All Questions
151 questions with no upvoted or accepted answers
1
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139
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Converse to Hausdorff-Young (or Riesz-Thorin) for finite cyclic groups?
Let $v$ be a vector $v \in \mathbb{R}^p$, with non-negative entries and $p$ prime. The Hausdorff-Young inequality gives bounds of the form:
$$\|\mathcal{F}v\|_a \le C_{a,b} \|v\|_b$$
where the ...
- 435
1
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0
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62
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Properties of the Fourier Transform of Countably Supported Functions on $[0,1)$
Identifying $\mathbb{R}/\mathbb{Z}$ with the interval $\left[0,1\right)$, let $C_{\textrm{coun}}\left(\mathbb{R}/\mathbb{Z}\right)$ denote the set of all functions $f:\mathbb{R}/\mathbb{Z}\rightarrow\...
- 1,284
1
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0
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119
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Integrable functions that may not satisfy the inversion Fourier formula
Let $f\in L^1(\mathbb{R})$. We define $\phi_f(x)=\int_{\mathbb{R}} \hat{f}(\zeta)e^{2\pi i\zeta x}d\zeta$ if the improper Riemann integral is finite otherwise, $\phi_f(x)=\infty$.
Does there exist ...
- 4,058
1
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0
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67
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Approximate identities on the unit disk and going beyond a power series' radius of convergence
Let $\left\{ a_{n}\right\} _{n\geq0}$ be a bounded sequence of complex numbers, so that the power series $f\left(z\right)=\sum_{n=0}^{\infty}a_{n}z^{n}$ has a radius of convergence $\geq1$. ...
- 1,284
1
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1
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294
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Regarding subspace generated by the polynomial multiples of outer functions
Let $\mathbb{D}$ and $\mathbb{T}$ denote the open unit disk and unit circle in $\mathbb{C}$ respectively. We write $Hol(\mathbb{D})$ for the space of all holomorphic functions on $\mathbb{D}.$ The ...
- 31
1
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353
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Eigenvalues of convolution matrices
Let $h: \mathbb{R}\to \mathbb{R}$ be a smooth function. Fix $0\leq s_1\leq \cdots \leq s_m\leq 1$ and $0\leq t_1\leq \cdots \leq t_n\leq 1$. Construct $A\in \mathbb{R}^{m\times n}$ by letting $A_{i,j}:...
- 315
1
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145
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Using Paley-Wiener Theorem to prove the decay of $G(x-y)$
This question is related to my previous one, where I was looking for some help to prove the decay of the lattice Green function:
\begin{eqnarray}
G(x-y) = \int_{[-\pi,\pi]^{d}}\frac{d^{d}k}{(2\pi)^{d}}...
- 1,305
1
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0
answers
40
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Example of periodic semidifferentiable function without absolutely convergent Fourier series
Is there an example of a periodic continuous function that is semidifferentiable (i.e the left derivative and the right derivative exist at each point), but
with a non-absolutely convergent Fourier ...
- 1,035
1
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0
answers
126
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Almost every where divergent Fourier series
Does there exist any continuous function $f:[\pi,\pi]\to \mathbb{C}$ whose Fourier series $\sum \hat{f}(n)e^{int}$ is almost every where divergent?
- 4,058
1
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0
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324
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Conditions for Poisson summation (for discontinuous functions)
Let $G$ be an locally compact abelian group with $\Gamma$ a discrete cocompact subgroup. I'm looking for precise conditions by which Poisson summation formula holds. That is, for some function $f$ on $...
- 3,799
1
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0
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122
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Discrete Wavelet Transform and Gaussian decay
I have a question regarding the possibility of constructing a Discrete Wavelet Transform based on a scaling function having Gaussian decay (and no more decay than that). More specifically, I am ...
- 11
1
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0
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237
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On the bound of the Stein-Wainger oscillatory integral
Let $\lambda\in \mathbb{R}$, $\phi\in C^\infty(\mathbb{R})$. We define the Stein-Wainger oscillatory integral by
$$I=p.v.\int_\mathbb{R} e^{i\lambda\phi(t)}\frac{dt}{t}.$$
Stein-Wainger [1] showed ...
- 11
1
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0
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146
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Functional equation with Fourier transform
What are the continuous functions $f$ such that on $\mathbb{R}^{+*}$:
$$f(x) - \frac{C}{x} \hat{f}(\frac{1}{x}) =x^{\alpha}$$
Where $\hat{f}$ is the Fourier transform of $f(|x|)$ and $C$ a constant....
- 1,199
1
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0
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50
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Comparison of (square) of a function and its Fourier transform in an integral
I am completely stuck on a comparison between $f(t)^2$ and $\hat{f}(t)^2$ in an integral.
Considering $f(t)$ of rapid decrease at infinity such that near zero: $f(t) \sim_0 t^{-\frac{1}{2}- \alpha}+o(...
- 1,199
1
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0
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141
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Characterisation of functions for which the Fourier transform commutes with a particular operator
Defining the operator $\phi$ by: $\phi(f(x))=\frac{1}{|x|} f(\frac{1}{x})$, and noting $\mathcal{F}$ the Fourier transform on the real line, can we characterize all the functions (with real variable ...
- 1,199
1
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0
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233
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Fubini: can we interchange integration order on this double integral (with Fourier series product)
Can we interchange the order of integration of following double integral ?
$$I = \int_{0}^{1} \int_{0}^{\infty} F(x,y) \overline{R(x,y)} - R(x,y) \overline{F(x,y)} \; dx \; dy$$
Where $F(x,y)= \...
- 1,199
1
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0
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100
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Convergence and boundedness in $L^\infty([0,T]\times \Omega)$ of Karhunen-Loeve expansion
Let $X:[0,T]\times\Omega\rightarrow\mathbb{R}$ be a stochastic process in $L^2([0,T]\times\Omega)$. Consider the Karhunen-Loeve expansion of $X$:
$$ X(t,\omega)=\mu_X(t)+\sum_{n=1}^\infty \sqrt{\nu_n}\...
- 141
1
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0
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124
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Inequality about the Fourier transform: $\Vert u \Vert_{L^k} \le \Vert \mathcal{F}(u) \Vert_{L^m}$ (where $1 \le m \le 2$ and $m,k$ Holder conjugates)
How can I prove the following inequality about the Fourier transform?
$$\Vert u \Vert_{L^k(\mathbb{R}^N)} \le \Vert \mathcal{F}(u) \Vert_{L^m(\mathbb{R}^N)}$$ for $1 \le m \le 2$ and $m,k$ Holder ...
user89890
1
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0
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202
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Space of analytic function and sequence space $l^p$
Let $\mathbb{D} = \{z:|z|<1\}$ be open unit disc in complex plane. Define space of analytic function:
$N^p=\{f:\mathbb{D} \to \mathbb{C} | f(z)=\sum_{n=0}^{\infty} a_n z^n, \sum_{n=0}^{\infty}|a_n|...
- 259
1
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0
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194
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Cotlar-Stein's Lemma and the Dirichlet kernel
It is well-known that Cotlar-Stein's Lemma can be used to prove the $L^2$ boundedness of the Hilbert transform. See e.g. $L^2$ boundedness of the Hilbert transform via Cotlar-Stein Lemma. Then using ...
- 171
1
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0
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127
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What is the analogue of expansive matrix for automorphisms?
We say an invertible $n \times n$ matrix with entries in $\Bbb R^n$ is expansive if the absolute values of all of its eigenvalues exceed $1$. An easy calculation also shows that if we consider a ball ...
- 41
1
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0
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92
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Perturbation in Besov space
$\|f\|_{B^{0}_{p,p}}=(\sum_{j\geq -1} \|\Delta_j f\|_p^p)^{1/p}$ is the Besov norm of $f$.
Here the Fourier transform of $\Delta_jf~(j\geq 0)$ is $\psi(2^{-j}\xi)\hat{f}(\xi)$ and $\psi$ is a smooth ...
- 515
1
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0
answers
125
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Transformation of kernel
I have the following problem at hand.
Define the kernel
$$K(x_1,x_2) = \int_{-1}^1\int_{-1}^1 \exp(-2\pi\jmath x_1 y_1)R(y_1,y_2)\exp(2\pi\jmath x_2 y_2)\mathrm{d}y_1\mathrm{d_2}.$$
Now, if $R(y_1,...
- 21
1
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133
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Condition for boundedness in stationary phase theorem
I am trying to understand theorem 7.7.1 in Hormander's Analysis of linear partial differential operators, vol.1.
Let $K \subset \mathbb{R}^n$ be a compact set, $X$ an open neighborhood of $K$ and $j, ...
- 93
1
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0
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102
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How do functions operates in a Fourier algebra $A^{q}(\mathbb T)$?
We put , $A^{q}(\mathbb T)= \{ f\in L^{q}(\mathbb T): \hat{f}\in \ell^{q}(\mathbb Z) \}.$
By Helson-Kahane-Katznelson-Rudin Theorem, it follows that,
"Let $F$ be a function on $\mathbb C$ and if $F(f)...
- 1,051
1
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0
answers
348
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On a property of Riemann Zeta function zeros
Lets consider the function : $$F(x) = \sum_{n=1} (xn)^{-s_0} e^{-nx} $$
with $s_0$ a zero of the Riemann Zeta function in the critical strip.
This sum is well defined for $x \in \mathbb{R}^{+*}$. It ...
- 1,199
1
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0
answers
99
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decomposition of tempered distributions by entire analytic functions
Let $\phi$ be a $C^{\infty}$ function on $\mathbb R^{n}$ with
$$ \operatorname{supp} \phi \subset \{\xi \in \mathbb R^{n}: |\xi|\leq 2, \phi(\xi)=1~~\text{if}~|\xi|\leq 1\}$$
Let $j\in \mathbb N$ ...
- 1,051
1
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0
answers
269
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$L^{p}(\mathbb R)\subset L^{1}(\mathbb R) \ast L^{p}(\mathbb R), (1< p< \infty)$?
Let $\mathbb T$ be a circle group.
In 1939, Salem, has shown that, every member of $L^{1}(\mathbb T)$ can written as a product(convolution) some other two members of $L^{1}(\mathbb T),$ that is, $L^...
- 1,051
1
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0
answers
164
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How to use, $(|u|^{2}u - |v|^{2}v)(s)= (u-v)|u|^{2}(s)+ v(|u|^{2}-|v|^{2}) (s)$; to prove contraction in a Banach space $C([0,T]; M^{p,1})$?
(May be this is very basic question for MO)
(For details or this question you may see the paper page no. 9, MR2506839, Local well-posedness of nonlinear dispersive equations on modulation spaces; ...
- 1,051
1
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0
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129
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Differentiability of $f*g$ on the circle, for integrable f, bounded g, and some decay of the Fourier coefficients of f
If $f\in L^1(\mathbb{T})$ and $g\in L^\infty(\mathbb{T})$ where $\mathbb{T}$ is the circle, such that $\hat{f}\in L^{p}(\mathbb{Z})$ for some $1\leq p<\infty$, do we have that $f*g$ is ...
- 647
1
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0
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318
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Fourier series/transform of an amplitude-limited sinusoid
I am trying to estimate the amplitude of an original unlimited sine wave from a measurement of the power spectral density (PSD) of an amplitude-limited version. I expect that I may be able to do so ...
- 11
0
votes
1
answer
117
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Validity of approximation method for von Mangoldt function
I'm working on a problem involving the pointwise almost everywhere convergence of multilinear ergodic averages with the von Mangoldt function inspired by this paper. Specifically, I'm looking at ...
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0
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89
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Maximal function on mixed $L^{p}$
Consider $ f_{j,k}$ to be a function in $L^{p}(l^{q}(l^{2}))$, that is
$$
\Vert f_{j,k} \Vert^{p}_{L^{p}(l^{q}(l^{2}))} = \int_{\mathbb{R}^{n}} \left( \sum_{k} \big[ \sum_{j} \vert f_{j,k}(x) \vert^{2}...
- 31
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53
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Vectors of complex exponentials span $\mathbf{C}^N$
Let $Q = [0,1]\times [0,1]$. Let $1\leq k < \infty$ and $\{(x_l,\xi_l)\}_{l=1}^{k}\subseteq Q$ be such that $(x_i,\xi_i)\ne (x_j, \xi_j)$ for $i\ne j$. Additionally, for $1\leq l \leq k$, let $n_l\...
- 171
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0
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94
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The asymptotic behaviour of the Fourier transform of a certain class of radially symmetric functions
Fix $\theta\in (-\pi/2,\pi/2)$ and let $a>0$. Suppose that $f:\mathbb{C}\rightarrow \mathbb{C}$ is analytic in $S:=\{z\in \mathbb{C}: |\arg{z}|<\pi/2\}$ and
$$|f(z)|\sim |z|^{-a},\qquad |z|\to \...
- 852
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0
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145
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Why is this function in $L^1$?
I had a question about a claim made in the paper "Group Invariant Scattering" and why it is true. Consider the function $h_j(x) = 2^{nj}\psi(2^jx)$, where $\psi$ is a function such that $\...
- 101
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75
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$|\partial $ as Fourier multiplier
I have the following nonlinear dispersive PDEs
$$i \partial_t u- \partial_x^2 u =|\partial_x| |u|^2$$
where $f$ is some nice complex-valued function.
I am trying to use the ansatz $u(t,x) = e^{i \...
- 159
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0
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72
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Fourier coefficient of close functions
Let $p$ be some prime. Let $\mathbb{Z}_p$ be the cyclic group of order $p$. Let $f, g \colon \mathbb{Z}_p \to \{\pm 1\}$ be two functions. Recall that the Fourier transform is defined as
$$ f(x) = \...
- 1
0
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0
answers
79
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Reverse Hölder for possibly singular matrix weights
I'm working on some nonlinear partial differential equations and I have been led to the following puzzle. Let $W(x)$ be a symmetric positive semidefinite-valued function of $x \in \mathbb{R}^d$ (a &...
- 981
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0
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75
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Extracting the point mass measure of some type of positive measures
Let us consider the measure algebra $M(\mathbb{R})$ consisting of all Radon measures on the reals.
Let $\delta_0$ be the point mass measure concentrated on 0, which is also the multiplicative ...
- 4,058
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votes
0
answers
299
views
Some density properties about Sobolev periodic spaces
Let $L>0$ fixed. Consider the space
$$
\mathcal{P}:=\{f: \mathbb{R} \longrightarrow \mathbb{C} \; ; \; f \: \text{is infinitely differentiable and periodic with period}\: L\}.
$$
For $r \in \mathbb{...
- 205
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0
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52
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Functions on dense subgroups of $\mathbb{R}^n$
Let $G$ be a finitely generated dense subgroup of $\mathbb{R}^n$, and $f$ be a character on $G$.
In the situation I'm looking at $f$ is either $1$ or $-1$ at any point.
Function $f$ can be extended to ...
- 2,772
0
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0
answers
74
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Parseval type lower bound on sum of squares of function projections
This is a followup to this earlier question
Let $f:\mathbb{Z}\rightarrow \{\pm 1\}.$ Assume that the support of $f$ is finite, say it is contained in $[1,N],$ it can even be taken to be $[1,N]$ if it ...
- 10.4k
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0
answers
60
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Solution of a functional equation with cosine transform
What are the functions verifying:
$$\int_0^{\infty} f(t) \cos(2\pi xt)=\lambda \frac{1}{x} f(\frac{1}{x})$$
With $\lambda$ a constant ?
(Functions $x^{-\alpha}$ with $0<\alpha<1$ are solutions ...
- 1,199
0
votes
0
answers
56
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Existence of a couple of functions solution of a differential equation (with additional constraint)
I would like to know if we can find a real function $v(x)$ and a complex function $f(x)$, such that they solve the following differential equation (with $\alpha$ a complex, $0<Re(\alpha)<1$):
$$...
- 1,199
0
votes
0
answers
227
views
Negative Sobolev norm of non-zero mean non-periodic function on bounded space
The usual formulation of $H^{-1}$ norm for a zero-mean periodic function on some domain $\Omega\in\mathbb{R}$ is as follows:
$\|f\|^2_{H^{-1}}=\sum\limits_{k\in Z, k\neq 0}\dfrac{\hat{f}^2_k}{k^2}$, ...
- 318
0
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0
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116
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The functional $L(\varphi)=\int_0^{2\pi}\frac{\sqrt{1-\varphi^2-(\varphi')^2}}{1-\varphi^2}d\theta$
Consider the $2\pi$-periodic inner product space $L^2[0,2\pi]$.
Let $F\triangleq\{f\in L^2[0,2\pi]|f(\theta)>0,(f(\theta),\cos\theta)=(f(\theta),\sin\theta)=0\}$.
Let $G\triangleq\{\varphi\in L^2[...
- 139
0
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0
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182
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Establishing an upper bound for a dyadic average of a function in $ {L^{p}}([0,1)) $
Suppose that $ f $ is $ 1 $-periodic and that $ f \in {L^{p}}([0,1)) $, where $ p > 1 $. Let
$$
(D_{n})_{n \in \mathbb{N}_{0}} =
\left( \left\{
I^{n}_{j} \stackrel{\text{df}}{=} \left[ \frac{j}{2^{...
- 1
0
votes
0
answers
533
views
Constructing an $\epsilon$-net for a Lipschitz subspace of $L^2$
Let $X$ be a subset of $L^2([0,1])$ which contains only Lipschitz function.
Also, the member of $X$ are uniformly bounded
$$
|x(t)| < M, \text{ for all $x \in X$ and $t \in [0,1]$}.
$$
Let $F: X \...
user24451
0
votes
0
answers
100
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Two distribution spaces ${\mathcal S}'/{\mathcal P}$ and ${\mathcal S}_\infty'$
Let ${\mathcal S}'$ be the set of all distributions.
Denote by ${\mathcal P}$ the set of all polynomials,
which is embedded into ${\mathcal S}'$ as a closed subspace.
Equip ${\mathcal S'}/{\mathcal P}$...
