0
votes
2answers
43 views

Multiplying and Dividing Series

For example, how do you compute the taylor series for $$e^x \sin x=\sum_{n=0}^{\infty} \frac {x^n}{n!} \sum_{n=0}^{\infty} (-1)^n\frac {x^{2n+1}}{(2n+1)!}$$ Of course I want the result to contain ...
-1
votes
0answers
18 views

solving a complex fourier series

while solving for co efficient of a complex fourier series i came across this form cos(0.5*n*pi)+i*sin(0.5*n*pi) is there any way to simplify this? Note: N is an integer
0
votes
0answers
12 views

Nyquist Frequency, filter, phase/amplitude

Problem The problem seems quite simple and I believe it is though I have no idea how to approach it. I have tried googling 'Nyquist frequency' but have not had any luck with problems similar to this. ...
3
votes
1answer
35 views

Lower bound on certain exponential sums and expressions related to them

Let $$G(\alpha, x) = \sum_{n\le x}e(\alpha n^2)$$ Clearly, $r_k(n)$, the number of representations of a number as the sum of $k$ squares is given by the following expression: $$r_k(n) = \int_0^1 ...
1
vote
0answers
82 views

Is $\int_0^\infty \sin(Kx)f_K(x)\mathrm dx$ of larger order than $\int_0^\infty \cos(Kx)f_K(x)\mathrm dx$?

Suppose we have a function $f$, such that $f$ is of some smoothness degree $m$, and $f,f^{(k)} \in L_1[0,\infty)$ $k=1,...,m$. Now if $f^{(k)}(0) = \lim_{x\rightarrow\infty}f^{(k)}(x) = 0$ for ...
0
votes
1answer
155 views

Equality of two series.

EDIT: Consider $m,n\in\mathbb{Z}$ with $\frac{m}{n}\in \mathbb{R} -\mathbb{Z}$ and $n>1$. Given integers $m,n$ I want to prove that $$\bigg\lfloor \frac{m}{n} ...
0
votes
0answers
23 views

Discrete Fourier Transform of the infinite series

I am reading this book and having hard time understanding how to get to eq(2) from eq(1) $$P(k,t) = e^{-\alpha t} \sum\limits_{l,m=-\infty}^\infty (-i)^m e^{ik(l+m)} I_l(\alpha t) I_m(i\beta t) ...
3
votes
3answers
213 views

Fourier Series for $|\cos(x)|$

I'm having trouble figuring out the Fourier series of $|\cos(x)|$ from $-\pi$ to $\pi$. I understand its an even function, so all the $b_n$s are $0$ $$a_0 = \frac 2 \pi \int_0^\pi |\cos(x)|\,dx = ...
3
votes
1answer
61 views

Ques from exam: sequence of functions and improper integrals

$P_n(x):R\rightarrow R$ is a sequence of functions defined by: $$P_n(x)= \frac{n}{1+n^2x^2}$$ f:R->C is continuous and 2pi periodic. We define: $$f_n(x)=\frac{1}{\pi}\int ...
2
votes
1answer
151 views

Fourier Series of $f(x) = 0$ from $(-\pi, 0)$, $x$ from $(0,\pi)$

I need to determine the fourier series of the following function, (using trig method, not complex) $$ f(x) = \begin{cases} 0 & \text{if } -\pi < x < 0, \\ x & \text{if } 0 < x < ...
3
votes
2answers
59 views

Find a sequence of positive functions with non-trivial properties in $L^1([-\pi,\pi])$ and in $L^2([-\pi,\pi])$

I was asked to exhibit a sequence of positive functions $\{f_n\}_{n\in\mathbb{N}}$ belonging to $L^2([-\pi,\pi])$ such that: $\{f_n\}_{n\in\mathbb{N}}$ is strongly converging to $0$ in ...
1
vote
1answer
54 views

Fourier Series Approximations of Functions

From a few examples of smooth functions, discontinuous functions and continuous functions which have a 'kink' (i.e. $|x|$ where left and right limits disagree)... I've seen that the fourier series ...
2
votes
1answer
96 views

Solve Basel problem with Fourier series of $[-\pi,\pi]\to \mathbb{R}:\theta \mapsto |\theta|$

A problem from Stein/Shakarchi's Fourier Analysis: Consider the function $f:[-\pi, \pi] \to \mathbb{R}:\theta \mapsto |\theta|$. Show $$\hat{f}(n)=\begin{cases} \pi/2& n=0 ...
1
vote
0answers
74 views

DFT example in textbook

There is an example of the use of the DFT formula in my textbook which I don't quite follow. The text goes as follows: Let us define the $N$-periodic and anti-Hermitian series $g_n$ where $g_n = f_n ...
3
votes
1answer
48 views

Show that for $0<t<1$, $\log\sin (t\pi)=\log(t\pi) + \sum_{n=1}^\infty\log\left(1-\frac{t^2}{n^2}\right)$

Show that for $0<t<1$, $$\log\sin (t\pi)=\log(t\pi) + \sum_{n=1}^\infty\log\left(1-\frac{t^2}{n^2}\right)$$ So I derived the following Fourier series: ...
9
votes
1answer
176 views

Is $\sum_{n=1}^{\infty}{\frac{\sin(nx)}n}$ continuous?

Considering the infinite series $\sum_{n=1}^{\infty}{\frac{\sin(nx)}n}$ , I can show that it is not convergent uniformly by Cauchy's criterion and that it is convergent for every $x$ by Dirichlet's ...
2
votes
2answers
51 views

$C$ such that $\sum_{k\in \mathbb{Z}^n} k_i^2k_j^2|a_{ij}|^2 \leq C \sum_{k\in \mathbb{Z}^n} \|k\|^4|a_{ij}|^2$

More generally, can we find $C_n>0$ such that $$\sum_{k\in \mathbb{Z}^n} k_i^2k_j^2|a_{ij}|^2 \leq C \sum_{k\in \mathbb{Z}^n} \|k\|^2|a_{ij}|^4$$ for all $\{a_k\}_{k\in \mathbb{Z}^n} \in ...
10
votes
1answer
195 views

Limit of maximum of $f_{n}(x)=\dfrac{1}{n}(\sin{x}+\sin{(2x)}+\sin{(3x)}+\cdots+\sin{(nx)})$

let $$f_{n}(x)=\dfrac{1}{n}(\sin{x}+\sin{(2x)}+\sin{(3x)}+\cdots+\sin{(nx)}),x\in R,n\in N$$ let $$a_{n}=\max_{x\in R}{(f_{n}(x))}$$ Find this limit $$\lim_{n\to\infty}a_{n}$$ My try: since ...
2
votes
1answer
41 views

Function with Fourier coefficients extend onto closed disk

Let $f$ be a continuous function with period $2\pi$. Define $$u(r,\theta)=\sum_{n=-\infty}^\infty r^{|n|}\hat{f}(n)e^{in\theta}$$ for $r\in[0,1)$, where $\hat{f}(n)$ is the $n$th Fourier ...
2
votes
1answer
51 views

Sum of exponentials with Fourier coefficient

Let $f$ be a continuous function with period $2\pi$. Define $$u(r,\theta)=\sum_{n=-\infty}^\infty r^{|n|}\hat{f}(n)e^{in\theta}$$ for $r\in[0,1)$, where $\hat{f}(n)$ is the $n$th Fourier coefficient ...
0
votes
1answer
38 views

Is there a countable Fourier transform for infinite sequences?

There's the discrete Fourier transform and the continuous one, but where's the one for infinite sequences. Let $(a_i) \subset \mathbb{C}$ be a sequence of complex numbers. The naive ways of defining ...
6
votes
1answer
79 views

percentage of numbers starting with $2$ in $\{2^n\}$

I have once heard a professor telling (during a course on Fourier theory) that there is a way to determine the numbers starting with a $2$ in the sequence $\{2^n\colon n\in\mathbb{N}\}$. I asked him ...
0
votes
0answers
45 views

splitting up summations of product

I want to check the validity of splitting up summations of products. I am using the DFT matrix and am trying to get a simplified expression of it . In essence, I am trying to prove the following lemma ...
2
votes
2answers
78 views

Absolute sum of Fourier coefficients

Let $f$ be a smooth and rapidly decaying function. Is it true that $\sum_n|\hat f(n)|\leq C \sum_n |f(n)|$ for some constant independent of $f$? Thanks! Can we say that $\sum_n|\hat f(n)|\leq C ...
14
votes
2answers
374 views

Closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$.

What is the closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$? We can use Fourier series of $e^{-bx}$ ($|x|<\pi$) to evaluate $\sum_{n=-\infty}^{\infty}\frac{1}{n^2+b^2}$. But this ...
0
votes
1answer
66 views

Wave-Function Series?

So I was basically exploring the function: $\displaystyle {\text{frac}(x)}$ which is the fractional part function and I noticed that it has a nice fourier series definition which is: ...
2
votes
3answers
187 views

Calculating the Fourier series of $x^{3}$

I was given as homework to calculate the Fourier series of $x^{3}$. I know, in general, how to obtain the coefficients of the series using integration with $$\sin(nx),\cos(nx)$$ multiplied by the ...
2
votes
2answers
153 views

A integral with polygamma

I was doing a integral, the last part is $$\int_0^{\frac{\pi}{2}}x^3\csc x\text{d}x$$ I ran this on Maple, it turns into polygammas...How we evaluate this? I think there should be a way to evaluate ...
1
vote
1answer
48 views

Need help with a integral

I was evaluating $$\int_0^{\frac{\pi}{2}}x\ln \cos x \, \text{d}x$$ I like to try with the fourier series $$\int_0^{\frac{\pi}{2}}\left(\sum_{k=1}^\infty\frac{(-1)^{k-1}\cos (2kx)}{k}x-x\ln 2\right) ...
12
votes
3answers
500 views

A log improper integral

Evaluate : $$\int_0^{\frac{\pi}{2}}\ln ^2\left(\cos ^2x\right)\text{d}x$$ I found it can be simplified to $$\int_0^{\frac{\pi}{2}}4\ln ^2\left(\cos x\right)\text{d}x$$ I found the exact value in the ...
2
votes
1answer
112 views

More on the generalized integral

Refer to my previous topic: A generalized integral need help I think we get this : $$\frac{\sin \theta}{1-2\cos \theta x+x^2}=\sum_{k=1}^{\infty}\sin (k\theta )x^{k-1}$$ Then $$\int_0^1\frac{\ln ...
2
votes
0answers
119 views

Do we have closed form for these series?

Continuous on the previous question, can we get a closed form for these? $$\begin{align} & \sum\limits_{n=1}^{\infty }{\frac{n+3}{{{n}^{3}}+\ln n}} \\ & \sum\limits_{n=1}^{\infty }{\left( ...
0
votes
1answer
102 views

A small question on fourier series

Why the series is divergent, but the equation holds? $$\sum\limits_{i=1}^{\infty }{\sin kx}=\frac{1}{2}\cot \left( \frac{x}{2} \right)$$
1
vote
0answers
64 views

Probability measures on $\mathbb{T}$ whose Fourier coefficients tend to 1

Let $\mu$ be a probability measure on the complex unit circle $\mathbb{T}$. Are the following two assertions equivalent? $\limsup_{n\to\infty}|\hat{\mu}(n)|=1$. There exists an increasing sequence ...
0
votes
1answer
84 views

Sum of series & sequences

I dont know how to evaluate the first one, for second one I can only show the sum is less than 2. $$\begin{align} & \prod\limits_{k=4}^{\infty }{\left( 1-{{\left( \frac{3}{k} \right)}^{3}} ...
5
votes
2answers
373 views

log sin and log cos integral, maybe relate to fourier series

I try to use the method of differentiation under integral sign for the first one And integrate it back, but I failed to find the constant $c$ .... Anyone hav other method? $$\begin{align} & ...
0
votes
1answer
179 views

How to find the coefficient of this Fourier sine series?

From $$1=\sum_{k\geq 1} a_k \sin((k\pi+\frac{\pi}{2})x),$$ I want to find $a_k.$ My unsuccessful approach is first multiplying both side by $\cos((k\pi+\frac{\pi}{2})x)$. That is, ...
1
vote
1answer
78 views

The Continuity of the Discrete Time Fourier Transform of Absolutely Summable Series

I saw on a book to following claim: Given an Absolutely Summable Series $ \sum_{n = -\infty }^{\infty}\left | x\left [ n \right ] \right | \leqslant \infty $, Namely, $ l_1 $ series it is possible to ...
1
vote
1answer
50 views

Show from every positive sequence decreasing to zero, we can build a sequence satisfying a concavity condition

Let $\{x_n\}_{n=-\infty}^{\infty}$ be a positive sequence decreasing to zero as $|n| \to \infty$. Show there is a sequence $\{y_n\}$ satisfying \begin{align} y_n >& x_n \tag{1}\\ ...
0
votes
0answers
83 views

Fourier series of function defined by trigonometric series.

I'm dealing with the problem, that the function defined by trigonometric series(that is, limit of symmetric sum of $e^{inx}$) I have shown that this function converges everywhere in pointwise sense. ...
0
votes
1answer
201 views

Show $\left\lvert\sum_{k=-n}^n \frac{\sin k t }{k}\right\rvert \le \pi + 2$ for all $n$ and $t$, $t \in [0,2\pi]$

Let $f(t)=(t-\pi)\chi_{(0,2\pi)}$, $t \in [0,2\pi]$, then the partial sum of the Fourier series of $f$ is $$ S_n(t)=- \sum_{0 < |k| \le n} \frac{\sin k t}{k}. $$ Show $|S_n(t)| \le \pi+2$ for all ...
3
votes
1answer
109 views

Convergence of the Real analysis

The question is find the Fourier series of "|cost| for all t". I already found the fourier series But now the question asks " At which values of $x$, does the series fail to converge to ? To what ...
6
votes
1answer
200 views

To show Taylor series of a Fourier transform $\hat{f }$ converges to $\hat{f}$

I got some trouble with the following question. Say $f$ is in $L^1(R)$ with compact support . I need to show (1) $\hat{f(\zeta)}$ is infinitely differentiable and all derivatives are continuous. ...
3
votes
1answer
73 views

Change domain on series - Counting aplitudes

If I have a function $$f(t) = y$$ where $t$ & $y$ are positive Integers for $t = \{1,2,3,4,5,6,7,8\} \to y = \{1,1,1,2,1,2,3,1\}$ How can I create a function $g(y)$ such that it counts the ...
0
votes
1answer
63 views

Help me to understand the Gaussian blurring (2)

Here is an unknown luminosity function $f(x,y)$ and its integration results: $$\begin{align*} p_{i,j} &= \frac{1}{\Delta_{i,j}}\iint\limits_{D_{i,j}} \! f(x,y) \, dx \, dy,\\ \Delta_{i,j} &= ...
10
votes
1answer
376 views

For a trigonometric polynomial $P$, can $\lim \limits_{n \to \infty} P(n^2) = 0$ without $P(n^2) = 0$?

Disclaimer: The original version of this question focused on $2^n$ in lieu of $n^2$. It is in the hope that the question is easier with $n^2$ that I changed it. I have an always-nonnegative (on the ...
1
vote
1answer
37 views

Some doubt on series.

If i have a series of the form ( say fourier or anything ) , For example lets consider $$\sum_{k\in \mathbb Z} \exp\left( \frac {-ik\pi}{L}\right) a_k(t) ,$$ is it always possible to split it down to ...
2
votes
1answer
147 views

About $\sum_{k=1}^\infty \frac{b_k}{k}$, where $b_k$ are Fourier coefficients

This is my first post here. I have some troubles with this property of the Fourier coefficients. Indeed, let $f(x)$ be a continuous real function, with compact support $[a,b] \subset (0,2\pi)$, and ...
0
votes
2answers
132 views

Summation of a complex series

Is there a way to perform the finite sum $\sum_{m = 1}^n \exp(2 \pi i k (\sqrt5) ^m)$?, m even. I am trying to show a specific sequence is not equidistributed, and so I'd like to show that Weyl's ...
8
votes
1answer
4k views

Criteria for swapping integration and summation order

I have a function (a potential from an electrostatic potential via a Fourier series) in the form of $$V(x, y, z)=\sum_n\sum_m \ a(x, n, m) b(y, n) c(z, m) \int\int f(u, v) d(u,n) e(v,m) du\, dv$$ ...