Starred posts

Koro

Apr 24, 2024 13:31

The centered Hardy Littlewood function M_c f and the centered HLW w.r.t. cubes M_c f' are comparable, i.e. there exist positive constants a and b such that
a M_c ' f <= M_c f<= b M_c' f (A)
Can we say the same about the uncentered HLW functions M_u and M_u ' ?
' indicates w.r.t. cubes
I know that M_c is weak (1,1) so by (A), I can say that M_c' is weak (1,1) as well.
But how do I conclude from (A) that M_u and M_u' are weak (1,1) as well?

Koro

Nov 17, 2023 07:43

Does Plancherel theorem hold for functions in Schwartz class $S(\mathbb R^n)$?
n>1
One of the key steps in proving the theorem is the following: define $g= \overline{f(-x)}$, then $\hat g(x)= \int g(y) e^{-2\pi i xy} dy= \overline{\int f(y) e^{-2\pi xy} dy} = \hat f(x)$. But I believe this is wrong.
one should instead have: $\hat g(x)= \int g(y) e^{-2\pi i xy} dy= (-1)^n \overline{\int f(y) e^{-2\pi xy} dy}$
right?
but this looks wrong even for n=1 case.
what went wrong, however? I have $\int_{\mathbb R^n} \overline {f(-y)} e^{-2\pi x.y}dy=(-1)^n \int \overline {f(y)} e^{2\pi x.y} dy$, by J…

Martin Sleziak

Jun 1, 2021 08:05

Just a reminder - to read MathJax/LaTeX in chat you can use the bookmarklet mentioned in this post on meta or go directly to robjohn's website: math.ucla.edu/~robjohn/math/mathjax.html

2

Calvin Khor

Jul 6, 2022 07:31

2

A: Can a self-adjoint operator have a continuous set of eigenvalues?

Just so we have the silly example here. Consider $\ell^2([0,1])$ meaning the set of SEQUENCES $u_{x}$ indexed by a number $x\in [0,1]$. So the scalar product is $$ \langle u, v \rangle = \sum_{x \in [0,1]} \overline{u_x} v_x. $$ In order for $u \in \ell^2([0,1])$, we have that $u_x \neq 0$ for a...

Calvin Khor

Jul 6, 2022 07:31

Is there a typo in this example? I'm guessing the operator should be $(Au)_x:= xu_x$?

SultanDeGranada

Dec 14, 2021 15:48

It's been almost a week since the last update so I took the liberty of asking some students at my university. At least four of us are willing to take up the project of studying TVS/Measure theory with a view towards both analysis and some technichal themes of Number Theory. If any of you or your collegues are interested please tell me as soon as possible so that we can set up a schedule.

Martin Sleziak

Oct 22, 2021 05:44

There is a new room related to analysis: chat.stackexchange.com/rooms/info/130704/basic-real-analysis

Martin Sleziak

Sep 5, 2021 07:57

Richard Borcherds has a course in complex-analysis on YouTube: youtube.com/playlist?list=PL8yHsr3EFj537_iYA5QrvwhvMlpkJ1yGN

Martin Sleziak

Jul 18, 2021 06:32

Dr Joel Feinstein's Functional Analysis: youtube.com/playlist?list=PL554B877A872B4F94 rdmc.nottingham.ac.uk/handle/internal/257

Martin Sleziak

Jun 11, 2021 10:14

Functional Analysis - Part 1 - Metric Space

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Martin Sleziak

May 9, 2021 06:07

Doctorate program: Functional Analysis - Lecture 9: The Hahn-Banach theorem

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Alphie

Apr 20, 2021 14:10

My question is why do we need to invoke Corollary 5.10 on page 163 to show that f_x is integrable? It seems to me that integrability follows from the definition of the integral since we have an L^1 Cauchy sequence of step maps converging the f_x for almost every x.

Alphie

Apr 19, 2021 15:54

Hello everyone this is my first time in a chat room on Math SE. Am reading the proof of Fubini's Theorem in Lang's book real and functional analysis and I have some questions. Anybody familiar with this book?

Snapdragon-X

Mar 13, 2021 09:48

Nice room name!

Bohemian relativist

Mar 13, 2021 09:32

@MartinSleziak I don't actually understand why "Rationals are dense in $\mathbb R$", which is actually the alternative statement of "the real numbers are the completion of the rationals", I think.

Michael B

Jan 21, 2021 10:43

I'm trying to understand weak convergence, could someone explain why the integral \int_0^{2\pi} sin(nx)g(x)dx converges to 0 for both the functions in L^2 ?

strawberry-sunshine

Jan 5, 2021 11:33

Alright, thanks for the advice. My question is related to: math.stackexchange.com/questions/3972718/…

user193319

Mar 30, 2018 14:00

Let $\mathcal{F}$ be a family of measurable functions over some measurable set $E \subseteq \Bbb{R}$. Define $\mathcal{F}^+ = \{f^+ \mid f \in \mathcal{F} \}$, and define $\mathcal{F}^-$ similarly. Is the following true: $\mathcal{F}$ is uniformly integrable if and only if both $\mathcal{F}^+$ and $\mathcal{F}^-$ are uniformly integrable. I have a proof, and I can't spot any errors in it, but I just want to make sure it isn't obviously false.

2

Martin Sleziak

Nov 28, 2017 22:00

For instructions how to render MathJax(TeX) in chat see this post on meta or go directly to robjohn's website.

2

Martin Sleziak

Sep 7, 2017 13:42

For instructions how to render MathJax(TeX) in chat see this post on meta or go directly to robjohn's website.

2

Martin Sleziak

Aug 26, 2017 16:59

For instructions how to render MathJax(TeX) in chat see this post on meta or go directly to robjohn's website.

2

Martin Sleziak

May 3, 2020 08:43

Just a reminder - to read MathJax/LaTeX in chat you can use the bookmarklet mentioned in this post on meta or go directly to robjohn's website: math.ucla.edu/~robjohn/math/mathjax.html

Martin Sleziak

Sep 18, 2019 10:30

I'll just repost this here in a more readable form: The Definite Integral Problem (with a twist)?

Martin Sleziak

Sep 18, 2019 10:29

yesterday, by More Anonymous

Not sure how relevant this is to the room ... But I'm posting it here because of ""measure theory"

https://math.stackexchange.com/questions/2888976/the-definite-integral-problem-with-a-twist

Martin Sleziak

Sep 9, 2019 09:43

Just a reminder - to read MathJax/LaTeX in chat you can use the bookmarklet mentioned in this post on meta or go directly to robjohn's website: math.ucla.edu/~robjohn/math/mathjax.html

Martin Sleziak

Aug 14, 2019 16:36

It seems that I was wrong in my claim about continuous functions: Example of a continuous function that is not Lebesgue measurable

user193319

Aug 4, 2019 15:29

Problem: Show that for every continuous not strictly monotone function $g$ on $\Bbb{R}$ there exists a non-measurable function $f$ on $\Bbb{R}$ such that $g \circ f$ is measurable...Attempt: If $g$ is not strictly monotone, then WLOG there are points $x_1 < x_2 < x_3$ such that $g(x_1) < g(x_2) \le g(x_3)$. Choose $A \subseteq (g(x_1),g(x_2))$ be a non-measurable set...My thought was to take $f = 1_A$, but I don't think this will work.

Martin Sleziak

Jul 19, 2019 06:18

For instructions how to render MathJax(TeX) in chat see this post on meta or go directly to robjohn's website

Martin Sleziak

Jun 10, 2019 19:18

Yet I see in the proof of Theorem 1.10.12 in Megginson's Banach Space Theory that the author uses, for example, a consequence of Open Mapping Theorem (Corollary 1.6.6).

user193319

May 23, 2019 12:37

I am reading the proof of the Implicit Function Theorem in Geometry and Topology by Bredon; I am having trouble with a few parts.

Martin Sleziak

May 13, 2019 03:03

For instructions how to render MathJax(TeX) in chat see this post on meta or go directly to robjohn's website.

user193319

Feb 22, 2019 13:00

Let $A$ be a Lebesgue measurable set of finite measure, and let $\{f_n : A \to \Bbb{R}\}$ be a sequence of measurable sets converging to $f : A \to \Bbb{R}$ pointwise. Does there exist $B \subseteq A$ such that $m(A \setminus B) = 0$ and $f_n$ converges uniformly to $f$ on $B$? I think I was able to prove this using Egoroff's theorem. I just want to verify that it is in fact true.

user193319

Feb 20, 2019 12:59

Problem: Let $f$ be a real valued function defined on a measurable domain $E$. Suppose that $f$ is continuous except at a finite number of points. Is $f$ measurable? Proof: Let $D \subseteq E$ be set of all point at which $f$ is discontinuous. Since $D$ is finite, $m(D) = 0$, so $f$ is measurable on $D$. Since $f$ is continuous on $E \setminus D$, it must be measurable on $f$. Hence, $f$ is measurable on the union $(E \setminus D) \cup D =E$.

Martin Sleziak

Dec 8, 2018 10:20

For instructions how to render MathJax(TeX) in chat see this post on meta or go directly to robjohn's website.

Michael Greinecker

Oct 2, 2018 08:53

If you want an explicit recommendation: "The Elements of Integration and Lebesgue Measure" by Bartle is great and has wonderful exercises that provide plenty of training possibilities.

Michael Greinecker

Oct 2, 2018 07:37

@BAYMAX Every continuous function is measurable.

Michael Greinecker

Sep 27, 2018 19:12

Anyone know a general topology book that goes deep into its subject and has a somewhat categorical flavor with lots of diagram chasing?

Martin Sleziak

Dec 4, 2014 05:28

For instructions how to render MathJax(TeX) in chat see this post on meta.

2

user193319

Aug 11, 2018 17:45

Problem: Suppose that the mapping $f : \Bbb{R}^n \to \Bbb{R}^n$ is a contraction. Define $g(x) = x - f(x)$ for all $x \in \Bbb{R}^n$. Show that the mapping $g : \Bbb{R}^n \to \Bbb{R}^n$ is both one-to-one and onto. Also show that $g$ and its inverse are continuous.

user193319

Aug 4, 2018 20:55

Problem: If $K$ is closed, bounded, and equisummable in $\ell^p$, where $1 \le p < \infty$, show that $K$ is compact.

Martin Sleziak

Jul 27, 2018 16:09

@BAYMAX For every linear transformation $T\colon \mathbb C^n\to\mathbb C^n$ there exists a basis such that the matrix of $T$ w.r.t. this basis is in the Jordan normal form.

Martin Sleziak

Jul 27, 2018 16:09

You simply take for each Jordan block also the basis vectors corresponding to this block.

Michael Greinecker

Jul 19, 2018 05:41

@BAYMAX Under Lebesgue measure, every nonempty open set has positive measure. But open dense sets can have arbitrarily small measure, as in the example I gave. A great thin book that relates measure and concepts such as being open dense is "Measure and Category" by John Oxtoby.

Michael Greinecker

Jul 19, 2018 05:33

If $U$ is an open set that includes $E^C$, then $U$ can certainly have infinite measure. For example, $U=\mathbb{R}^2$ would work. The point is that even though $E^C$ is a dense subset of $\mathbb{R}^2$, it need not have large measure.

Michael Greinecker

Jul 19, 2018 05:33

A common confusion of beginners is that open dense sets must be the whole space or almost the whole space.

Michael Greinecker

Jul 10, 2018 08:36

@BAYMAX Not every function has a Lebesgue integral, but if you have a convergent sequence of integrable functions that lies between two integrable functions, the limit function is integrable and its integral is the limit of the integrals; that's basically the dominated convergence theorem.

BAYMAX

Jul 10, 2018 03:13

Why do we need a separate method of integration known as Lebesgue integrtion?

user193319

Jul 7, 2018 19:35

Dumb question: If $Y$ is a subspace of $X$, and $K \subseteq Y$ is compact in $Y$'s subspace topology, then I cannot necessarily conclude that $K$ is compact in $X$, right?

Martin Sleziak

May 31, 2018 06:32

I've seen that book mentioned a few times. Most recently, somebody recommended it in a comment to my question: What is history behind Smith-Volterra-Cantor sets?

user193319

May 18, 2018 13:05

Does anyone know what Ian means by diagonalization in his comment on this question: math.stackexchange.com/questions/2039788/…

Modern Abstract Analysis

Modern Abstract Analysis