All Questions
5,909 questions
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A problem about the quotient space of an extended Dirichlet space
Let $(\mathscr{E},\mathscr{F})$ be a recurrent Dirichlet form on $L^2(X;m)$ and $\mathscr{F}_e$ the corresponding extended Dirichlet space, then $1\in\mathscr{F}_e$ and $\mathscr{E}(1,1)=0$. Let ${\...
CommunityBot
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votes
0
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71
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Existence of local minimizer
For a $f\in C^3$ function, if there is a sufficiently small $\epsilon$
$$\| \nabla F(x) \| < \epsilon$$
and a sufficiently large $\alpha$ where
$$\lambda_{\min}[\nabla^2 F(x)] \ge \alpha$$
Can ...
- 17.6k
6
votes
1
answer
217
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F-sigma subset of plane meeting every circle at 3 points
Is there an $F_{\sigma}$-set (countable union of closed subsets of plane) $S \subseteq \mathbb{R}^2$ that meets every circle at 3 points?
- 18.6k
1
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1
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244
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Grauert's semicontinuity theorem over the real field
I need to know the following: let $f:\rightarrow {\mathbb R}$ be a real-analytic function defined in a neighbourhood of a point in an analytic manifold. If the fiber $f^{-1}(0)$ is simply connected, ...
- 26.3k
-2
votes
1
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158
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About local maxima of multivariable polynomials
Lets say I have a real valued function which is writable as a polynomial in terms of Frobenius norms of a pair of matrices as in it is of the form, $f_B(A) = f(||A||_F^2, ||AB||_F^2, ||A^TAB||_F^2)$ ...
- 2,246
1
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1
answer
109
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Share of fortunate people in some pie splitting setting
(This question is a follow-up on an older one.)
A huge pie is divided among $N$ guests. The first guest gets $\frac{1}{N}$ of the pie. Guest number $k$ guest gets $\frac{k}{N}$ of what's left, for ...
- 188k
2
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0
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60
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Finding a function in contour integration involving Riemann mapping
Let $T$ be a rectifiable Jordan curve in $\mathbb{C},$ $G$ be the interior of $T,$ and $\Phi$ be a conformal map of the unit disk $\mathbb{D}$ onto $G.$ Let $\mathcal{P}_{n}$ be the space of algebraic ...
3
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1
answer
151
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The weakest condition guarantees some Separation-type of convex sets in Banach spaces
Classical Hahn-Banach Separation theorem plays a vital role in many branches of Analysis, Like functional Analysis, Convex Analysis, Variational Analyis, Theory of ODEs, optimal control and ...
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-3
votes
1
answer
227
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Equation $\ \binom xn'\ =\ \log(n)$ [closed]
Problem: Solve equation
$$ \binom xn'\ =\ \log(n) $$
Here prime stands for the derivative with respect to $x$.
Observe that:
$\quad$ for integer $n$ large,
the approximate solution is $\ ...
- 54.2k
26
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4
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$\binom{x}{2}+\binom{x}{4}+\cdots+\binom{x}{2u}$ is a convex function on $[0,+\infty)$?
Let $f(x)=\binom{x}{2}+\binom{x}{4}+\cdots+\binom{x}{2u}$, where $u\in\mathbb{Z}^+$ and $\binom{x}{l}=\frac{x(x-1)\dots(x-l+1)}{l!}$ for all $l\in\mathbb{Z}^+$.
Then can we prove $f(x)$ is a convex ...
- 3,255
4
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0
answers
261
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Is the following integral positive or not?
Let $n$ be a given even positive integer. We have the following integral
\begin{eqnarray}
&&\int_0^1\cdots\int_0^1\prod\limits_{i=1}^n\prod\limits_{j=1}^n(x_i-y_j)dx_1\cdots dx_ndy_1\cdots ...
- 1,997
18
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5
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3k
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Bernoulli sum meets golden number
Let $B_n$ denote the Bernoulli numbers and let $\phi=\frac{1+\sqrt{5}}2$ be the golden ratio.
I encountered the following infinite sum and would like to ask:
Question. Is this true? If so, any ...
- 13.4k
2
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1
answer
104
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Limit of biggest share of the pie
A huge pie is divided among $N$ guests. The first guest gets $\frac{1}{N}$ of the pie. Guest number $k$ guest gets $\frac{k}{N}$ of what's left, for all $1\leq k\leq N$. (In particular, the last guest ...
- 109k
17
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1
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989
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Can two-point sets be Borel?
Recall that a two-point set is a subset of the plane which meets every line in exactly two points. Such a set was first constructed by Mazurkiewicz in 1914.
I wonder if the following question of ...
- 18.6k
7
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1
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876
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A curious definite integral
I was playing around with $\mathcal{I}=\int_0^1\text{frac}({\frac{1}{x^n}}) dx$, where $\text{frac(.)}$ is the fractional part function, and I discovered that
$$
\mathcal{I} =
\begin{cases}
\frac{1}{...
- 178
4
votes
3
answers
668
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Regularity for the roots of (characteristic) polynomials with given multiplicity
A classical result states that roots of a polynomial are continuous functions of its coefficients.
This is, for exemple, a direct consequence of Rouché's theorem.
Using the implicit function ...
- 108k
2
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1
answer
1k
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Pointwise convergence implies uniform convergence?
Let $K$ be an integral kernel of a bounded operator $S:L^2(\mathbb{R}^n) \rightarrow L^2(\mathbb{R}^n) $ defined like
$$(Sf)(x)= \int_{\mathbb{R}^n}K(x,y)f(y)dy.$$
Assume that $K\in C^{\text{bounded}...
- 24.2k
2
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3
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627
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For every monotonically-decreasing non-negative function $ f $, does there exist a function $ g $ so that $ f g $ is integrable? [closed]
Let $ f $ be a monotonically-decreasing non-negative function satisfying $ \displaystyle \lim_{x \to \infty} f(x) = 0 $. Is it true that the following claim holds?
Claim: There exists a function $ ...
- 12.9k
5
votes
1
answer
333
views
Do monotone functions on the interval have an "Alexander duality" property?
Let $X,Y$ be two copies of the unit interval $[0,1]$. Consider functions $X\rightarrow Y$ and $Y\rightarrow X$ both as subsets of the cartesian product $X\times Y$. (More precisely: identify a ...
CommunityBot
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singular integral operators
Let $(\Omega,g)$ be a compact domain with smooth boundary and suppose that $g$ is smooth. Let $g_D$ and $g_N$ denote the Dirichlet and Neumann green functions for the Laplace-Beltrami operator.
My ...
- 4,115
3
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0
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214
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Is flatness of Wigner Ville Distribution of error function in Fourier Approximation possible? Is it required?
For a real valued function $f(t)$ I want to check the information left, after taking a Fourier partial sum/integral. Let $\hat{f}$ be its Fourier transform and let $$e_{\omega}(t) = f(t) - \int\...
- 698
4
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1
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851
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Paley–Wiener theorem for functions with exponential decay
I feel like this should be well-known, but haven't been able to find any reference so far. Consider the set of all smooth functions on $\mathbb{R}$ such that $$\sup_{x\in \mathbb{R}} |e^{\alpha x} f^{(...
- 24.2k
-1
votes
1
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208
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Does this function belong to $L^2(\mathbb{D})$?
Edit: After the answer of Prof. Eremenko to the previous version, I realized that a weaker assumption works for the main motivation of this post. so I revise the question.
The unit ...
- 366
5
votes
0
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331
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Best Approximation in Operator/non-Frobenius Norm
Since the Frobenius norm on matrices is generated by an inner product, solving the optimization/approximation problem of approximating an operator $X$ with a scalar multiple of another operator $Y$
$$\...
- 153
6
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3
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626
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Alternating power series $\sum_{k\geq 0}(-1)^k z^{(2k+1)^2}$
Suppose that $f(x):=\sum_{k\geq 0}(-1)^ke^{-(2k+1)^2x}$ has a holomorphic continuation to a neighborhood of $0$, that is, $f(x)=\sum_{n\geq 0}a_n x^n$ for $x> 0$ small. I want to know the value of $...
- 24.2k
9
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1
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636
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Is there a characterization of the Hausdorff measures?
It is known that there is a unique measure on the Borel $\sigma$-algebra of $\mathbb{R}^n$ such that the measure of the rectangle $\prod_i [a_i,b_i[$ is $\prod_i (b_i-a_i)$. This is the Lebesgue ...
- 702
2
votes
3
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557
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A consequence of convexity
Let $f:\mathbb{R}\to\mathbb{R}$ a convex decreasing function. Let $x_0 < x_1 < x_2$.
Studying the behaviour of the difference quotient, it is clear that
$$f(x_0)-f(x_2) \leq M (f(x_0)-f(x_1))$$
...
- 45k
10
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1
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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 ...
- 109k
10
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3
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913
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Inequality for functions on [0,1]
Let $a\in (0,1), \;\;\psi_a(x):=\prod_{j=0}^\infty (1-a^{2j+1}x).$
Question. Is it true that, for all $x\in [0,1]$ and all $k\in\mathbb{N},$ the following inequality holds:
$$\frac{x^k}{(1-a)(1-a^3)\...
- 1
7
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1
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489
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When the value of a function in a point is equal to its integral average over the point's neighborhood?
It is well-known that the harmonic functions have this remarkable Averaging Property: if $f$ is harmonic in a domain $U \subset R^n$, then, for any point $x \in U$, $f(x)$ is equal to the integral ...
- 91
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0
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76
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Which sets support which spectra?
I know (and this is of course rather elementary) that an isolated point in the spectrum of a self-adjoint operator $T$ always belongs to the point-spectrum.
I would like to ask: Are there similar ...
- 173
3
votes
1
answer
172
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The sign of the following integral
Given $\theta\in [0,2\pi)$ and let $a,b$ be two nonnegative integers. Consider the following integral:
$$I=\int_0^{\pi}\sin^{2a+1}x\sin^{2b+1}(x+\theta)dx.$$
Since
$$\int_0^{\pi}\sin^mx\cos^{2n+1}xdx=...
- 109k
7
votes
1
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233
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Hausdorff dimension and sigma finiteness
If a function $ f : \mathbf{R} \to \mathbf{R} $ is $\mathscr{C}^{0,\alpha}$ for every $ 0 < \alpha < 1 $ then its graph has Hausdorff dimension $1$.
I would like to see an example of such a ...
- 2,093
3
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1
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530
views
Spectrum of self-adjoint operator
As a non functional analyst, I stumbled over the following question:
Given a self-adjoint Operator $T:D(T) \subset H \rightarrow H.$ Assume we know that $T$ has some eigenvalue $\lambda$ which is ...
- 5,005
3
votes
0
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92
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Arithmetic progressions inside non meager sets
If $A \subseteq \mathbb{R}$ is non-meager Borel set, then $A$ contains arithmetic progressions of every finite length. I know that this is false if we do not assume that $A$ is Borel. In particular, ...
- 31
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109
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Weak convergence of series representing the log characteristic function
Disclaimer. I already asked this question on math.stackexchange.com without any answers or comments as of yet.
In which weak sense does the series representation of the log-characteristic function ...
- 101
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1
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151
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A Bi-Lipschitzian application
We say that $\Omega$ is a star-shaped domain (with respect to the origin) of $\mathbb R ^n$ if :
$$\Omega = \{x\in \mathbb R ^n : \left \| x \right \| < g(\frac{x}{\left \| x \right \|})\}\; \...
CommunityBot
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1
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1
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One trig "survives" a binomial summation: why?
I've seen many trigonometric identities but here is one that I encountered for which I did not find a reference.
In case you wonder where this came from, I was investigating certain $q$-series in ...
0
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1
answer
662
views
A polynomial and its reciprocal expansion [closed]
Suppose $f(x)=\prod_{k=1}^n(x-a_k)$ where all $a_k>0$.
Expand the function $\frac1f$ at $\infty$ so that
$$\frac1{f(x)}=\frac{b_n}{x^n}+\frac{b_{n+1}}{x^{n+1}}+\cdots.$$
Does it follow that each $...
- 47.4k
1
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0
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182
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The real method of interpolation and operator ideals
Let $\overline{A} \mbox{ and } \overline{B}$ be n+1-tuples of Banach spaces and $T:\overline{A}\rightarrow \overline{B}$ be an interpolation operator; let $J(\overline{A})$ and (the corresponding, ...
6
votes
2
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681
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integral depending on a parameter
Let $f$ be a real-valued continuous function on the interval $[0,1]$ and satisfy the following estimate
$$
\left|\int_0^1 f(t) e^{st}dt\right|\le Cs^{\frac12},\quad s>1,
$$
where the constant $C$ ...
- 43.2k
5
votes
1
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329
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Constructive approximation of Hölder functions using kernel functions
Suppose I have a function $f \in \mathcal C^{\alpha, L}([0,1])$, where
$\mathcal C^{\alpha, L}([0,1])$ is the space of $\alpha$-smooth Hölder
functions with norm $L$. I am interested in efficiently ...
- 8,071
16
votes
1
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487
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Bull's-eye Riemann sum
Let $f:[a,b] \to \mathbb{R}^2$ be a continuous curve on the plane.
Question: Are there numbers $a \leq x \leq c \leq y \le b$ such that $$(c-a)f(x)+(b-c)f(y) = \int_a^b f(t) \, dt \ ?$$
In other ...
- 4,713
4
votes
0
answers
139
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The class of all iterated antiderivatives of rational functions
Consider the following property of a function $f$:
There exists a non-negative integer $n$ such that the $n$'th derivative of $f$ is a rational function.
Question 1: Is there a name in the ...
- 4,713
6
votes
1
answer
670
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$C^k$ version of Hadamard's Lemma. Differentiability of the remainder
Let $f:\Omega\to\mathbb{R}$ be of $C^k$ class, let $0\in\Omega\subset\mathbb{R}^n$ and let $\Omega$ be star shaped at $0.$
From Hadamard's Lemma we know that we can write function $f$ as
$$f(x)=f(0)+...
- 4,713
6
votes
2
answers
881
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Summation of bounded sequences
Let $b_{n,j}\in \mathbb{C}$ for each $n,j\in \mathbb{N}$. I was wondering if there is some characterization of those $b_{n,j}$ such that for all bounded sequences $s_j\in \mathbb{C}, j\in \mathbb{N}$, ...
- 24.2k
5
votes
1
answer
694
views
Number of zeros of a real analytic function
Let $f(x,y,t):[-1,1]^3\to \mathbb{R}$ be a real-analytic function. Assume that for any fixed $x,y$, $f(x,y;t)$ is not a constant function $[-1,1]\to \mathbb{R}$. Since the zeros of a non-constant real-...
- 24.2k
20
votes
3
answers
1k
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mixing convex and concave for convexity
Let $n\in\mathbb{N}$ and $0<x<1$ be a real number. Is the following a convex function of $x$?
$$G_n(x)=\log\left(\frac{(1+x^{4n+1})(1+x^{4n-1})(1+x^{2n})(1-x^{2n+1})}{(1+x^{2n+1})(1-x^{2n+2})}\...
- 61.9k
15
votes
3
answers
903
views
Tauberian theorem $\sum_{k=1}^{\infty}e^{-\lambda_{k}t}c_{k} \xrightarrow{t\to 0} \sum_{k=1}^{\infty}c_{k} $
I am trying to prove or disprove
$$\sum_{k=1}^{\infty}e^{-\lambda_{k}t}c_{k} \xrightarrow{t\to 0} \sum_{k=1}^{\infty}c_{k} ,$$
where $\sum c_{k}<\infty, \sum c_{k}^{2}<\infty\text{ and }\frac{\...
- 5,474
9
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1
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570
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Elements of $L^p$ and nice representatives of equivalence classes
Considering $L^p$ $( 1 \leq p < \infty)$ as a normed vector space, each element of $L^p$ is actually an Equivalent class. Take $[f] \in L^p $ as an Equivalent class, What is the Nicest possible ...
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