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4 votes
1 answer
144 views

Asymptotic decay rate of an oscillator integral

Question: I want to evaluate the decay estimate of the integral $I^d(t; v) = \int_0^{\sqrt{d}\pi} dr \, r^{d-2} \int_0^\pi \sin(tr) e^{i\sqrt{d}vtr\cos\theta} \sin^{d-2}\theta \, d\theta $ for ...
1 vote
1 answer
121 views

An asymptotic integral with complex phase

Suppose that $D\subset \mathbb R^2$ is the closed unit disk and that $f\in C^{\infty}(D)$. Assume that for all $\lambda \in (1,\infty)$ there holds $$ \left|\int_D f(x^1,x^2)\, e^{\lambda (x^1+ix^2)}\,...
0 votes
1 answer
102 views

On weighted Fourier transforms

Suppose that $f\in L^{\infty}((0,1))$ and that there exists $c_1,c_2>0$ such that $$ \left|\int_0^1 e^{i \xi x} e^{-|\xi|^{-1}x}f(x)\,dx \right| \leq c_1 e^{-c_2|\xi|} \quad \forall\, |\xi|>1.$$ ...
6 votes
2 answers
336 views

On frequency decay of an integral transform of a function

Suppose $f \in C^{\infty}_c((-1,1))$ and assume that there exists constants $a,b>0$ such that $$ \bigg|\int_{\mathbb R} f(t) \,e^{\tau t^2+i\tau t}\,dt\bigg| \leq a\,e^{-b|\tau|},$$ for all $\tau \...
8 votes
1 answer
638 views

Rate of decrease of the Fourier transform of standard mollifiers

What is the the rate of decrease of $|\widehat{f_p}(t)|$ (as $t\to\infty$), where $p\in(0,\infty)$, $$\widehat{f_p}(t):=\int_{\mathbb R} e^{itx}f_p(x)\,dx,$$ and $$f_p(x):=e^{-1/(1-x^2/p)^p}1(|x|<\...
1 vote
0 answers
76 views

Second question on a real sequence

I am thinking again about the real sequence $\{a_n\}_{n\ge1}$ which decays faster than any algebraic speed, that is, $\lim_{n\to \infty}n^pa_n = 0$ for every integer $p$. Actually, $a_n$ can be ...
2 votes
0 answers
162 views

Hilbert transform on weighted Sobolev spaces

Let $\mathscr H\,f$ denote the Hilbert transform of a function $f \in L^2(\mathbb R)$. We know that $\mathscr H$ is an isometry on $L^2(\mathbb R)$, but I want to know to what is the mapping ...
1 vote
0 answers
107 views

Comparison of two Fourier transforms

I am looking for $\delta>0$, such that $$ \delta \int_{-\infty}^{\infty} \exp(its) { \Gamma\{2(it+1)/3\}\over \Gamma\{(it+1)/2\} }dt \le \\ \int_{-\infty}^{\infty} \exp(its) { \Gamma (it+1)\over \...
1 vote
1 answer
179 views

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 ...
2 votes
0 answers
136 views

To find a positive function with compact spectrum

Let $e_1=(0,1)^T$, $$ S=\left\{x\in \mathbb{R}^2\Big| \frac{|\langle x, e_1\rangle|}{|x|}>\delta>0\right\}, $$ is a cone in $\mathbb{R}^2$. I want to find a non-trivial smooth function ...
2 votes
1 answer
361 views

more on "sinc-ing" integrals and sums

This is a follow up on the MO question here. I kept being fascinated and bemused by these functions. Denote $\text{sinc}(x)=\frac{\sin x}x$. Experiments suggest that $$\sum_{n=1}^{\infty}\text{sinc}^...
7 votes
1 answer
397 views

Fourier expansion of Takagi-function (everywhere non differentiable function).

Let us consider Takagi-function defined by $T(x) \colon\!= \sum_{n=0}^{\infty}s(2^nx)/2^n$, where $s(x) \colon\!\!= \underset{n \in {\Bbb Z}}{\mathrm{min}} \,|x-n|$. $T(x)$ has its period $1$, so ...
1 vote
1 answer
3k views

In Fourier Transforms: Positive Definite Functions, Bochner's Theorem, and Derivatives

I've been reading about Bochner's Theorem lately, but when I apply it to the derivative of a function, I seem to get a contradiction with the theorem. "Bochner's theorem states that a positive ...
2 votes
0 answers
354 views

What is this effect in Fourier/additive synthesis called?

Hi, I have re-synthesized a cyclic function additively, and I added a fixed offset to the frequency of each partial. So if the function was $\sum a_{n} sin(2 \pi x * n)$ and its frequencies were $n*f_{...