All Questions
Tagged with projection real-analysis
25 questions
4
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0
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$L^2$ vs $L^\infty$ projection
Let $\mathbb P_N$ the space polynomials of degree at most $N$ on $X=[-1,1]$. What is
$$\sup_{f\in L^\infty(X)\setminus \mathbb P_N}\frac{\|f-P_2[f]\|_{\infty}}{d_\infty(f,\mathbb P_N)},$$
where $d_\...
- 1,207
1
vote
1
answer
56
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If $f: U \to \mathbb{R}^3, U \subset \mathbb{R}^4$ an open set, is $\mathcal{C}^1$ in U and has rank 3, then its modulus does not have a maximum
I study mathematics as a hobby and therefore do not have access to a professor to check if my work is correct, so I am coming here for help. Can you please check if my answer is correct?
The exercise ...
- 59
0
votes
1
answer
1k
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Definition of radial projection
In my notes there is a place where "we project radially onto the ball $\partial B(x,r) \subseteq \mathbb{R}^n$", but this map is not precisely defined. I would like to confirm whether I ...
- 3
2
votes
1
answer
59
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Projection of linear combination of another projection
Let $(X,\langle \cdot,\cdot \rangle)$ a Hilbert Space, and $C\subseteq X$ an non empty, closed and convex set.
Show that $\forall x\in X$ and $\alpha \in [0,1]$:
$$\pi_{C}(\pi_{C}(x)+\alpha(x-\pi_{C}(...
- 23
1
vote
1
answer
25
views
Why can we reformulate $ \lvert \lvert Y-P_{[X]}Y\rvert \rvert ^{2}-\lvert \lvert Y-P_{[X_{0}]}Y\rvert \rvert ^{2}$ in the following way?
Let $[X]$ and $[X_{0}]$ denote the span of $X$ and $X_{0}$, respectively. Further, let $P_{[X_{0}]}$ denote the orthogonal projection onto the subspace $[X_{0}]$, and $P_{[X]}$ denote the orthogonal ...
- 1,838
0
votes
1
answer
96
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Showing existence of a projective [duplicate]
Let $X$ be a normed space and $Y$ be a finite dimensional subspace of $X$.
Show that there is a projective $P\in B(X)$ such that $Im P=Y$.
Hint: First Solve for $dimY=1$ then generalize the solution ...
- 762
3
votes
1
answer
561
views
Spectrum of a projection
Let $X$ be a banach space and $P\in B(X)$ a projection.
Show that $\sigma(P) \subset {0,1}$.
In which case $\sigma(P)={0}$?
In which case $\sigma(P)={1}$?
The definition of spectrum:
For a banach ...
- 567
0
votes
1
answer
59
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On uniquely determining a bounded surface in 3D space when the planar projections are given
Let $f(x,y,z)=0$ is a bounded closed surface defined in $D^3$ Where $D$ is a finite closed interval included in real number set. The planar projections of the surface are given:
On $z=0$ plane the ...
- 432
1
vote
1
answer
84
views
Theorem about mean value of iterations of isometries
Let $\mathcal{H}$ be a Hilbert space and let $T\in \mathcal{L}(\mathcal{H})$ be an isometry, and for every integer n let
$$A_n=\frac{1}{n} \sum_{j=0}^{n-1}T^j.$$
Then $A_nv$ converges to $P_Sv$ as n ...
- 1,348
1
vote
2
answers
41
views
$F_{1} \times F_{2}$ is a closed subset then $F_{1}$ is a closed subset of $\mathbf{R}^{m}$ and $F_{2}$ is a closed subset of $\mathbf{R}^{n}$
I am reading Axler's MIRA and found the following question:
Suppose $F_{1}$ is a nonempty subset of $\mathbf{R}^{m}$ and $F_{2}$ is a nonempty subset of $\mathbf{R}^{n}$ Prove that $F_{1} \times F_{2}$...
- 227
1
vote
0
answers
76
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A case when $fg=0$ as well as $f\bar{g}=0$
I wonder how the second last sentence can be deduced. That is, $(f-Pf)\cdot Pg=0$. Here, $e_\alpha(t) = e^{-i\alpha t}.$ I don't believe that such things as whether a complex vector space is closed ...
- 513
1
vote
1
answer
119
views
Rudin's Real and Complex Analysis, Section 9.16
In Section of 9.16 from Rudin's RCA, it says
Let $\hat{M}$ be the image of a closed translation-invariant subspace $M \subset L^2$, nder the Fourier transfrom. Let $P$ be the orthogonal projection of ...
- 807
0
votes
1
answer
286
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How to find Skew Projection Operator onto Plane parallel to some vector?
I was trying to solve previous year question paper of competitive exam
In that I observed some strange question which I have not encountered before.
They had given one equation of plane and told to ...
- 6,773
0
votes
1
answer
162
views
The classical projection theorem
I am going over a proof of the classical projection theorem which states the following:
Let $H$ be a Hilbert space and $M$ a closed subspace of $H$. Corresponding to any vector $x \in H$, there is a ...
- 313
0
votes
2
answers
59
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Finding continuous projection from $L^2([0,2\pi),\mu)$ to $M:=\overline{\mathrm{span}\{e^{int}: n=0, 1, 2,\ldots\}}$
Task:
Consider $L^2([0,2\pi),\mu)$ with $d\mu=\frac{1}{2\pi}dx$.
Let $$e_n(t):=e^{int},\ n \in \mathbb Z$$
$$M:=\overline{\mathrm{span}\{e_n: n=0, 1, 2,\ldots\}}$$
Find a continuous projection ...
- 117
3
votes
2
answers
264
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Lipschitz Constant of Standard Projection
Let $D> d\geq 1$ be positive integers, let $X\triangleq \{x \in \mathbb{R}^D: x_{i}=0\, \forall d<i\leq D\}$, and define the canonical projection
$$
\pi^D_d(x)\triangleq (x_i)_{i=1}^D\mapsto (...
1
vote
1
answer
86
views
Help with exercise regarding orthogonal projection
I am trying to solve the following problem:
Let $\mathcal{H}$ be a Hilbert space and $V\subset\mathcal{H}$ a closed nontrivial subspace. Let $\{e_{k}\}_{k=1}^{\infty}$ be an orthonormal basis for $\...
- 454
0
votes
3
answers
123
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Orthogonal Projections onto lines
Let $P_{\theta}: \Bbb{R}^2 \to \Bbb{R}$ the function $$P_{\theta}(x_1,x_2)=x_1\cos{\theta}+x_2\sin{\theta}, \text{ } \theta \in [0,\pi)$$
$P_{\theta}(x_1,x_2)$ is the orthogonal projection of $x=(x_1,...
- 22.2k
3
votes
1
answer
79
views
How to map $\alpha x$ and $(1 - \alpha)x$ for $\alpha \in (0,1)$ to the same point
Can you please help me with a problem from real function analysis?
Let $p$ be a continuous, increasing function such that $p(s) \in (0,1)$ for $s > 0$. Let $f$ and $g$ be two functions defined on ...
- 33
0
votes
1
answer
64
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Approximating the basis of a specific function
We are given a continuous function $g: A \to B $, where $A, B$ are compact subsets of $\mathbb{R}$.
We define a function $f(x) := g(b_1x)+g(b_2x)+...+ g(b_mx)$, where $b_i < 1$ and $b_ix$ is a ...
- 329
2
votes
0
answers
110
views
Obtaining directional derivatives from the gradient of the projection onto the positive semidefinite cone.
I am currently having a hard time with the notation for derivatives of spectral functions.
In 2006, Malick and Sendov in (DOI: 10.1007/s11228-005-0005-1) have derived an explicit form for the second ...
- 442
1
vote
0
answers
18
views
Covering from Dense Projection
Let $V$ be a Banach space, $W$ a (strict) subspace of $V$, and $U$ a dense proper subset of $V$. When is does there exist a (linear) projection $P^V_W:V\twoheadrightarrow W$, such that
$$
P_W^V(U)=W
....
- 6,828
0
votes
1
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671
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Some queries related to proof of -"Every Regular space X with a countable basis is metrizable"
We shall prove that X is metrizable by imbedding X into a metrizable space Y;that is by showing that X is homeomorphic to some subspace of Y.
Step 1: We prove the following:-
There exists a countable ...
- 3,599
1
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0
answers
139
views
Bilinear form and projection on a convex, closed set
Let $H$ be an euclidean Hilbert space, $K\subseteq H$ non-empty, closed, convex. Let $\beta$ be a continuous bilinear form satisfying $(\vert \beta(x,x)\vert\geq b\Vert x\Vert^2,\; x\in H)$ for some $...
- 1,230
1
vote
1
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691
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In quadratic programming, is the projection onto constraints optimal?
Consider the following quadratic program
$$ x^* := \arg \min_{ x \in X } \ \left\{ x^\top x + c^\top x \right\} \text{ subject to } Ax=b $$
where $X \subset \mathbb{R}^n $ is a non-empty, convex, ...
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