I have a question and I have a definition $\{\varphi_n(t)=\frac{1}{\sqrt{2\pi}}e^{int}\}$.I know that $n\in \mathbb Z$ for this question. does int means integer?

I couldn't find any reference with a web search.

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    $\begingroup$ I'm guessing here, but I do not think it is one thing. The function is a function of $t$, $n$ is an integer, and $i$ is either the complex number $i$ or some other constant. So $i\cdot n\cdot t$. $\endgroup$ – Em. Jun 11 '16 at 6:56

The exponent $int$ is indeed a product of the imaginery number $i$, an integer $n$ and a variable $t$. By Euler's formula, we have $$e^{int} = \cos(nt) + i\sin(nt).$$

What is important about this sequence of functions $\{\phi_n(t)\}_{n\in\mathbb{Z}}$ is that they form a orthonormal Schauder basis for the inner product space $L^2(0,2\pi)$, i.e.

$$\int_0^{2\pi} \phi_m(t) \overline{\phi_n(t)} \, dt = \begin{cases} 1 & \text{ if } m = n,\\ 0 & \text{ otherwise.}\end{cases}$$

and for any $f \in L^2(0,2\pi)$, there exists a sequence $(f_j)_{j=1}^\infty \subset \langle \phi_n \rangle$ such that $$\int_0^{2\pi} \Big( f(t) - f_j(t) \Big)^2 \, dt \to 0 \text{ as } j \to \infty. $$

  • $\begingroup$ My question contains $\int_{-\pi}^\pi$ whithin $L_2$. Can you kindly expand on the meaning of $\langle \phi_i ,\phi_j \rangle = \delta_{i,j}$? Does it means $\langle \phi_i ,\phi_j \rangle =1$ for $i=j$ and $\langle \phi_i ,\phi_j \rangle =0$ for $i \neq j$? $\endgroup$ – havakok Jun 11 '16 at 7:17
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    $\begingroup$ Since the trigonometric functions are $2\pi$ periodic, the same happens when you domain is $(-\pi, \pi)$. For the second question, yes, I have clarified in my answer. $\endgroup$ – Empiricist Jun 11 '16 at 7:20

I believe from context $\varphi_n(t)=\frac{1}{\sqrt{2\pi}}e^{int}=\frac{1}{\sqrt{2\pi}}(\cos(nt)+i\sin(nt))$


$n \in \mathbb Z$

with argument $t$.


$\text{int}$ here $i$ is iota and $e^{int}$ =$ \cos nt +i\sin nt $

Edit: if $i$ does not refer to iota then it can represent any number, although usually $i^2=-1$.

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    $\begingroup$ The $i$ doesn't carry through into the argument of the cosine or sine functions. $\endgroup$ – okrzysik Jun 11 '16 at 7:04
  • $\begingroup$ @okrzysik then it would be the representation of a number . $\endgroup$ – Shona Jun 11 '16 at 7:06
  • $\begingroup$ @okrzsik thanks i corrected it ☺ $\endgroup$ – Shona Jun 11 '16 at 7:15
  • $\begingroup$ @Jon Mark Perry thanks for the edit .☺ $\endgroup$ – Shona Jun 11 '16 at 8:19

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