Intereting Posts

tangent space at some point of a quasi-projective variety
The only two rational values for cosine and their connection to the Kummer Rings
The longest sequence of numbers with a certain divisibility property
Did Galois show $5^\sqrt{2}$ can't solve a high-order integer polynomial?
Martingale and Submartingale problem
Dimension of the space of algebraic Riemann curvature tensors
Any $p + 1$ consecutive integers contain at least two invertible elements modulo $p!!$ if $p$ is odd
Calculate the sum of infinite series with general term $\dfrac{n^2}{2^n}$.
Product of two negative numbers is positive
what is Prime Gaps relationship with number 6?
Monotonic Function Optimization on Convex Constraint Region
Prove that $\dfrac{a}{b^2+5}+ \dfrac{b}{c^2+5} + \dfrac{c}{a^2+5} \le \dfrac 12$
Proving equivalence relations
Asymptotics of $\prod_{x=1}^{\lceil\frac{n}{\log_2{n} }\rceil} \left(\frac{1}{\sqrt{n}} + x\left(\frac{1}{n}-\frac{2}{n^\frac{3}{2}} \right)\right) $
Proof that ${2p\choose p}\equiv 2\pmod p$

In this excercise we show how the symmetries of a function imply certain properties of its Fourier coefficients. Let $f$ be a 2$\pi$ Riemann integrable function defined on $R$.

(a) Show that the Fourier series of the function $f$ can be written as

$$f(\theta) \thicksim \hat{f}(0) + \sum_{n\ge1}[\hat{f}(n) + \hat{f}(-n)]\ cos n\theta + i[\hat{f}(n) – \hat{f}(-n)]\ sin n\theta. $$

(b)Prove that if $f$ is even, then $ \hat{f}(n)= \hat{f}(-n) $

, and we get a cosine series.

- Proving that the characters of an infinite Abelian group is a basis for the space of functions from the group to $\mathbb{C}$
- A Fourier transform of a continuous $L^1$ function
- Deriving Fourier inversion formula from Fourier series
- Solving the heat equation with Fourier Transformations
- $C$ such that $\sum_{k\in \mathbb{Z}^n} k_i^2k_j^2|a_{ij}|^2 \leq C \sum_{k\in \mathbb{Z}^n} \|k\|^4|a_{ij}|^2$
- Fourier transform as a Gelfand transform

(c)Prove that if $f$ is odd, then $ \hat{f}(n)= -\hat{f}(-n) $, and we get a sine series.[This hint is helpful here Series Of Sines.

(d)Suppose that $f(\theta + \pi )$ = $f(\theta)$ for all $\theta \in R$. Show that $\hat{f}(n)$ = 0 for all odd n.(Its answer is here Effect of Symmetry of a Function on its Fourier Series)

(e)Show that $f$ is real-valued iff $\bar{\hat{f}}(n) =\hat{f}(-n)$ for all n.

For (e), we have two steps to prove 1- If $f$ is real valued, then $\bar{\hat{f}}(n) =\hat{f}(-n)$ for all n.Which is sooo easy by using the definition of Fourier Coeffient.

2- If $\bar{\hat{f}}(n) =\hat{f}(-n)$ for all n, Then $f$ is real valued.I reached this step,$$\int_{-\pi}^{\pi}\overline{f}(\theta)e^{in\theta} d\theta = \int_{-\pi}^{\pi}f(\theta)e^{in\theta} d\theta$$, Does this means that f is real valued ? If so what is the justification.

- Why do the first spikes in these plots point in opposite directions?
- What is the relation between analytical Fourier transform and DFT?
- Question about proof of Fourier transform of derivative
- How does knowing a function as even or odd help in integration ??
- Find the Fourier transform of $\frac1{1+t^2}$
- Translation of a certain proof of $(\sum k)^2 = \sum k^3 $
- $C$ such that $\sum_{k\in \mathbb{Z}^n} k_i^2k_j^2|a_{ij}|^2 \leq C \sum_{k\in \mathbb{Z}^n} \|k\|^4|a_{ij}|^2$
- Computing the Fourier series of $f = \cos{2x}$?
- Concrete FFT polynomial multiplication example
- Fourier transform of $\frac{1}{f(t)}$

Let’s solve (a).

The Fourier series of the function $f$ is

$$

\sum_{n=-\infty}^{\infty}\hat{f}(n)e^{in\theta}

$$

By definition, this is

$$

\sum_{n=0}^{\infty}\hat{f}(n)e^{in\theta} + \sum_{n=1}^{\infty}\hat{f}(-n)e^{i(-n)\theta}

$$

Taking out the $n=0$ term in the first series, this is

$$

\hat{f}(0)+\sum_{n=1}^{\infty}\hat{f}(n)e^{in\theta} + \sum_{n=1}^{\infty}\hat{f}(-n)e^{i(-n)\theta}\tag{1}

$$

Applying Euler’s formula we have

$$

e^{in\theta}=\cos(n\theta)+i\sin(n\theta)\\

e^{i(-n)\theta}=\cos((-n)\theta)+i\sin((-n)\theta)

$$

Since $\cos$ is an even function and $\sin$ is an odd function, this is

$$

e^{in\theta}=\cos(n\theta)+i\sin(n\theta)\\

e^{i(-n)\theta}=\cos(n\theta)-i\sin(n\theta)

$$

Substituting in $(1)$, we obtain

$$

\hat{f}(0)+\sum_{n=1}^{\infty}\hat{f}(n)[\cos(n\theta)+i\sin(n\theta)] + \sum_{n=1}^{\infty}\hat{f}(-n)[\cos(n\theta)-i\sin(n\theta)]

$$

But this equals

$$

\hat{f}(0)+\sum_{n=1}^{\infty}\left\{\hat{f}(n)[\cos(n\theta)+i\sin(n\theta)] + \hat{f}(-n)[\cos(n\theta)-i\sin(n\theta)]\right\}

$$

which is nothing more than

$$

\hat{f}(0)+\sum_{n=1}^{\infty}\left\{[\hat{f}(n) + \hat{f}(-n)]\cos(n\theta) + i[\hat{f}(n) – \hat{f}(-n)]\sin(n\theta)\right\}

$$

- Circular logic in set definition – Tautology?
- Show that the direct sum of a kernel of a projection and its image create the originating vector space.
- What is the difference between ⊢ and ⊨?
- What is the intuition behind the Wirtinger derivatives?
- Finite choice without AC
- A team of 11 players with at least 3 bowlers and 1 wicket keeper is to be formed from 16 players; 4 are bowlers; 2 are wicket keepers.
- Banach space in functional analysis
- Is there any generalization of the hyperarithmetical hierarchy using the analytical hierarchy to formulas belonging to third-order logic and above?
- Find polynomial $f(x)$ based on divisibility properties of $f(x)+1$ and $f(x) – 1$
- A Curious Binomial Sum Identity without Calculus of Finite Differences
- Banach-Tarski paradox: minimum number of pieces to get from a sphere of any size to a sphere of any other size?
- Show that $\gcd\left(\frac{a^n-b^n}{a-b},a-b\right)=\gcd(n d^{n-1},a-b)$
- $\operatorname{Aut}(S_4)$ is isomorphic to $S_4$
- Must $1+X$ be irreducible in any polynomial ring $R$?
- Uses of the incidence matrix of a graph