Tagged Questions

Questions on periodic functions, functions $f(x)$ that satisfy the identity $f(x+c)=f(x)$, for some nonzero $c$.

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Fourier Coefficients of periodic function

Consider a Function $f\in L^2(\mathbb{T})$. Is there any lower bound for the decay of the Fourier coefficients $$\hat{f}(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi} f(t) e^{-int} dt$$ known? There are a lot ...
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How to find period of this periodic function?

How can I find a period of this function? $$2\sin{3x} + 3\sin{2x}$$ Is here any way how to sum both sinuses?
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How to expand the Fourier series for $f(x)=\max \{0, \frac{\pi}{2}-\lvert x\rvert \}$?

My Question: My Goal is to determine the Fourier series for $f(x)=\max \{0, \frac{\pi}{2}-\lvert x\rvert \} \quad$ for $x \in [-\pi, \pi ]$ This function is $2\pi$-periodic. My Approach: i found ...
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$\cos(x)+\cos(x\sqrt{2})$ is not periodic

Show that the function $$f(x)=\cos(x)+\cos(x\sqrt{2})$$ is not periodic. I tried $x = a$ and $a\sqrt{2}$. I am guessing that the method of contradiction would be of some help over here. What else ...
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Does weak convergence (of a periodic function) imply weak-* convergence for the derivative?

Lets assume we have $b(x,t) \in L^\infty$ periodic in x and $\frac{db}{dt} \in L^{p}, p\in\mathbb{N}$. $b(x,t)$ converges weakly to an $f(t)=\int b(x,t)dx \in L^q$ for all $1<q<\infty$. ...
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Boundedness and Uniform Continuity

Prove that a continuous periodic function on $\mathbb{R}$ is bounded and uniformly continuous on $\mathbb{R}$. Given the continuous periodic function $f:\mathbb{R} \to \mathbb{R}$ for some period ...
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Laplace transform of a periodic function

Knowing that $$L[f(t)]=\frac1{1-e^{-sp}}\int_0^{p} e^{-st}f(t)dt$$ $p$ indicates the period of the function If $f$ is a continuous function by segments in $[0,\infty)$ and $F(s)=L[f(t)]$ exists for ...
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Solving a Forced Oscillations Differential Equation Problem

A building consists of two floors. The fi rst floor is attached rigidly to the ground, and the second floor is of mass m = 1000 slugs (fps units) and weighs 16 tons (32,000 lb). The elastic frame of ...
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Type of periodicity in champernowne constant.

Digits of Champernowne constant are aperiodic, else it will be rational. Fine! But it is not random because I can write a program which will give me the position of every digit. E.g. I can calculate ...
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Set of periods of a real-valued function

Let $~f : E \rightarrow \mathbb{R}, ~ E \subset \mathbb{R}~~$ be a periodic fucntion and $S = \{ T \in \mathbb{R} ~ : ~ \forall x\in \mathbb{R} ~~ f(x+T) = f(x) \}$ be the set of all periods of ...
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Is a broken clock right twice a day?

I was thinking about this expression, and I wondered if it holds true when the clock is slow. I can imagine a slow clock which is not right at all in the span of 12 hoursâ€”imagine a clock which ticks 5 ...
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number of zeros of complex waves

Does anybody know about any type of methods how to calucalte/estimate the number of the zeros of complex waves (periodic functions as superposition of many harmonic waves) within a given period [0,x] ...
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Periodic polynomial?

I was thinking if it was possible to create a polynomial that would be periodic all over the reals, since polynomials can be periodic on an interval. I then I found out the following function: ...
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Trigonometric Identities - Assignment

How do I simplify Cos(5 theta)? I got as far as Cos(2theta + 3theta). Do I then say Cos(2theta + 3theta) = Cos(2theta) + Cos(3theta)? In that case, how do I get Cos(3theta)?
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Negative periodic functions

Can any one tell me how to find a periodic function $f(t)<-k$, $k$ is strictly positive constant, but I don't need functions where we just add a negative number to an usual periodic function like: ...
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How can a Bézier curve be periodic?

As I know it, a periodic function is a function that repeats its values in regular intervals or period. However BĂ©zier curves can also be periodic which means closed as opposed to non-periodic which ...
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periodic solution of differential equation

let be the ODE $-y''(x)+f(x)y(x)=0$ if the function $f(x+T)=f(x)$ is PERIODIC does it mean that the ODE has only periodic solutions ? if all the solutions are periodic , then can all be ...
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Proof that a periodic function is bounded and uniformly continuous.

I need to show that if $f:\mathbb{R}\to \mathbb{R}$ is continuous and $\forall x \in \mathbb R, f(x+1)=f(x)$, then: $f$ is bounded, $f$ is uniformly continuous, there exists $c\in \mathbb{R}$ such ...