Questions tagged [norms]

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2
votes
1answer
75 views

Estimate of Hölder Norms (Littlewood–Paley theory)

I'm currently studying Littlewood–Paley theory and its application to norm estimate/PDEs by reading Muscalu and Schlag's textbook, where I encountered the following norm estimate problem: Recall that ...
2
votes
0answers
103 views

Green's identity with a different norm

Let $\Omega \subset \mathbb{R}^n$ be a domain with a smooth boundary $\Gamma$. Suppose that $f, g \colon \mathbb{R}^n \to \mathbb{R}$ are of class $C^\infty( \overline{\Omega})$. Then Green's first ...
2
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0answers
94 views

$L^p$ estimate of a multiplier operator

I'm studying harmonic analysis by myself and I encountered the following claim about multipliers: consider a sequence of complex numbers $\{m_{n}\}_{n \in \mathbb{Z}}$ that satisfies: $$\sum_{n \in \...
12
votes
3answers
227 views

Probability of $\ell_1$-norms of vertices of the rotated Hamming cube

Let $O$ be a $d$-dimensional rotation matrix (i.e., it has real entries and $OO^T = O^TO = I$). Let $\mathbf{x}$ be a uniformly random bitstring of length $d$, i.e., $\mathbf{x} \sim U(\{0,1\}^d)$. In ...
-2
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1answer
62 views

A generalized norm function in $\mathbb{R}^n$ [closed]

We defined a new norm. The norm of $x \in \mathbb{R}^n$ is defined as $$ N_P(x) = \min \{t \geq 0 : x \in t\cdot P\} \enspace,$$ where $P$ is a centrally symmetric and convex body centered at the ...
1
vote
1answer
36 views

Double space integral formulation of homogeneous Sobolev norm

Define for $u\in C_c^\infty (\mathbb R^n), 0<s<1$ the integral $$ I_s(u) = \int_{(x,y)\in \mathbb R^{n+n}} \frac{(u(x+y)-u(x))^2}{|y|^{d+2s}} dxdy. $$ I wish to prove that for some $C=C(s)>1,$...
0
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0answers
73 views

What is the standard terminology for the quantity $\|\nabla f\|_{L^2(\mu)} := \sqrt{\int_{\mathbb R^d}\|\nabla f(x)\|^2d\mu(x)}$?

Let $f:\mathbb R^d \to \mathbb R$ be a continuously differentiable function and let $\mu$ be a probability measure on $\mathbb R^d$. Question. What is the standard teminology for the quantity $\|\...
4
votes
1answer
325 views

Embedding of a Banach space into a Hilbert space

Let $\mathbb H$ be a Hilbert space and let $\mathbb B$ be a Banach space continuously embedded in $\mathbb H$ and distinct from $\mathbb H$. Is it true in general that $\mathbb B$ is an $F_\sigma$ of ...
0
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1answer
55 views

Variance of spectral density is related to the gradient of signal?

Define the frequency variance as: $$ \sigma^2 = \int^\infty_{-\infty}\omega^2 P(\omega) d\omega$$ Where $P(\omega)$ is the spectral density function, which is the same as normalized power. Therefore, $...
0
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0answers
23 views

Analytic formula for $D(x,y) := \sup_{z \in B_p} \|z-x\|_1 - \alpha\|z-y\|_1$, where $\alpha \ge 0$ and $B_p$ is the unit $L_p$-ball

Let $\alpha \in [0,\infty)$ and $p \in [1,\infty]$, and consider the function $D_\alpha:B_p \times B_p\to \mathbb R$ defined by $$ D_{\alpha,p}(x,y) := \sup_{z \in B_p} \|z-x\|_1 - \alpha\|z-y\|_1, $$...
0
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0answers
43 views

Invariant matrix norm over integers

Let $A$ be an arbitrary matrix and $R$ a special matrix of the same order. We know that $\|RA\|=\|A\|$ when $R$ is unitary and $\|.\|$ is Frobenius/spectral norm. In case over integers, do we have ...
1
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0answers
47 views

Traceless low rank approximation of a symmetric matrix by SVD

I have a symmetric matrix $M\in \mathcal{S}^n$ with rank $\mathbf{r}>2$. We can arrange its singular values by $$(\sigma_1=|\lambda_1|)\geq (\sigma_2=|\lambda_2|)\geq \dots \geq (\sigma_r=|\...
3
votes
1answer
69 views

Condition number for matrix of eigenvectors of a diagonalizable matrix

Let $A$ be a diagonalizable matrix, i.e., $A=SDS^{−1}$. For any matrix $A$, condition number is defined as $\kappa(A)=\|A\| \|A\|^{-1}$. For $A$ being a diagonalizable matrix, define $G_A=\{{S: S^{-1} ...
2
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0answers
51 views

Decomposition of the Orlicz norm into sequential norm

I am bearing seeking for a sequential decomposition of the norm in Orlicz space. Let me state what is known in the particular case of Lebesgue space $L^p(\Bbb R^d)$. Given $u\in L^p(\Bbb R^d)$ let $$n\...
0
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1answer
60 views

Weighted metric (or semi-metric) with incorporated distance between Dimensions

Im trying to construct a distance Measure between two vectors, that takes into account also the distance between the Dimensions. I will illustrate my Problem with some examples: $x,y \in \mathbb{R}^n$,...
4
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1answer
93 views

Uniform smoothness inequality for Schatten norms

I've previously asked this question on stack exchange. I'm looking for a proof of the inequality $$ \left[ \frac12(\left\|A+B\right\|_p^p + \left\|A-B\right\|_p^p)\right]^{2/p} \leq \left\|A\right\|_p^...
3
votes
1answer
50 views

Estimate of the norm of the radial part of a function

Consider a function $u\in L^2(\mathbb R^N)$, and another function $\varphi$ which is the unique solution to the Poisson equation $\Delta \varphi = u$ vanishing at $\infty.$ We know that the radial ...
1
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1answer
88 views

Norm of a matrix with clustered eigenvalues

On page 271 of Trefethen and Bau's Numerical Linear Algebra, it is constructed a matrix $$A=2I_{m\times m}+0.5\cdot\frac{\text{rand}(m)}{\sqrt{m}}$$ for $m=200$, where rand(m) is an array with $m\...
7
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2answers
447 views

Bounding supremum norm of Lipschitz function by L1 norm

Consider $f:[0,1]^d \to \mathbb{R}$. Suppose that $f$ is $L$-Lipschitz w.r.t. the Euclidean norm. Can we provide an upper bound on $\|f\|_\infty$ in terms of $\|f\|_1 := \int_{[0,1]^d} |f(x)|dx$ ? In ...
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0answers
50 views

Double dual of an extended seminorm

I have already ask this question on MSE but it didn't received any answer during the weekend. So I thought I will ask it here too. Please let me know if it is bad practice and/or which of MO and MSE ...
11
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1answer
364 views

Subtracting the weak limit reduces the norm in the limit

Question Let $X$ be some reflexive Banach space. Suppose $x_n$ is some sequence in $X$ that weak converges to some $y \neq 0$. Is it the case that $$ \limsup \|x_n - y\| < \limsup \|x_n\| ?$$ ...
2
votes
2answers
114 views

Minimum Euclidean squared norm in the convex hull of points with rational coordinates

This is probably known, but I have not located a reference. Let $P$ be the convex hull of $k$ points in $\mathbb R^n$ with rational coordinates. Consider the Euclidean square norm function $F:P\to\...
1
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0answers
44 views

Is the product of two Banach algebras given by the injective cross-norm itself a Banach algebra?

I understand that you can take the tensor product of Banach spaces in many different ways by specifying different norms; of particular interest to me are the cross-norms. The projective and injective ...
2
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0answers
105 views

Are there any known algebras or vector spaces, where absolute value, modulus or norm is connected to the factors of $\pi$ or $e^{-\gamma}$?

I am currently working on an algebra of divergent integrals and series, and all the elements of that space consist of a regular part (which is a real or complex number) and irregular part (which is ...
4
votes
0answers
121 views

Condition on kernel convolution operator

I am studying O'Neil's convolution inequality. Let $\Phi_1$ and $\Phi_2$ be $N$-functions, with $$ \Phi_i(2t)\approx \Phi_i(t), \quad i=1,2 $$ with $t\gg 1$ and let $k \in M_+(\mathbf R^n)$ is the ...
2
votes
1answer
173 views

How to find upper and lower bound

Let $\Sigma \in S_{++}^n$ be a symmetric positive definite matrix with all diagonal entries equal to one. Let $U \in \mathbb{R}^{n \times k_1}$, $W \in \mathbb{R}^{n \times k_2}$, $\Lambda \in \mathbb{...
0
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0answers
58 views

Ratio of maximum to minimum value

Let $y = X \beta + \epsilon$, where $y \in R^{n}$, $X \in R^{n \times p}$, $\beta \in R^{p}$ and $\epsilon \in R^{n}$. Let $X = USV^\top$ be the SVD of the $X$. Let $u_i$ be the rows of $U$, then ...
1
vote
1answer
296 views

How to minimize l1-norm constrained by “infinity norm”

Let $A \in \mathbb{R}^{m \times n}$ and $b \in \mathbb{R}^m $. I have the following two problems: P.1. \begin{equation} \underset{x\in\mathbb{R}^n}{\text{minimize}} \| Ax-b \|_1 \\ \text{s.t. } \| x \...
2
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0answers
108 views

Conditions on the inequality with a gauge norm

Let $\Phi(x)=\int_0^x \phi(y)\,dy$, $x \in \mathbb{R}_+$, be an N-function, and let $u$ be locally inferable on $\mathbb{R}_+$. Consider the gauge norm $$ \rho_{\Phi,u}(f)=\inf\{\lambda>0: \int_{\...
0
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0answers
76 views

Generalization of Banach-Lamperti theorem to element-wise nonlinear transformations

According to the Banach-Lamperti theorem, every linear isometry $T$ of $\ell_p = \ell_p(\mathbb{N})$ (with $1 \leq p < \infty,𝑝\ne 2$) is of the form $T:(a_n)↦(\epsilon(n)a_{\sigma(n)})$, where $\...
5
votes
1answer
207 views

Recover approximate monotonicity of induced norms

Let $A$ some square matrix with real entries. Take any norm $\|\cdot\|$ consistent with a vector norm. Gelfand's formula tells us that $\rho(A) = \lim_{n \rightarrow \infty} \|A^n\|^{1/n}$. Moreover, ...
15
votes
2answers
994 views

In infinite dimensions, is it possible that convergence of distances to a sequence always implies convergence of that sequence?

This is a cross-posted on MSE here. Let $(X,d)$ be a metric space. Say that $x_n\in X$ is a P-sequence if $\lim_{n\rightarrow\infty}d(x_n,y)$ converges for every $y\in X.$ Say that $(X,d)$ is P-...
2
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0answers
45 views

Removing integral from norm by inequality

My first question on Math Overflow. For my Mathematics Bachelor thesis I am looking at a paper called "Deep Limits of Residual Neural networks" by Matthew Thorpe and Yves van Gennip. (arxiv....
0
votes
1answer
110 views

Prove the optimal solution to maximizing nuclear norm with constraints is attained at corner points of feasible region

The nuclear norm (trace norm) of a matrix $X \in \Bbb R^{m \times n}$ is defined as $$\|X\|_* := \sum_{i=1}^{\min(m,n)} \sigma_i(X)$$ where $\sigma_i(X)$ are the singular values of $X$. The ...
3
votes
1answer
240 views

Seminorm which is zero on dense subset

Let $X$ be a Banach space and let $\hat{X}$ be a dense subset of $X$. If $p$ is a seminorm on $X$ such that $p(x) =0 $ for all $x \in \hat{X}$, does $p(x) =0$ for all $x\in X$ (is $p$ the trivial ...
9
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0answers
235 views

Integral points on elliptic curve and the Lee norm

This question is based on small experiments I have done in Sagemath and if it is not research level, I will move it to MSE: Let $E$ be an elliptic curve defined with coefficients in $\mathbb{Z}$. The ...
3
votes
2answers
210 views

Norm on tensor product of fields

Let $F$ be an algebraically closed field of characteristic $p$ equipped with an absolute value $|\cdot|:F \rightarrow \mathbb{R}_{\ge 0}$ with respect to which $F$ is complete. Define $|\cdot|_{prod}$...
-1
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2answers
205 views

inequivalent norms [closed]

I am thinking about the following question: Let $X$ be a Banach space, say separable, e.g., $l_p$ or $c_0$. When can I say that there exist inequivalent complete norms on $X$?
2
votes
1answer
258 views

Separable Banach spaces isometric to quotient of a Banach space

We know that every separable Banach space is isometrically isomorphic to a quotient space of $(\ell^1,\|.\|_1)$. We also know that the norm defined by $\|x\|=(\|x\|_1^2+\|x\|_2^2)^{1/2}$ for all $x\in ...
1
vote
1answer
162 views

Does kernel regression preserve monotonicity?

Consider the Kernel regression estimator: $$\hat{y}(x)=\frac{\sum_{i=1}^n{K(x-x_i)y_i}}{\sum_{i=1}^n{K(x-x_i)}},$$ where $x,x_1,\dots,x_n\in\mathbb{R}^d$, $y_1,\dots,y_n\in\mathbb{R}$, where $K:\...
2
votes
1answer
204 views

Conditions such that norm of matrix vector can be written as the derivative of the norm of the vector for some convex fonction

Problem statement: Let $A$ be a matrix $\mathbb{R}^{d \times d}$, I want to find some conditions on $A$ such that there exists a differentiable convex function $f: \mathbb{R_{+}} \rightarrow \mathbb{...
2
votes
0answers
126 views

Is this an error in Loomis and Sternberg?

In Loomis and Sternberg's Advanced Calculus section 3.3 Continuity, they make this comment (just before Theorem 3.2, [pp. 128-129 in my copy): A linear map $T : V \rightarrow W$ is bounded below by ...
6
votes
1answer
494 views

Cartesian product of Banach spaces: all norms such that the inclusion is an isometry are equivalent?

Let $\mathcal{A}$ be an arbitrary (typically infinite-dimensional) Banach space with norm $\|\cdot\|_{\mathcal{A}}$ and let $\mathcal{A}^{n}$ be its Cartesian product. I came across the following ...
0
votes
1answer
121 views

$L^p$ norm inequalities with respect to strongly-log-concave densities

Let $\pi(x)=\frac{e^{-f(x)}}{\int_{\mathbb{R}^d}e^{-f(u)}du}$ be a strongly-log-concave distribution, i.e., $f(x):\mathbb{R}^d\rightarrow R$ is an $m$-strongly convex function. Also, $f(x)$ has $L$-...
2
votes
1answer
77 views

When does the map from a normed vector cone to its double dual preserve norms?

If $V$ is a normed vector space then the natural map from $V$ to its double dual $V''$ is norm-preserving as follows from Hahn-Banach theorem. This is well-known. Now assume that P is just a vector ...
2
votes
1answer
191 views

Is the union of l^p a Banach space under some norm?

As a set of sequences, take the union of $\ell^p$, $p\geq 1$. As $p$ increases, the $\ell ^p$ space is larger, with strict inclusion. However, this infinite union is strictly contained in $c_0$, ...
4
votes
0answers
268 views

analytic approximations of the min and max operators

Question: What is the state of the art on analytic approximations of $\min$ and $\max$? My hunch is that numerical analysts probably have a better solution than the one I propose here. For any $\...
0
votes
0answers
148 views

Spectral norm of difference of quadratic matrices restricted to a subspace

Say that we have two matrices $X$ and $Y$ of dimensions $(T \times N)$ with $N < T$ and $rank(X)=rank(Y)=N$. Furthermore, define a $(T \times k)$ dimensional matrix $D$ with $k<N$ and $rank(D)=k$...
-1
votes
1answer
79 views

Sequence converging to different limits with respect to two different _complete_ norms

Do there exist a real vector space $X$ complete with respect to norms $|\cdot|$ and $\|\cdot\|$ and a sequence $(x_n)_{n\in \mathbb N} \subset X$ such that there exist $x,y\in X$: $x\ne y$, $|x_n - x|\...
7
votes
1answer
606 views

Operator norm of square root of matrix vs original

If I have a nonsymmetric matrix whose operator norm is $\leq 1$ and square root it, does its operator norm remain below $1$? More formally, I want to know whether there is always at least one square ...

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