2
votes
2answers
64 views

Convergence for all $\theta$ of a sum with periodic function

How can I show that: $$ \sum_{n \geq 1} \dfrac{\sin(n\theta)}{n} $$ converges for all $\theta \in \mathbb{R}$?
3
votes
1answer
57 views

Will Fourier Series converge even if you only use Prime Integer frequencies?

So there is a Fourier Series for a function $f(x)$ with period $P$: $$ f(x) = \frac{A_0}{2} + \sum_{n=1}^{N} A_n \cdot \cos \left(\frac{n 2 \pi x}{P} + \phi_n \right) $$ Let $\frac{2 \pi x}{P} = t$ ...
1
vote
1answer
77 views

How to find the Fourier series of $f(x)=x$?

I got a question which is too simple: Find the Fourier series of $f(x)=x$ in the interval $[-\pi,\pi]$ and show that this function doesn't converge to its Fourier series. I found the series as ...
1
vote
1answer
42 views

Degree of smoothness of real functions and Fourier series

I was wondering, why is the degree of smoothness $S$ of functions an integer? Why can't there be functions with $S=2/3$ ? The way we determine how smooth a function is by how many continuous ...
2
votes
1answer
77 views

Fourier Series/Parseval's Theorem

I have pretty much completed this question and have found the Fourier representation to be; $$ f(x) =\frac A2 +\sum_{n=0}^\infty 2A\frac{\cos(((2n-1)(\pi x))/2f_o)}{\pi(2n-1)} $$ Now I don't ...
3
votes
0answers
32 views

Difference between almost everywhere convergence of whole Fourier series and a subseries of $L^2$ functions

Let $f \in L^2(\mathbb{R})$ and $F_{N}f$ denote the $N$-th partial sum of its Fourier series. Then $||F_{N}f -f||_{L^{2}} \rightarrow 0$ as $N \rightarrow \infty$. But this implies there exists a ...
1
vote
0answers
34 views

Finding the number of derivatives for series problems

I have the following problem: How smooth are the following functions? That is, how many derivatives can you guarantee them to have? $$a)\;\;\;\;\; ...
0
votes
1answer
106 views

Pointwise convergence of Fourier sine series and uniform convergence of Fourier cosine series.

Let $\overline{f}$ be a function on the whole real line, such that $\overline{f}$ is continuous and differentiable everywhere, and its derivative $\overline{f}'$ is also continuous everywhere. Now, ...
1
vote
1answer
119 views

Convergence of the Fourier serie of $f(x)=e^{2\pi i \alpha x}$

I have some difficulties with the last part of an old exam exercise. For the 1-periodic function $f$ defined on $[0, 1[$ by $f(x)=e^{2 \pi i \alpha x}$ with $0<\alpha <1$. I have found that its ...
2
votes
1answer
336 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 < ...
2
votes
2answers
98 views

Expansion and convergence of $\sum_{m=1}^{\infty}\frac{\sin(2\pi n x^{1/m})}{ m}$

Consider the series: $$\sum_{m=1}^{\infty}\frac{\sin(2\pi n x^{1/m})}{ m}\;\;\;\;n\in\mathbb{N}$$ Other than formal manipulation of the Taylor series of the $\sin$ function, is there a way to expand ...
0
votes
1answer
97 views

Fourier series convergence for sum of Schwartz class functions

Let $f$ be a Schwartz class function. Let $F(x)=\sum_{n\in\mathbb{Z}}f(x-2\pi n)$. Then $F$ is periodic of period $2\pi$. How can we show that the Fourier series of $F$ converges to $F$ pointwise ...
1
vote
1answer
60 views

Convergence of Fourier series for piecewise constant function

Let $f(x)=1$ for $x\in (0,\pi)$ and $f(x)=0$ for $x\in (-\pi,0)$. Furthermore, extend $f$ to be periodic of period $2\pi$. I calculated the Fourier series of $f$ to be ...
1
vote
1answer
72 views

Show absolute and uniform convergence of a Fourier series

Hello and good evening! The Fourier series of $f(x):=\lvert x\rvert$ on $[-\pi,\pi]$ is $$ f(x)=\frac{\pi}{2}-\frac{4}{\pi}\sum_{n=1}^{\infty}\frac{\cos((2n-1)x)}{(2n-1)^2}. $$ I have to examine if ...
2
votes
1answer
106 views

Prove that the Fourier series of $\dfrac{1}{f}$ is absolutely convergent.

I have a problem: Let $f$ be a continuous function on the unit circle $(\Gamma)$: $$\Gamma=\{e^{i\theta}: \theta\in [0, 2 \pi]\}$$ Assume that $f \ne 0$ on $\Gamma$, and the Fourier ...
2
votes
2answers
174 views

Show a Fourier series converges uniformly

I need to show that the Fourier Series of |x| in the interval $(-\pi, \pi)$ converges uniformly to |x| in $[-\pi, \pi]$. I know that |x| = $\frac{\pi}{2}$ + ...
0
votes
1answer
120 views

What does it mean for a function to converge at a point of discontinuity?

In Fourier analysis, if $x$ is a point of discontinuity of $f(x)$, then $f(x)=\frac{f(x^+)+f(x^-)}{2}$. How is this Convergence? Uniform convergence? Pointwise convergence?
1
vote
0answers
177 views

Taylor series of Fourier series of triangle wave

Odd triangle wave $\text{t}(x)$ with angles at $(2x+1)\in\mathbb{Z}$ can be represented by Fourier series: ...
3
votes
1answer
266 views

Convergence of: $\sum\limits_{n=1}^{\infty} \frac{(-1)^{n+1}\sin(nx)}{n}$

Need help with checking: $\sum\limits_{n=1}^{\infty} \frac{(-1)^{n+1}\sin(nx)}{n}$ for point-wise convergence and uniform convergence of: ${-\pi} \leq x \leq {\pi}$.
0
votes
0answers
106 views

finding the fourier coefficients of $f(x) = \sum_{n=1}^{\infty} \frac{\cos(nx)}{n^2}$

This is what i know so far: The given series uniformly converges by the M-test and that i can swap the integration and the sum when calculating the coefficients. Apparently i am supposed to use the ...
4
votes
2answers
274 views

The relationship between Fourier coefficients of function $f$ and its continuity

How to prove that if Fourier series of function $f$ converge uniformly, then function is continuous?
16
votes
1answer
577 views

Series which are not Fourier Series

How to show that $$ \sum_{n=2}^\infty \frac{\sin{(nx)}}{\log n} $$ not the Fourier series of any function? I have shown that the series is convergent by Dirichlet test. Let $a(n)=\frac{1}{\log ...
0
votes
2answers
405 views

When convergence in mean implies uniform convergence?

With Fourier series, I'm confused about Bessel's inequality and Parseval's identity. I understand that Bessel's inequality becomes Parseval's equality if and only if both integrals $ ...
3
votes
0answers
158 views

Uniform convergence of a series

This problem came from the Krantz text ($2^{nd}$ ed. ch. 9, prob. 17): Prove that the series $\displaystyle\sum_{j=1}^{\infty }{\frac{\sin{(jx)}}{j}}$ converges uniformly on compact intervals that do ...