppose $f_k\to f$ a.e. and there exists $C>0$ such that $||f_k||_p\leq C$ for all $k$. Prove that for all $g\in ...
1
vote
1answer
18 views

if $f$ is in weak $L^p$ and $\phi$ is $C_0^{1}$ then $f \ast \phi$ is in weak $L^p$

Okay, so I'd like to know if what I wrote in the title is true. Suppose that $f \in L^{p,\infty}(\mathbb{R}^n)$ (weak $L^p$ space) and $\phi \in C_0^1(\mathbb{R}^n)$ [or even $C_0^{\infty}$ if it ...
2
votes
1answer
53 views

Simple Functions: Uniform Convergence

In the proof to proposition 4.2 of 'The Riemann Integral' it is stated that the net of simple functions converges uniformly for continuous functions. This question aims to prove this in a general ...
1
vote
0answers
68 views

Strong Notion of Integral

Is there a strong(!) notion of integral that can face all of those issues: Singularities Decay Modes Oscillations Measure Spaces Locally Convex Spaces For example combining decay modes with ...
0
votes
1answer
17 views

Lebesgue integral question using du Boise-Reymond lemma

This question was inspired a previous question of mine. If we are given that $\Omega \subset \mathbb{R}^{n}$ is open and bounded and $$\int_{\Omega}fv dx = 0$$ where $f \in C(\Omega)$ and $v \in ...
0
votes
0answers
27 views

does the limit of the ratio of $p+1$ norm and $p$ norm equal to $\infty$ norm

Suppose that $f\in L^1(\Omega,\Sigma,\mu)\cap L^\infty(\Omega,\Sigma,\mu)$. Then I have proved that for any $1\leq p\leq \infty$, $f\in L^p(\Omega,\Sigma,\mu)$. Moreover, I have proved ...
1
vote
0answers
39 views

Convergence of product of continuous functions and test functions

I suspect the following result is true but I"m not sure how to go about proving: It is given that $\Omega \subset \mathbb{R}^{n}$ is an open bounded, connected domain.(Not sure if theses conditions ...
0
votes
1answer
31 views

derivate of indicator function

What is the derivative of the indicator function: \begin{equation} f(x)=\begin{cases} 1 & x^{\min} x\leq x^{\max}\\ -\infty &\mbox{otherwise}? \end{cases} \end{equation} thank you
1
vote
1answer
28 views

questions about $L^p$ space with $0<p\leq 1$ parallel to the case $1<p$

Question (1). Riesz-Fischer Theorem: For $1\leq p\leq \infty$, $L^p(\mu)$ is complete. Corollary of proof: Let $1\leq p\leq \infty$. If $(f_n)_{n=1}^\infty$ is a sequence coverging to $f$ with ...
0
votes
0answers
26 views

Question about function and primitive

I have a function$f$ such that: $f:\Omega\times \mathbb{R}\rightarrow \mathbb{R}$ is continuous , there exist $C>0$ and $\theta>2$ such that $|f(t,u)|\leq C(1+|t|^{\theta-1}~ a.e t\in ...
2
votes
1answer
32 views

Exercise on L^p spaces

Let $f$ be a function of $L^p([0,2]) \>\> \forall p \in [1, \infty )$ and suppose $||f||_p \leq 1$. Show that $f$ belongs to $L^{\infty}([0,2])$ and $||f||_{\infty} \leq 1$.
3
votes
1answer
69 views

Inequality for integral => Inequality for integrand

I have that for any measurable set $\Omega\subset\mathbb{R}^d$ with $|\Omega|<\infty$ \begin{align}\sqrt{\int_\Omega f(x) dx }\leq \sqrt{c\cdot|\Omega|} + \sqrt{\int_\Omega g(x) dx }.\end{align} ...
1
vote
1answer
39 views

a question about $L^p$ functions on domains in Euclidean spaces

Let $\Omega$ be an open set in $\mathbb{R}^n$ and $f\in L^p(\Omega)$, $1\leq p<\infty$. Define $||f||_{p,\Omega}=\inf\{||f-a||_p: a\in\mathbb{R}\}$. Prove that there exists $a\in\mathbb{R}$ such ...
1
vote
0answers
40 views

Holomorphic Functional Calculus

Framework: Consider a Banach space: $$(E,\|\cdot\|)$$ Given an unbounded operator: $$T:\mathcal{D}(T)\to E\qquad\mathcal{D}(T)\subseteq E$$ together with its resolvent map: ...
1
vote
0answers
44 views

Measure theory integration question involving continuous function

Quick measure theory question. Given that $\Omega \subset \mathbb{R}^{n}$ and $f$ is continuous on $\Omega$. How would you show that if $$\int_{\Omega}f \, dx = 0$$ Then $f = 0$ everywhere? Thanks ...
1
vote
1answer
127 views

Solving Volterra integral equation of first kind with a Gaussian diffusive evolution kernel

I am trying to solve following Voltera integral equation for $P(t|t')$ numerically: $$ \rho(1,t|0,t') = \int_{t'}^{t} dt'' \rho(1,t|1,t'') P(t''|t') $$ where $$ \rho(x,t|x',t') = ...
6
votes
1answer
140 views

Measure theory questions applied to Second Order PDE

Most of the questions are more measure theory and integration related but I need to give some context, so I will now. Consider the quasilinear 2nd-order partial differential equation ...
8
votes
2answers
460 views

Reinventing The Wheel - Part 1: The Riemann Integral [closed]

Preface The core of any notion of integral is some sort of weighted sum: $$\sum b\mu(A)$$ Depending on wether the domain or range is decomposed these split into Riemann and Lebesgue type ones: ...
2
votes
0answers
29 views

An abstract integration problem from a mathematical finance calibration problem

I would massively appreciate help on this problem which relates to me trying to calibrate my financial model to market data. It can be stated without reference to any finance, this is my abstract ...
0
votes
0answers
38 views

Impulsive Boundary value problems

I have this paper They consider this impulsive problem i dont understand this : Proof. First, suppose that $x\in E\cap C^2[J',R]$ is a solution of problem $(1.5)$. It is easy to see by ...