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10 votes
1 answer
833 views

This is not a dyadic cosine-product

The double-angle formula, $\sin2x=2\sin x\cos x$, turns the scary-looking integral $$\int_0^{\infty}dz\prod_{k=1}^{\infty}\cos\frac{z}{2^k}$$ into fun once you realize $\prod_k\cos\frac{z}{2^k}=\frac{\...
T. Amdeberhan's user avatar
8 votes
2 answers
1k views

Is the Fourier transform of $e^{-|x|^n}$ positive?

Let $$\Phi(x) = \int_{\mathbf{R}^n} e^{-|y|^n +i (x,y)} dy.$$ Is $\Phi$ positive everywhere in $\mathbf{R}^n$? Could someone helps me answer this question or gives a reference for it? Thanks.
nguyen0610's user avatar
42 votes
7 answers
5k views

How should an analytic number theorist look at Bessel functions?

(And a related question: Where should an analytic number theorist learn about Bessel functions?) Bessel functions occur quite frequently in analytic number theory. One example, Corollary 4.7 of ...
Frank Thorne's user avatar
  • 7,347
22 votes
2 answers
2k views

When are Fourier coefficients monotonic?

Given some sufficiently smooth function $f$ what conditions would be sufficient for its Fourier coefficients, as defined by $$ \hat{f}(n) := \int_{0}^{2\pi}\cos(nx)f(x)\ dx, \quad \text{for } n = 1,2,\...
spaceman's user avatar
  • 595
20 votes
1 answer
1k views

Fourier transform of $f_a(x)= a^{-2}\exp(-|x|^a)$, $a \in (0,2)$, is decreasing in $a$

Can one show that Fourier transform of $$ f_a(x) = a^{-2} \exp(-|x|^a), \qquad a \in (0,2)$$ is decreasing in $a$? I have a solution for $a \in (0,1]$ which cannot be used for $a\in (1,2)$.
Tanya Vladi's user avatar
17 votes
2 answers
5k views

Positive-Definite Functions and Fourier Transforms

Bochner's theorem states that a positive definite function is the Fourier transform of a finite Borel measure. As well, an easy converse of this is that a Fourier transform must be positive definite. ...
Alex R.'s user avatar
  • 4,952
16 votes
4 answers
11k views

Fourier transform of Analytic Functions

Forgive me if this question does not meet the bar for this forum. But i would really appreciated some help. I'm trying to construct a function according to some conditions in the frequency domain of ...
jonalm's user avatar
  • 317
7 votes
2 answers
455 views

On a monotonicity property of Fourier coefficients of truncated power functions

Is it true that $$a_{k,n}:=\int_0^{2\pi}x^k\cos(nx)\,dx$$ is nonincreasing in natural $n$ for each $k\in\{0,1,\dots\}$? This question is related to this previous one. Twice integrating by parts, one ...
Iosif Pinelis's user avatar
5 votes
2 answers
484 views

Optimizing a smoothing function with the Prime Number Theorem in mind

Let $f:[0,\infty)\to \mathbb{R}$ be a function with $f(x)=1$ for $0\leq x\leq 1$. Write $Mf$ for the Mellin transform of $f$. Let $c>0$, $T>10^6$ be constants. We are interested in minimizing ...
H A Helfgott's user avatar
  • 20.2k
27 votes
2 answers
5k views

What can be said about the Fourier transforms of characteristic functions?

What can be said about the Fourier transform of the characteristic function $1_A$, where $A\subset \mathbb{R}^n$ is of finite Lebesgue measure? In particular, What properties are common to ...
Joni Teräväinen's user avatar
25 votes
5 answers
6k views

When I can safely assume that a function is a Laplace transform of other function?

If I have a function and I want to represent it as being the Laplace transform of another, that is, I want to be sure that there is $\hat{f}(s)$ such that my function $f(x)$ can be written as: $$f(x) =...
Rorsa's user avatar
  • 923
24 votes
3 answers
3k views

Can Hölder's Inequality be strengthened for smooth functions?

Is there an $\epsilon>0$ so that for every nonnegative integrable function $f$ on the reals, $$\frac{\| f \ast f \|_\infty \| f \ast f \|_1}{\|f \ast f \|_2^2} > 1+\epsilon?$$ Of course, we ...
Kevin O'Bryant's user avatar
10 votes
6 answers
6k views

Fourier transform of (real) exponential

Is it possible to make sense, in distributional sense, of the Fourier transform of the exponential function (defined over the whole real line)?
johny's user avatar
  • 101
10 votes
1 answer
1k views

Basic examples of induction on scales arguments

An important ingredient in recent progress on Euclidean harmonic analysis has been that of "inductions on scales". A few examples are the papers of Wolff, Tao, and Bourgain and Guth. Here is a (vague)...
Mark Lewko's user avatar
8 votes
2 answers
3k views

$L^p$-norm of Fourier series in terms of coefficients, $p \neq 2$

It is known that the $L^2$-norm of a Fourier series equals the $l^2$-norm of the coefficients. Are there similar results in the case of $L^p$-norm for $p\neq 2$? Can it be expressed explicitly in ...
Housen's user avatar
  • 176
7 votes
3 answers
648 views

Simple closed curves and the coefficent of $\exp(i\theta)$ in the associated Fourier series

Given a continuous map $f:S^1\to \mathbb{C}$ from the unit circle to the complex numbers, one can form its Fourier series $\sum_{n=-\infty}^\infty a_n\exp(in\theta)$. I want to stick with those $f$ ...
David Epstein's user avatar
7 votes
1 answer
6k views

Can the Poisson summation formula break?

The Poisson summation formula states if $f: \mathbb{R} \to \mathbb{R}$ then $\displaystyle \sum_{n \in \mathbb{Z}} f(n) = \sum_{n \in \mathbb{Z}} \hat{f}(n) $ where $$\hat{f}(\xi) = \int_{\mathbb{R}}...
john mangual's user avatar
  • 22.8k
6 votes
2 answers
5k views

Can I relate the L1 norm of a function to its Fourier expansion?

I would like to express the integral of the absolute value of a real-valued function $f$ (over a finite interval) in terms of the Fourier coefficients of $f$. Failing that, I would like to know of any ...
Gregory Putzel's user avatar
6 votes
6 answers
3k views

The maximum of a real trigonometric polynomial

Given the coefficients $a_0,\ldots,a_N$, $b_1,\ldots,b_N$ of a real trigonometric polynomial: $ f(x) = a_0 + \sum_{n=1}^N a_n \cos(nx) + \sum_{n=1}^N b_n \sin(nx) $ is there any efficient way to ...
Vincenzo's user avatar
  • 531
6 votes
1 answer
1k views

How important is the Atangana-Baleanu fractional derivative, the main recent development in fractional calculus?

In 2016 a new definition of a fractional derivative was announced in this paper, which has since had more than 100 citations. This derivative, the Atangana-Baleanu derivative, is the main recent ...
willhart's user avatar
  • 113
5 votes
2 answers
202 views

Monotonicity of a parametric integral

For real $x>0$, let $$f(x):=\frac1{\sqrt x}\,\int_0^\infty\frac{1-\exp\{-x\, (1-\cos t)\}}{t^2}\,dt.$$ How to prove that $f$ is increasing on $(0,\infty)$? Here is the graph $\{(x,f(x))\colon0<...
Iosif Pinelis's user avatar
4 votes
0 answers
242 views

Fefferman's article: Pointwise convergence of Fourier series, II

I have some problems reading Pointwise convergence of Fourier series by Fefferman https://www.jstor.org/stable/1970917 I got stuck in Chapter 6, Lemma 5. In the proof he split the $\mathcal P'$ into ...
Thomas Yang's user avatar
2 votes
1 answer
433 views

bounding the absolute value of a trigonometric polynomial

Consider a function $f:[0,1]\rightarrow \mathbb{C}$ and points $t_0,t_1,\ldots,t_n\in[0,1]$ \begin{equation*} f(t)=\prod_{k=1}^n\frac{(e^{2\pi i t}-e^{2\pi i t_k})}{(e^{2\pi i t_0}-e^{2\pi i t_k})} \...
mohi's user avatar
  • 859
1 vote
1 answer
112 views

A bilinear estimate with a simple one-dimensional oscillatory integral kernel

Let $f\in L^{p}(\mathbb{R})$, $1\leq p\leq 2$. I am trying to show that $$\int_{\mathbb{R}}\int_{\mathbb{R}} \,K(y,z)\, \frac{f(y)f(z)}{y^{\frac{1}{2\,p^{\prime}}}\,z^{\frac{1}{2\,p^{\prime}}}}\,dy\,...
Medo's user avatar
  • 852
1 vote
1 answer
307 views

Convexity of discrete Fourier transform

Let $f : [0,2\pi] \to \mathbb{R}$ be a continuous convex function on $(0,2\pi)$ which is singular about $0$ and $2\pi$ but finite when evaluated at the boundaries. Assume also that $f$ is symmetric ...
spaceman's user avatar
  • 595
1 vote
2 answers
140 views

Extending a discrete singular kernel

Let $\{\phi(n)\}_{n\in\mathbb Z}$ be a sequence of complex numbers with the following properties: $\phi(0)=0$ and $|\phi(n)|\leq \frac{C_1}{|n|}$ for all $n\neq 0$ and $C_1>0$ is independent of $n....
A beginner mathmatician's user avatar
0 votes
1 answer
111 views

Averaged Parseval Relation for Sampling a Function on Integers

This was asked a long time ago on math.stackexchange with no answers. Let $f:\mathbb{Z}\rightarrow \mathbb{C}.$ Assume that the support of $f$ is finite, say it is $[1,N],$ and that $\mid f\mid$ is ...
kodlu's user avatar
  • 10.4k