Intereting Posts

Euclidean domain $\mathbb{Z}$
Show that $\forall (x,y)$ in the first quadrant: $\frac {x^2+y^2}{4}\leq e^{x+y-2}$
How confident can we be about the validity of the classification of finite simple groups?
absolute value inequalities
Differentiability of $x^2\log(x^4+y^2)$ at $(0,0)$
Proving that $ f: \to \Bbb{R} $ is Riemann-integrable using an $ \epsilon $-$ \delta $ definition.
Difference between Fourier series and Fourier transformation
What converges this series?
Understanding this proof that a polynomial is irreducible in $\mathbb{Q}$
Can someone explain Gödel's incompleteness theorems in layman terms?
Arranging identical balls
Is a compensated Poisson process uniformly integrable
Conformal Maps onto the Unit Disc in $\mathbb{C}$
Expected number of steps till a random walk hits a or -b.
Insertion sort proof

I’m considering a proof of the convergence of the Fourier series. It begins by considering the full Fourier series of the periodic extension of $\phi$ defined on $[-\ell, \ell]$.

The full Fourier series is

$$

\sum_{n=-\infty}^\infty C_ne^{\frac{in\pi x}\ell}, \quad C_n = \frac{1}{2\ell}\int_{-\ell}^\ell \phi(x)e^{\frac{-in\pi x}{\ell}}dx.

$$

- Uniqueness of Fourier Coefficients
- Fourier series for $\sin^2(x)$
- Heuristic\iterated construction of the Weierstrass nowhere differentiable function.
- Proof for Sturm Liouville eigenfunction expantion pointwise convergence theorem
- Prove Parseval Identity for $f \in C(\Bbb T) 2\pi$ periodic continuous functions
- Fourier Analysis

Then we attempt to bound $|C_n|$ as

$$

|C_n| \leq \frac{1}{2\ell}\int_{-\ell}^{\ell} \left|\phi(x)e^{\frac{-in\pi x}{\ell}}\right| dx = \frac{1}{2\ell} \int_{-\ell}^\ell \left|\phi(x)\right|dx \leq \sup_{x \in [-\ell, \ell]} \left|\phi(x)\right|

$$

I do not understand the justification for the first equality. Why can we drop the $|e^{\frac{in \pi x}{\ell}}|$ term from the integrand?

- Difficulty in understanding a part in a proof from Stein and Shakarchi Fourier Analysis book.
- Pointwise but not uniform convergence of a Fourier series
- Fourier-Series of a part-wise defined function?
- Short form of few series
- Closed form of a series (dilogarithm)
- How to show that if all fourier coefficient of a function is zero, then the function is zero function?
- Absolute convergence of Fourier series of a Hölder continuous function
- “Counterexample” for a weaker version of Riemann–Lebesgue lemma
- Why does this Fourier series have a finite number of terms?
- Showing that complex exponentials of the Fourier Series are an orthonormal basis

For a real number $y$, $|e^{iy}| = 1$ since

$$|e^{iy}| = \sqrt{e^{iy}\overline{e^{iy}}} = \sqrt{e^{iy}e^{-iy}} = \sqrt{e^{i(y-y)}} = \sqrt{e^0} = \sqrt{1} = 1.$$

$|e^{it}| = 1$ if $t$ is real.

- Is the Cartesian product of an infinite number of $\mathbb{Z}^+$ countable?
- Is there $a,b,c,d\in \mathbb N$ so that $a^2+b^2=c^2$, $b^2+c^2=d^2$?
- Prove the number of red sides are always larger than $\frac{n^{2}-2n}{2}$
- Two definitions of locally compact space
- Prove by induction that $a^{4n+1}-a$ is divisible by 30 for any a and $n\ge1$
- Definition of set.
- Expected Value of Maximum of Two Lognormal Random Variables with One Source of Randomness
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- Change of variables formula for Riemann and Lebesgue integration
- Sum of cubes proof
- Is $e^{e^9}$ an integer?
- Find exact value of $\cos (\frac{2\pi}{5})$ using complex numbers.
- Need some help with this recurrence equation
- Explain the 1 + 2 + 3 in $ \frac{1 + 1 + 1 + \cdots}{1 + 2 + 3 + \cdots} = \lim_{n \to \infty} \frac{1}{(n+1)/2} $
- Splitting Field over a Field