All Questions
10,934 questions
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A basic estimate of exponential sums
Demeter in his book "Fourier Restriction, Decoupling, and Applications" (P287) used the following estimate:
\begin{equation}
\sup_{0\leq n\leq q}\bigg|\sum_{m=0}^n e^{2\pi i\frac{a}{q}m^2}\...
- 351
10
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2
answers
504
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Generalizations of the Robbins lemma and Gaussian integration by parts
This is getting no attention, so I'll try this here:
The Robbins lemma, named after Herbert Robbins, says that if $X\sim\operatorname{Poisson}(\lambda)$ and $g$ is a function for which $\operatorname{...
10
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3
answers
1k
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Historical developement of analysis and partial differential equations (especially in the 20th century)
Q: Is there a set of some comprehensive surveys or monographs describing (in
technical detail) the historical development of the various
subareas of analysis and partial differential equations?
I'...
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1
answer
1k
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Dual space of continuous Banach-space-valued functions
Let $X$ be a Banach space and $K$ some compact Hausdorff space. I am interested in the dual space of the Banach space
$$C(K; X) = \lbrace f: K \to X, \ f \text{ is continuous}\rbrace, \qquad \lVert ...
- 381
10
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2
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281
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Weak* continuity of positive parts
I'm a little embarrassed to be asking this, but surely there is a simple argument that I didn't see?
Let $(f_\lambda)$ be a net in $l^\infty$ which converges weak* to $f \in l^\infty$. We do not ...
- 42.8k
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1
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373
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Real rank zero of group $C^*$-algebras
The concept of real rank zero of a $C^*$-algebra is introduced as non-commutative analogue of dimension ( topological dimension ). For example, it shown (by Brown-Pedersen) such that, if $X$ is a ...
- 399
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2
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1k
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The intuition behind the Hilbert projective metric and the Perron Frobenius Theorem
Recently I have read a proof of the Perron Frobenius Theorem for positive aperiodic matrices. In this proof, the trick is to put a metric in the "positive quadrant" of $\mathbb{R}^n$, $\mathbb{R}^{n}_+...
user11178
10
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2
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926
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Isomorphisms between spaces of test functions and sequence spaces
I am in the process of writing some self-contained notes on probability theory in spaces of distributions, for the purposes of statistical mechanics and quantum field theory. Perhaps the simplest ...
10
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1
answer
2k
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Counting norms on an infinite dimensional vector space
It is known that whenever E is a finite dimensional real vector space, there is only one norm on E up to equivalence (actually one non discrete vector space topology).
Is it known what happens when E ...
- 101
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1
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893
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Is there something wrong with Hörmander's theorem on stationary phase method
It is well-know that the Bessel function has the asymptotic expansion $J_n(\omega) \sim \left( \frac 2 {\pi \omega} \right)^{1/2} \left( \cos \left(\omega -\frac 1 2 n \pi - \frac 1 4 \pi\right) - \...
- 153
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2
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5k
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Approximate a probability distribution by moment matching
Suppose we want to approximate a real-valued random variable $X$ by a discrete random variable $Z$ with finitely many atoms. Suppose all moments of $X$ is finite. We want to match the moments of $X$ ...
- 1,839
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1
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349
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On a variant of Carlson’s theorem
My question is on whether or not there exists some monotone strictly decreasing sequence of positive numbers $c_1>c_2>\ldots$ such that given any $f$ which is a uniformly bounded holomorphic ...
- 4,143
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1
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3k
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Trace of integral trace-class operator
I have seen many answers to the converse question (which seems to be difficult in general), but I would like to ask the following:
Let $T: L^2 \rightarrow L^2$ be a trace-class operator that is also ...
user69109
10
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1
answer
802
views
Restriction of irreducible unitary representation to normal subgroup of finite index
Let $G$ be a Lie group (or more generally a locally compact group), let $N$ be a closed and normal subgroup of $G$ of finite index. Let $H$ be an infinite dimensional complex Hilbert space, and let $\...
- 333
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1
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494
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Optimal exponent in the Lojasiewicz-Simon gradient inequality
Lojasiewicz's theorem asserts that if $F: \mathbb{R}^n\to \mathbb{R}$ is a real-analytic function in a neighborhood of its critical point $0$, then there exist constants $\theta\in (0,1/2]$, $\gamma\...
user81607
10
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1
answer
598
views
What happens if we rotate the kernel of an integral operator?
Given an integral operator $K$ on $L^2(\mathbb R)$ with kernel $k(x, y)$, consider the integral operator $L$ on $L^2(\mathbb R)$, whose kernel has the form $k(\alpha x+\beta y, \gamma x+\delta y)$, ...
- 452
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3
answers
1k
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Efficient computation of "discrete infimal convolution"
This question arises from an application to graphical models in probability theory, but I have abstracted that part out so only algebra remains. Let $\mathbb{R}$ denote standard field of real numbers ...
- 8,501
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2
answers
2k
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When is a space of measures a measurable space?
Let $X$ denote a measurable space, that is, a set equipped with a $\sigma$-algebra $\Sigma(X)$. Let $M(X)$ denote the space of real-valued measures over $X$. This is a vector space over the real ...
- 8,512
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4
answers
1k
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References: Infinite dimensional Lie algebras
What I really want are properties (if it is abelian, nilpotent, solvable, simple, or semisimple; Cartan subalgebras...) of the Lie algebra of smooth functions on a symplectic manifold $(M,\omega)$; ...
- 641
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1
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869
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Complement of a subspace which is a cartesian product
Let $H$ be a Hilbert space and $U$ a closed subspace of $H\times H$ .
Does then exist closed subspaces $V$ and $W$ of $H$ such that $H\times H =
U \oplus (V\times W)$ ?
See also Perturbations of an ...
- 2,753
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2
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1k
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Are operators with trivial spectrum nilpotent in a sense?
Being far from analysis, I recently learned about the Invariant subspace problem and came up with the following (perhaps simple or well-known) question.
Let $H$ be a separable complex Hilbert space ...
- 32.4k
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2
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843
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Implicit function theorem with continuous dependence on parameter
Let $X,Y$ be Hilbert spaces and $P$ a topological space$^1$ and $p_0\in P$.
Let $f:X\times P\to Y$ be a continuous map such that
for any parameter $p\in P$, $f_p:= f|_{X\times \{p\}}:X\to Y$ is ...
- 2,533
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1
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439
views
Interpolation between $L_1^0$ and $L_2^0$
Let $L_p^0$ be the mean zero functions in $L_p(G)$, where, say, $G$ is an infinite compact group endowed with normalized Haar measure. Suppose that $T$ is a bounded linear operator on $L_1$ that maps $...
- 31.5k
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783
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When do tensor products of C*-algebras commute with colimits?
Let $I$ be a filtered poset, which you should think of as being huge. Let $A_i$ be an $I$-diagram of $C^{\star}$-algebras and let $A$ be the colimit of this diagram; if necessary, we can also assume ...
- 976
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315
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Weakly metrizable sets in normed spaces
A similar question was asked on MSE without getting an answer.
In the proof of lemma 1.2 of Asplund operators and holomorphic maps the author (my attempt to contact him failed because the only e-mail ...
- 16.4k
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203
views
Verbal description, or terminology, for the ${\mathcal L}_p$-spaces of Lindenstrauss and Pelczynski
This question is intended for Banach-space specialists and so I will not repeat all the definitions here. My aim is to find out how the Banach space community refers to such spaces in discussions, and ...
- 25.8k
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366
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Are all compact subsets of Banach spaces small in a measure-theoretic sense?
Definition. A subset $K$ of a topological group $X$ is called measure-continuous if there exists a $\sigma$-additive Borel probability measure $\mu$ on $X$ such that for every compact subset $C\subset ...
- 41.9k
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591
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Grothendieck spaces and total subspaces of the dual
There is probably an embarrassingly simple counter-example to my question but I couldn't figure it out myself. Let me give it a try here.
A Banach space $X$ is Grothendieck if weak*-convergent ...
- 11.3k
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1
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503
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Is there an "exponential law" for differentiable maps between smooth manifolds?
Although it seems like a textbook question, I was not able to find a textbook or even a research article answering the following question:
Let $M$, $N$ and $P$ be finite-dimensional smooth manifolds ...
- 843
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1
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680
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A problem concerning $L^2([0,1]\times[0,1])$
Trying to solve a conjecture in differential geometry, I am leaded to the following problem (which may seem weird to a analyst). I wonder if anyone know some techniques that happen to solve it.
Let $...
- 1,804
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1
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283
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A function is of bounded variation if and only if the errors of its best approximation by trigonometric polynomials satisfy $\sum\frac{e_n}n<\infty$?
Let $\mathcal P_n$ be the set of trigonometric polynomials of degree less than or equal to $n$ and let $\lVert\cdot\rVert_\infty$ be the supremum norm. The error of the best approximation of $f$ of ...
- 288
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454
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Open Questions about Wasserstein Space and PDE
While working on my thesis, I encountered the idea of OMT and started reading some more (like Villani's book). In particular, I came across a PhD thesis by Martial Agueh. I thought it was interesting ...
- 427
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1
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433
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Shift invariant subspaces of $l^1$
There is a simple characterization of shift-invariant closed subspaces of $l^2$: for any measurable subset $S$ of $\mathbb{T} = \mathbb{R}/2\pi\mathbb{Z}$, the set of elements of $l^2$ whose Fourier ...
- 42.8k
10
votes
2
answers
960
views
Stone-Weierstrass for cones
A version of the Stone-Weierstrass Theorem asserts: If A is a linear subspace of C(K), the set of continuous functions on a compact space, and if A is a subalgebra that contains the constant functions ...
- 101
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1
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594
views
Density of smooth function in Hilbert spaces
I am looking for a simple reference to the following fact:
If $f:\Omega\to\mathbb{R}$ is continuous, where $\Omega\subset H$ is an open subset of a separable Hilbert space $H$, then for any $\...
- 28k
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1
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429
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Direct sums of operator spaces
I am interested in the $\ell^1$ analogue of direct sums for Operator spaces, e.g. Operator Space Dictionary. Briefly, and operator space is either a concrete subspace of $B(H)$, the operators on a ...
- 18.7k
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1
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515
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Complemented subspaces in the dual of James' space $J$
James' space $J$ is subprojective; i.e., every infinite dimensional (closed) subspace of $J$ contains an infinite dimensional subspace which is complemented in $J$. This fact can be found in Corollary ...
- 4,461
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1
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573
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Tannakian formalism for topological Hopf algebras
Tannaka-Krein duality allows, under the appropriate assumptions, to reconstruct a Hopf algebra from its category of modules. This method was found to be powerful for instance in the work of Etingof-...
- 1,609
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761
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Is a Bessel function larger than all other Bessel functions when evaluated at its first maximum?
Let $\mathcal{J}_{n+1/2}$ be the Bessel function of order $n+1/2$. Let $j'_{n+1/2,1}$ denote the first zero of its derivative, which is also the location of the first maximum of $\mathcal{J}_{n+1/2}$.
...
- 135
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2
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6k
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Characterizing the Dual of $W_0^{s,p}$
I am interested in literature/results characterizing the dual of the fractional Sobolev space $W^{s,p}(\Omega)$, where $\Omega \subset \mathbb{R}^N$ is open, bounded, and smooth, $0< s<1$, and $...
- 882
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3
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1k
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ordered exponential of unbounded operators
Let $H$ be a Hilbert space,
and let $A_t$ be a family of unbounded positive (self-adjoint) operators on $H$ parametrized by $\mathbb t\in R_{\ge 0}$. Consider the ordinary differential equation
$$
\...
- 43.2k
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3
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1k
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Non-linear "Fourier analysis"
Call a function of the following form a beep: $e^{-(\frac{x-\alpha}{\beta})^2}\sin(\rho x+\theta)$. Given a real-valued function $f\in L^2(R)$ and a number $n$, I'm interested in the approximating $f$...
- 17.6k
10
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2
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426
views
Density of the linear span of products of harmonic polymomials
Let $\mathcal{H}$ denote the space of all harmonic polynomials with complex coefficients in $n$ variables $x_1,\ldots, x_n$ in $\mathbb{R}^n$. I'm trying to show that the linear span of the set $\...
- 577
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514
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Positivity of power of positive PSD matrices
Background: Let $M$ be an $n\times n$ matrix with nonnegative entries. It is immediate that for any integer $k$, $M^k$ has nonnegative entries.
Suppose now that, on top of having nonnegative entries, ...
- 505
10
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1
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253
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Approximation via finite rank Cameron-Martin projections
Let $(W, \|\cdot\|_W)$ be a real separable Banach space equipped with
a non-degenerate Gaussian Borel measure $\mu$. Let $H \subset W$ be
the corresponding Cameron-Martin Hilbert space (also known as ...
- 29.8k
10
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1
answer
1k
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Absolute continuity reflected in Fourier coefficients?
Imagine $\mu$ and $\nu$ are two Borel probabilty measures in the interval $[0,1]$.
We say that $\mu$ is absolutely continuous with respect to $\nu$, if for every measurable set $A$ such that $\nu(A)=...
- 445
10
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1
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492
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Which W*-algebras are the duals of C*-coalgebras?
A Banach algebra (assumed associative and unital) is precisely a monoid object in the monoidal category of Banach spaces, short linear maps, and the projective tensor product. A Banach coalgebra is ...
- 2,754
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1
answer
734
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An exponential polynomial with at least one bounded positivity component
In a forthcoming paper on nodal domains of Gaussian random functions, we (I and Misha Sodin) have a statement that is, roughly speaking, the following: if bounded nodal domains are possible at all, ...
- 61.9k
10
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4
answers
783
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Does a quantitative version of Fredholm theory exist?
Let $X$ be a Banach space and $K:X\to X$ be a compact operator. If $I+K$ is injective then it is onto and hence the inverse $(I+K)^{-1}$ is bounded. What kind of qualitative or quantitative ...
- 9,007
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1
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635
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What's the nearest algebraic theory to inner product spaces?
Following the references to the accepted answer to Is the category of Banach spaces with contractions an algebraic theory? one discovers that there is an algebraic theory (infinitary) which is closely ...
- 26.8k