29
votes
1answer
581 views

For which maps $S^1\to S^1$ is the winding number defined?

There are two classes of maps $S^1\to S^1$ for which I know how to define the winding number: • Continuous maps: Using the unique path lifting property of the universal covering map $\mathbb R\to ...
0
votes
0answers
53 views

Under condition of Zygmund is the following inequality true?

Let $f:R\rightarrow R$ be a continuous function and satisfies the following Zygmund condition $$ |f(x+\delta)+f(x-\delta)-2f(x)|\leq \text{const} \frac{|\delta|}{(\log\frac{1}{|\delta|})^{\alpha}}, ...
5
votes
1answer
192 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 ...
2
votes
2answers
100 views

Seeking a class of functions for which sums approximate integrals well

Is there a "natural" class of integrable functions $f: {\mathbb R} \rightarrow {\mathbb R}$ for which it is true (and, preferably, not too hard to prove!) that $\sup_{0 \leq a < h} |h S(a,h) - I|$ ...
4
votes
0answers
85 views

Error of midpoint method for differentiable functions

Is it the case that for every differentiable function $f$ on $[0,1]$ (with finite one-sided derivatives at the endpoints), the midpoint method of estimating $\int_0^1 f(x) \: dx$ has error $o(1/n)$? ...
0
votes
0answers
45 views

How to choose negative definite function $\lambda (x)$, so that $\lambda^{-1} \in L^{1}(\mathbb R)$?

We define, $$\lambda (x) =\int_{0}^{\infty} \frac{\sin^{2}x \alpha}{\alpha^{2}} d\mu (\alpha), (\mu(0)=0) $$ where $\mu (\alpha)$ is a non-decreasing function such that the integral converges for ...
1
vote
1answer
81 views

A question which belongs to a class of Zygmund functions

Let $f$ be an absolutely continuous, periodic with period 1 and satisfies the condition $$ |f(x+\delta)+f(x-\delta)-2f(x)|\leq \text{const}\frac{\delta}{(\log\frac{1}{\delta})^{\epsilon}}, ...
1
vote
1answer
86 views

Compactly supported smooth function with Laplace transform bounded on a cone

My question is if it is possible to find a compactly supported smooth function $\varphi:\mathbf{R}\to \mathbf{R}$ s.t. the following integration $\int_{\mathbf{R}}\varphi(t)e^{itx}e^{tx}dt$ stays ...
2
votes
0answers
85 views

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 ...
2
votes
0answers
169 views

Is $f$ an absolutely continuous function? [closed]

Let $$ f(x)=\sum_{n=1}^{\infty}\frac{\sin(2^{n}\pi x)}{n\cdot2^{n}}, \,\,\,\,\,\, x\in [-1, 1]. $$ Is $f$ an absolutely continuous function? If yes how can I show it? If not how about on total ...
2
votes
0answers
78 views

Is Modulation space is close under absolute?

It is clear that, for $1\leq p \leq \infty$, $f\in L^{p}$ iff $| f| \in L^{p}.$ By the usual modulation space $M^{p, q}(\mathbb R^{d})$, $1\leq p, q \leq \infty, d\in \mathbb N $, we mean the ...
0
votes
0answers
84 views

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 ...
2
votes
0answers
187 views

Obtaining a pointwise bound on the convolution of two singular measures

I am confused about a passage in the paper by T. Tao A sharp bilinear restriction estimate for paraboloids. We are in Section 7, near equation (34) (pag.16 of the arxiv). Notations and ...
2
votes
1answer
206 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})} ...
0
votes
2answers
684 views

Derivative indicator function

I am wondering what is the derivative of the following function with respect to $x(t)$ in sense of distributions. $$ I\left(\int_0^t x(\tau)d\tau \leq c\right) $$ where $I$ is the indicator function ...
5
votes
1answer
313 views

Weakest assumption for pointwise convergence of Fourier series

This should be a quick one, but so far books, my brain, and the internet have not produced a clear answer. Or maybe it's subtle and exposes a weakness in my understanding of FS! Suppose ...
5
votes
2answers
651 views

Property/Relations using Fourier series/transform, which give complete information about all the jump singularities of a function.

Consider a function which has only jump singularities of the form of the function itself or one of its derivatives jumping. Now let $\hat{f}(k)$ be its Fourier transform/series. We know the decay of ...
2
votes
2answers
371 views

Does the Fourier series of an $L^1$ function converge to the function *weakly* in $L^1$?

Let $f$ be a periodic $L^1$ function, and $S_n[f]$ the $n$-th partial sum of its Fourier series. I am aware that $S_n[f]$ might not converge toward $f$ in $L^1$ (i.e., in norm). However, does it at ...
0
votes
1answer
134 views

On a limit at the boundary of $\mathbb{D}$ related to complex and harmonic analysis

Let $p(z,t)=\frac{1}{2\pi}.\frac{1-|z|^2}{|z-t|^2}$ be the Poisson kernel on the open unit disk $\mathbb{D}$, fix $0<\alpha<1$ . Let $a\in \partial\mathbb{D}=S^1$ be fixed. Then my question is : ...
0
votes
1answer
755 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 ...
3
votes
1answer
346 views

Ask for theory about the weighted L^2(R^d) space.

Dear MOs, I am now considering the following norm: $$ ||f||_{H}^2 := \iint f(x) H(x,y) f(y) d x d y\:. $$ where the integral is over the whole space $R^{2d}$ and $H(x,y)$ is some non-negative ...
2
votes
0answers
76 views

Searching for inequalities relating a convolution-type integral of functions of modulus less than but close to one.

Suppose $f(x,y)$ and $g(x,y)$ are both measurable functions from $[0,1]\times[0,1]\to \mathbb{C}$ with $|f|,|g|<1$, and let $h(x,y)=\int_{0}^1 f(x,z)g(z,y) \ dz$. (So $|h(x,y)|<1$ also.) ...
2
votes
0answers
197 views

Continuous function sort

If you have a real-valued function f(x), positive, continuous and bounded on some interval, then what kind of transform would convert this to a monotonic function g(x) on that interval analogously to ...
8
votes
2answers
680 views

Fourier transform of x2 invariant measure

Let $T:\mathbb{R}/\mathbb{Z}\rightarrow \mathbb{R}/\mathbb{Z}$ be the map defined by $T(x)=2x$, and suppose that $\mu$ is a $T$ invariant and ergodic Borel probability measure on the space, which is ...
4
votes
2answers
481 views

$L^1$ norm of the Fourier transform of a truncated Gaussian

I asked this question on Math StackExchange recently but the only useful comment I got was that this could be a good question for Math Overflow. Here it goes: Consider the Gaussian $G(x):=e^{-x^2}$ ...
0
votes
1answer
534 views

Lebesgue's Majorized Convergence Theorem

Can anyone point me to an explanation and a proof of this theorem? For reference, it is mentioned in Kolmogorov's almost everywhere divergent function in $L$ as given in Zygmund, volume I. In the ...
21
votes
9answers
10k views

Is square of Delta function defined somewhere?

Hello, every one. I am wondering whether any one knows that whether the square of Dirac Delta function is defined some where? In the beginning, this question might look strange. But by restricting ...
15
votes
1answer
2k views

Convergence of Fourier Series of $L^1$ Functions

I recently learned of the result by Carleson and Hunt (1968) which states that if $f \in L^p$ for $p > 1$, then the Fourier series of $f$ converges to $f$ pointwise-a.e. Also, Wikipedia informs me ...
2
votes
0answers
284 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 ...