All Questions
3,842 questions with no upvoted or accepted answers
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231
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Pure greedy algorithm
I study pure greedy algorithms in different basises. I am interested in 1 one question: is there such a Riesz basis $D$ in Hilbert space and $f\in H$ such that
$\|f-G_m(f,D)\|>Cm^{-1/2}\lvert\{f}\...
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votes
0
answers
244
views
Checking whether this would be bounded
It may be better to post this question here. Assume that $M$ is an $m$ by $m$ ($m$ is an even number) symmetric
positive-semi-definite matrix with exactly $m/2$ positive eigenvalues
and every entry of ...
- 1
0
votes
0
answers
183
views
Continuity of the Shadow of a Nondecreasing Function
So I'm working a lot with monotone nondecreasing functions $f : [0,1] \rightarrow [0,1]$, and I'm defining a certain discrete dynamics on them. Here nondecreasing means $x < y \Rightarrow f(x) \leq ...
- 485
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0
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186
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Properties of Eigenfunctions of a Kernel
I'm a newbie and may be this question is bit simple for you but pardon me if it's too simple and provide me some references.
I've and Kernel function $K(x,y)$
$f(x)=(Kg)(x)=\int_{\Omega}K(x,y)g(y)dy$...
- 17
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0
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164
views
Can we separate Toeplitz matrices for negative and positive eigenvalues?
Consider a Toeplitz matrix T which has both positive and negative eigenvalues. Can we prove that there exist two Toeplitz matrix T1 and T2 such that T1+T2=T and T1 has only one positive Eigenvalues ...
- 9
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0
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255
views
Convergence of a function in a metric space to its metric.
Given a metric space $(\mathbb{A},d)$ in $\mathbb{R^n}$ with a metric $d$ being the Euclidean metric:
If $\lim_{t \rightarrow \infty}||A_{t+1}-A_t||\rightarrow 0$ is a convergent sequence where $A$ ...
- 101
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0
answers
73
views
A constrained prolongement
Let $\Omega$ be a domain of $R^n$, let $\omega$ be open subset of $\Omega$ and let $\theta \in W^{2,\infty}(\omega).$
I am wondering about the existence of a function $\tilde{\theta} \in W^{2,\infty}...
- 25
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0
answers
143
views
description of a convex set of functions
Hi everyone,
I have a question about the characterization of a set of functions.
Let $\Phi$ a set containing all the functions $\phi(x): \mathbb{R}_+\rightarrow \mathbb{R}_{+}$ that satisfy the ...
- 69
0
votes
0
answers
267
views
Convergence rate of fourier partial sum
Hi there,
I am looking for the proofs of convergence rate in Section 4 of the following artical
http://en.wikipedia.org/wiki/Convergence_of_Fourier_series
in wikipedia. Could someone please point ...
- 13
0
votes
0
answers
136
views
Finding peaks and determining noise
Hello ,
Im having one matrix which is product of two FFT transforms of one fits image ( astronomical image ). In that matrix you could find 3 peaks. One largest in center, and two around central ...
- 101
0
votes
0
answers
227
views
Hermite function expansion
Let $f$ be a continuous function on $\mathbb{R}$ with compact support and unique maximum. Form the functions
$$
F_{n,k}(x)=f^n\left(x-\frac{k}{2^n}\right), k \in Z, n>0
$$
I am wondering if one ...
- 71
0
votes
0
answers
150
views
$n$-th derivative of the prolate spheroidal function
For a given real number $c>0$ define functions $\left(\psi_{k,c}(\cdot)\right)_{k\ge0}$, as an eigenfunctions of the Sturm-Liouville operators $L_c$ defined
$$
L_c(\psi)=(1-x^2)\frac{d^2\psi}{dx^2}...
- 71
0
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0
answers
272
views
L_2-norm representation
Let
$$
f^{\alpha}_+(x)=\frac{1}{\Gamma(\alpha+1)}\sum_{k\ge 0}(-1)^k{\alpha+1 \choose k}(x-k)^{\alpha}_+,
$$
where $\alpha > -\frac 12$.
I am wondering if one can get nice representation of $L^2$-...
- 71
0
votes
0
answers
395
views
The ratio of two strictly increasing functions
Given:
\begin{equation}
f_1(a)=\sum_{i=1}^{k^*-1} \left(\begin{array}{c}
K \\\
i \\
\end{array} \right) \left(-1-\frac{1}{ar}\right)^i
\end{equation}
\begin{equation}
f_2(a)=\sum_{i=1}^{k^*-1} ...
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0
answers
606
views
partial differential equations with mixed boundary conditions
hi,
does anyone know some good references (books, papers) on partial differential equations with mixed boundary conditions ?
actually I am intrested in the following: Let $f(x)=(f_{1}(x),...,f_{n}(...
- 89
0
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0
answers
601
views
Fourier transform in n-dim Euclidean and Minkowski space
As far as I understood, the Fourier decomposition of a function $\boldsymbol{F}\colon\mathbb{R}^{n}\to\mathbb{R}^{m}$ where $\mathbb{R}^{n}$ is endowed with the Euclidean inner product $\left<\cdot,...
- 347
0
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0
answers
184
views
Extension of closed linear functionals...
If f is a closed linear functional defined on a dense subspace of a Banach space X, and, consider f1 which is an extension of f to X, is there a way to show that f1 is also closed without invoking the ...
- 55
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0
answers
352
views
Circlular Convolution for two finite sequences of different length.
Hello.
I just want to know how the circlular Convolution for two finite sequences of different lengths is defined.
As I have found in MIT lectures for the sequences of the same length we have
$C=\...
- 31
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votes
0
answers
104
views
Differential equation with switched parameters and boundary conditions in integral form
Sorry for the title, I didn't find a better description (showing that I have no idea for the solution). Feel free to put in a better title and change the tags if you can grasp a view on the problem.
...
- 43
0
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0
answers
138
views
Notion of simplicity of a function(al)
Given a function (functional actually) $f(x,g(x))$, can a notion of simplicity be attached with respect to the function $g(x)$? (all functions and args are real).
Specifically, intuitively one could ...
- 59
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0
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436
views
cokernels of semi-Fredholm operators
I did not find a reference for the following question, so I will pose it here. I think the answer should be elementary.
Let $F:X\rightarrow Y$ be a semi-Fredholm operator between Banach spaces, i.e. $...
- 2,935
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answers
144
views
Are all of compact support functions of $A(G)$ in its abstract Segal algebras?
Let $G$ be a locally compact group. We know that if $G$ is abelian and $\cal F$ implies the Fourier transform, for every Segal algebra of $G$ say $S^1(G)$, ${\cal F}S^1(G)$ is an abstract Segal ...
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0
answers
155
views
General form of a symplectic map
A symplectic automorphism of a Hilbert space has the form $T=U(\cosh S+J\sinh S)$ for a unitary $U$, an antilinear involution $J$ and a positive operator $S$. In fact a version of this goes through in ...
- 1,411
0
votes
0
answers
298
views
High dimensional beta integral (question following the previous post)
Hello,
This post is a question following the previous post. In one dimensional case, we have
$$
\int_0^x |y|^{1-\alpha} |x-y|^{1-\beta} d y = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)} |...
- 1,649
0
votes
0
answers
388
views
Global index of convexity/concavity of a function
We are looking for a global index of the convexity/concavity of a function.
For concreteness, how can I formalize the intuitive notion that a function $f$ is more convex than $g$ where $f,g:[0,1]\...
- 111
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0
answers
301
views
Lifting of product of a Banach algebra
Let $A$ be a non unital Banach algebra. The product induces a bounded linear map $T:A \otimes_{\gamma} A\to A$ where $\otimes_\gamma$ denotes the Banach projective tensor product.
A lifting of $T$ is ...
- 1,222
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0
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724
views
What is the function $\sin(n \omega) / (n \sin \omega)$?
During my work, I encounter the function like $\frac{\sin(n \omega)}{n \sin \omega}$. I'm puzzled and knew nothing about this function before.
Given integer $n>1$, my question is how to find a ...
- 173
0
votes
0
answers
362
views
Gradient of the energy functional in $H^{1,2}$-norm
I have to use estimates for the gradient of the energy functional on the free loop space of a fixed compact manifold $Q$. As such, one considers $H^{1,2}$-maps of the circle into $Q$. The energy ...
- 2,935
0
votes
0
answers
320
views
A result about Fredholm operator
When I read the article "Index Theory" in Handbook of global analysis, I meet a result as below(Corollary 2.13):
If every $F_0\in \mathcal {F}(H_1,H_2)$, there is an open neighborhood $U_0\subseteq \...
- 381
0
votes
1
answer
226
views
Subspaces of $H^{\infty}(\mathbb{D})$ which contains a nontrivial weak* closed subalgebra
Let $H^{\infty}(\mathbb{D})$ denotes the Banach space of bounded holomorphic functions in the unit disc. Consider the weak* topology on $L^{\infty}(\mathbb{T})$
that it inherits as the dual of $L^{1}(\...
- 175
0
votes
1
answer
440
views
Variation on Fatou's lemma for Sobolev norms
Recall that Fatou's Lemma says that for every sequence $f_n$ of non-negative measurable functions
$$\int \liminf_{n\to \infty} f_n \ d\mu\leq \liminf_{n\to \infty} \int f_n\ d\mu \ .$$
If I am not ...
- 1
0
votes
1
answer
221
views
Sort-of extension of Young inequality to arbitrary measures
Hello folks,
Suppose we're working in ${\mathbb T}^q$, the $q$-times Cartesian product of $[-\pi,\pi]$ with the endpoints identified. So, $ L^p := L^p({\mathbb T}^q) $.
The Young inequality may be ...
- 1
-1
votes
0
answers
94
views
Why define Schwartz by supremum rather than limit?
The Schwartz space is defined as the set of all indefinitely differentiable functions such that the supremum over the free variable of any (order) derivative times any (order) power is finite.
However,...
- 107
-1
votes
0
answers
53
views
convergence of convolution in Bochner space
I want to prove a well-known fact in $L^p(R^n)$ namely that, the convolution of an element in $L^p$ with an element of $L^1$ is in $L^p$
let: if $u∈L^p (R;X) , f∈L^1 (R)$ and $X$ is Separable and ...
-1
votes
1
answer
86
views
how take weak derivative of norms in hilbert spaces?
Let the following be hilbert spaces with dens inclusions $V ↪H=H^* ↪V^*$. Where $H^*$ and $V^*$ are the duals. $H$ has the product $(*,*)$ and $V×V^*$ has the product $⟨*,*⟩$.
Let $u∈L^2 ([0,T];V); ...
-1
votes
1
answer
246
views
Determine the singular values of a compact operator in terms of the eigenvalues of an alternating tensor product of operators
Let $H$ be a $\mathbb R$-Hilbert space, $A\in\mathfrak L(H)$ be compact and $$|A|:=\sqrt{A^\ast A}$$ denote the square-root of $A$. By definition, the $k$th largest singular value $\sigma_k(A)$ of $A$ ...
- 167
-1
votes
1
answer
114
views
Construct a probability function on the operator monotone functions, $g(t)=t g(t^{-1})$, fitting certain values
To immediately pose the question of interest to us, without first expanding upon its (quantum-information-theoretic) origin—we seek a univariate function $f$, for which we have the ("two-qubit ...
- 723
-1
votes
2
answers
440
views
Motivation for weak solution of a PDE (initial condition)
The following question came to me when reading the famous paper of ALT and LUCKHAUS: "Quasilinear elliptic-parabolic differential equations"
When looking at a (nonlinear degenerate) PDE like
$$ \...
- 1
-1
votes
1
answer
152
views
Question regarding to the basis of L^p space via compact self adjoint operators. ( eg: inverse of -laplacian )
Do eigenfunctions of inverse of elliptic operator (eg: Laplacian) form basis of $L^P(\Omega)$ ? For p=2 we know the answer is yes, I am looking for p>2.
More generally, is it true that eigenfunctions ...
-1
votes
1
answer
143
views
Help with notations from 2D to 3D FFT representations as 1D FFT
I have this question on mathematics forum too, Notations, I thought of posting here, which ever place I get an answer, I will try to close it in the other.
I need some help and clarifications for my ...
- 105
-3
votes
0
answers
69
views
Exercise generalizing (related to) Hölder's inequality
I came across this exercise and feel absolutely stuck:
Let $p, q, r \in (1, \infty]$ be such that $1/p + 1/q = 1 + 1/r$. Suppose that $F : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ satisfies ...
- 1
-4
votes
1
answer
302
views
A Question in Fourier Analysis proposing a conjecture
Let $f$ be a $2\pi$ periodic BV function whose derivative is also BV.Let the amount of jump at a point $x$ is denoted as $\lfloor f \rfloor (x) = f(x+0)-f(x-0)$ Define function $J:\mathbb{R} \to\...
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