All Questions
Tagged with fourier-transform real-analysis
66 questions
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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
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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
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2
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333
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What gives a "Parseval like" equation mixing cosine and sine Fourier transforms ?
Noting $\mathcal{F}^c$ the cosine transform and $\mathcal{F}^s$ the sine transform defined on real functions by:
$$\mathcal{F}^c [f (x)]=\int_0^{\infty} f(t) \cos(xt) dt $$
$$\mathcal{F}^s [f (x)]=\...
- 1,199
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Function and Fourier transform vanish on an interval
I'm no expert on these things (and this may not be cutting edge research level; it's really motivated by this MSE question), but it seems that there are non-zero measures (and also functions (?), I ...
- 24.2k
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Inequality about the Fourier transform: $\Vert u \Vert_{L^k} \le \Vert \mathcal{F}(u) \Vert_{L^m}$ (where $1 \le m \le 2$ and $m,k$ Holder conjugates)
How can I prove the following inequality about the Fourier transform?
$$\Vert u \Vert_{L^k(\mathbb{R}^N)} \le \Vert \mathcal{F}(u) \Vert_{L^m(\mathbb{R}^N)}$$ for $1 \le m \le 2$ and $m,k$ Holder ...
user89890
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Is flatness of Wigner Ville Distribution of error function in Fourier Approximation possible? Is it required?
For a real valued function $f(t)$ I want to check the information left, after taking a Fourier partial sum/integral. Let $\hat{f}$ be its Fourier transform and let $$e_{\omega}(t) = f(t) - \int\...
- 698
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1
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460
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Fourier transform either changes sign infinitely often far out or is continuous at $x=0$
I am reading a book "Fourier Series and Integrals" by Dym & McKean.
There is an exercise (Page 106):
Exercise: Check that if $f$ is a real, even, summable function and
if $f(0+)$ and $f(0-)$...
- 371
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Fourier transform surjective on $L^p(\mathbb{R}^n)$ for $p \in (1,2)$?
I know that $F_2:L^2 \rightarrow L^2$ is of course unitary, whereas $F_1:L^1 \rightarrow C_0$ is injective but not surjective. This can be seen by looking at the dual map.
Riesz-Thorin gives us that ...
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Is there a $C_c^{\infty}( \mathbb{R}^d)$ function whose Fourier transform we can explicitly write down?
I noticed that although $C_c^{\infty}$-functions are dense in some quite large spaces and well understood (especially their Fourier transform) I have never encountered an explicit example of a ...
- 181
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1
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625
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What are the spaces for which the Fourier transform is an automorphism? [closed]
this is well-known that the Fourier transform is an automorphism of $L^2(\mathbb R)$ and also of $\mathcal S(\mathbb R)$ (Schwartz space). Is there any other spaces of functions of one real variable ...
- 615
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808
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Inverse Fourier transform of $\frac{1}{\sqrt{\xi_1} + \xi_2}$
Consider the inverse Fourier transform of $\frac{1}{\sqrt{\xi_1} + \xi_2}$. My question is, how can we conclude about the decay properties, support and smoothness of the inverse Fourier transform? I ...
2
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1
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460
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Finite trigonometric polynomial
I noticed by numerical and some explicit calculations for a few examples that for real-valued finitely supported functions $\phi \in L^2(\mathbb{R})$ we have that
$T(x):= \sum_{n \in \mathbb{Z}} |\...
- 231
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A question about pointwise convergence of Fourier transform in $N$-dimensions
I am retreating back on this statement, after some explorations and calculation
Bow to Willie and others who were skeptical on this. Main difficulty can be seen in this reference. But I must mention ...
- 698
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2
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Is this statement which relates the Fourier transform of a function to its singularities correct?
I am working on a problem, which would possibly relate the Fourier transform/series with the jump singularities of the function where the function itself or one of its derivatives jump. ((some kind of ...
- 698
2
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814
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Quantifying the “flatness” of functions which are the Fourier transforms of positive functions
Short version of question: I'm trying to understand the extent to which a function is prevented from being "flat" as a result of being the Fourier transform of a positive function. That is, the extent ...
- 21
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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 ${\mathcal S'}/{\mathcal P}$...
