Intereting Posts

Let $f:\mathbb{R}\longrightarrow \mathbb{R}$ a differentiable function such that $f'(x)=0$ for all $x\in\mathbb{Q}$
Baby Rudin: Chapter 1, Problem 6{d}. How to complete this proof?
When are complete intersections also local complete intersections?
How to know whether an Ordinary Differential Equation is Chaotic?
Existence of submersions from spheres into spheres
Set of all Inner Automorphisms is a subgroup of the set of all Automorphisms of a group $G$
A concise guide to basic probability
Question on morphism locally of finite type
Simulate simple non-homogeneous Poisson proces
Solving inhomogenous ODE
General Solution to $x^2-2y^2=1$
Real Roots and Differentiation
A question with infinity
If the limit of a multivariable function is identical along a certain class of paths, can we claim the existence of the limit?
The Cantor set is homeomorphic to infinite product of $\{0,1\}$ with itself – cylinder basis – and it topology

If $f$ is continuous real-valued function, does the Riemann Lebesgue Lemma give us that $\int_{m}^k f(x) e^{-inx}\,dx \rightarrow 0\text{ as } n\rightarrow \infty$ for all $m\le k$? Specifically, is this true for any continuous function, whether it’s periodic or not?

Edited.

- Is $\sum_{n=1}^{\infty}{\frac{\sin(nx)}n}$ continuous?
- Proving $\sum_{n =1,3,5..}^{\infty }\frac{4k \ \sin^2\left(\frac{n}{k}\right)}{n^2}=\pi$
- How can apply the $L^p$ norm in a circle to $L^2$ norm in a square?
- Poisson's summation formula
- Why is the Fejér Kernel always non-negative?
- How to interpret Fourier Transform result?

- Tensor products of functions generate dense subspace?
- Fourier Series $\sin(\sin(x))$
- Fourier transform on $1/(x^2+a^2)$
- Generalising Parseval's Identity using the Convolution Theorem
- When is a Fourier series analytic?
- Is this Fourier like transform equal to the Riemann zeta function?
- Singular asymptotics of Gaussian integrals with periodic perturbations
- Evaluating the integral $\int_{-\infty}^\infty \frac{\sin^2(x)}{x^2}e^{i t x} dx$
- What is the relation between analytical Fourier transform and DFT?
- Fourier Transform calculation

If $f$ is Lebesgue integrable on $[r,s]$, then the result is still true. You can show it by approximation. For $\epsilon > 0$, there exists a continuously differentiable function $g$ on $[r,s]$ such that $\int_{m}^{k}|f-g|dx < \epsilon/2$. And, as $n\rightarrow\pm\infty$,

$$

\int_{r}^{s}e^{-inx}g(x)dx = \left.\frac{e^{-inx}}{-in}g(x)\right|_{x=r}^{s}-\int_{r}^{s}\frac{e^{-inx}}{-in}g'(x)dx \rightarrow 0

$$

This really has nothing to do with integers. You can assume $m,k,n$ are all real if you like, and you get the same result:

$$

\left|\int_{r}^{s}e^{-inx}f(x)dx\right| \le \int_{r}^{s}|f(x)-g(x)|dx + \left|\int_{r}^{s}e^{-inx}g(x)dx\right| < \epsilon

$$

if $|n|$ is chosen large enough that the second integral on the right is bounded by $\epsilon/2$.

- A question on convergence of derivative of power series
- Reference request, self study of cardinals and cardinal arithmetic without AC
- Distribution of Universal Quantifiers
- Finding a combinatorial argument for an interesting identity: $\sum_k \binom nk \binom{m+k}n = \sum_i \binom ni \binom mi 2^i$
- composition of covering maps
- $S_4$ does not have a normal subgroup of order $8$
- Help determining best strategy for game?
- What does “extend linearly” mean in linear algebra?
- Problem proving: $V = \ker T \oplus \operatorname{im}T$
- Conjugate cycles
- How to find logarithms of negative numbers?
- Finding the minimum number of selections .
- Why is $Sp(2m)$ as regular set of $f(A)=A^tJA-J$, and, hence a Lie group.
- Proving the inequality $|a-b| \leq |a-c| + |c-b|$ for real $a,b,c$
- Iterated Integrals – “Counterexample” to Fubini's Theorem