In mathematics, a square-integrable function, also called a quadratically integrable function or
L
2
{\displaystyle L^{2}}
function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite. Thus, square-integrability on the real line
(
−
∞
,
+
∞
)
{\displaystyle (-\infty ,+\infty )}
is defined as follows.
One may also speak of quadratic integrability over bounded intervals such as
[
a
,
b
]
{\displaystyle [a,b]}
for
a
≤
b
{\displaystyle a\leq b}
.
An equivalent definition is to say that the square of the function itself (rather than of its absolute value) is Lebesgue integrable. For this to be true, the integrals of the positive and negative portions of the real part must both be finite, as well as those for the imaginary part.
The vector space of square integrable functions (with respect to Lebesgue measure) form the Lp space with
p
=
2
{\displaystyle p=2}
. Among the Lp spaces, the class of square integrable functions is unique in being compatible with an inner product, which allows notions like angle and orthogonality to be defined. Along with this inner product, the square integrable functions form a Hilbert space, since all of the Lp spaces are complete under their respective p-norms.
Often the term is used not to refer to a specific function, but to equivalence classes of functions that are equal almost everywhere.
The James Webb Telescope is in one of Earths Lagrange points, (I believe it's in L2) How does the moons gravity affect this? Do they have to make course corrections?
So we are finding the L2 Lagrange point, specifically the distance from the earth, or d in this instance. I have used the equation above and I have come out with 1.5 * 10^9 meters as d, or L2's distance from the earth. Can anyone verify this, is the equation correct and is my final distance...
Why is the Webb placed at L2 (Lagrange point 2)?
I read some articles that said it would be perpetually shaded being on the opposite side of the sun and moon, but that makes no sense when the Webb orbit is so huge?
So why is it at L2?
When the Webb is at a point on its L2 orbit, (not at L2), what direction is the centrifugal force vector compared to the direction of the combined earth-sun gravity vector on the opposite side? Is the direction of this centrifugal vector ALWAYS parallel to the sun-earth plane? or is it always...
I was already puzzled by the concept of orbiting a Lagrange point and then I find out it's about the same size orbit as the Moon. I am thinking that if there was no Moon that the Earth and the Sun are far enough away to be treated as points and so that there would be an exact distance further...
Maybe this is more general discussion, but I am excited / nervous about the upcoming launch of the JWST.
https://jwst.nasa.gov/content/webbLaunch/countdown.html
I can't wait to see the observations this endeavor will bring!
The diagram in the article seems to say that the Webb will orbit around the unstable L2 point. At any distance near L2 but not exactly at L2 the Webb will tend to move further from L2, unless Webb has an engine and fuel to maintain it in the special orbit. I was not able to find any description...
I do not know if there is a solution to the problem.
With the aid of CAD Software, I get X = 3.5 when L1=7.082, L2=0.684, L3=0.876. I don't know how accurate this is.
My intuition says there should be a expression for X but I am yet to solve it. Any ideas?
Hello everyone,
I'm currently going through Strauss "introduction to differential equations" and i can't get around a certain proof that he
gives on chapter 11.5 page(327 (2nd edition)).Specifically, the proof refers to a certain version of Fredholm's alternative theorem.
Assume that we are...
I'm confused (what else is new) about L2.
While watching a video from PBS Digital Spacetime about the latest data drop from Gaia Space Telescope, Matt O'Dowd showed a CGI animation of the telescope leaving Earth then circling/orbiting L2 perpendicular to the Earth/sun plane.
I thought that the...
The photograph above this sentence is a photograph of a simple 2 leg circuit of alternating current with a light switch and a light bulb. I know how alternating current constantly switches direction from line 1 to line 2 and then goes from line 2 to line 1, etc. In the diagram in the above...
Hi there,
I am also familiar with Hilbert spaces and Functional Analysis and I find your question very interesting. I agree that the Fourier transform is a powerful tool for analyzing LTI systems and diagonalizing the convolution operator. As for your question about whether the same can be...
Homework Statement
In the picture, m1 is 20kg, and the wedge pushes up with a force of 300 N. If the length L1 is 2 meters, solve for m2 and L2.
Homework Equations
I'm not sure
The Attempt at a Solution
The picture shows some sort of beam balancing on a fulcrum that is not centered. On the...
Homework Statement
Three long straight parallell wires that conduct electricity called L1, L2 and L3. The angle at L2 is 90 degrees and the distance between L2 and L1, L3 is 0,50 meters. The current through the wires L1 and L3 is 300 A while the current in L2 is 600 A. Decide the resulting...
The Wiki article shows 5 Lagrange points. I can “see” how the points L1, L4 and L5 points would be balanced by the gravitation of the two bodies, but not the L2 and L3.
For L2 and L3, it looks to me like the combination of the Sun’s and Earth’s gravity increase pull and make less stable. So...
Sorry I'm a little rusty with my math and proof logic, and this feels like a dumb question, but oh well! The Euclidian norm of a vector in ℝ3 is \|{v}\| = \sqrt{x^2 + y^2 + z^2} where \|{v}\| \geq 0. I'm trying to show that there is always an infinite number of solutions for arbitrary...
hi, am major new on quantum mechanics. please help me understand. is the real wave function
Ψ2Px= [Ψ2p+1 +Ψ2p-1]1/2 an eigen function of L2 or Lz?
if so, how is it?
and if so kindly explain the values of l and m
thanks
Homework Statement
Show that the set {sin(nx)} from n=1 to n=∞ is orthogonal bases for L^2(0, π).
Homework EquationsThe Attempt at a Solution
Proof: Let f(x)= sin(nx), consider scalar product in L^2(0, π)
(ƒ_n , ƒ_m) = \int_{0}^π ƒ_n (x) ƒ_m (x) \, dx = \int_{0}^π sin(nx)sin(mx) \, dx =...
Homework Statement
The free particle wave packet in question is $$\psi=ce^{-(r/r_0)^2}$$
Homework EquationsThe Attempt at a Solution
I've been going through books and class notes but I really have no idea where this came from. I'm thinking that if I can decompose this in plane waves I could...
Homework Statement
Homework Equations
How can I start the proof? Shall I use the Poincare inequality?
The Attempt at a Solution
Well, I know that this norm is defined by , but still I don't know how to start constructing the proof?
Hi there,
I have the following thread:
1) m2 < m1 and L2 < L1
2) abs(m2-m1)=abs(L2-L1)
the values of m2, m1 and L1 are known, what equation can I use to find the value of L2?
thank you
Let:
gn(x) = 1 in [1/4 - 1/n2 to 1/4 + 1/ n2) for n = odd
1 in [3/4-1/n2 to 3/4 + 1/n2) for n = even
0 elsewhere
Show the function converges in the L2 sense but not pointwise.
My issue is in how I should use the definition of...
If I have the following operator for $H=L^2(0,1)$:$$Tf(s)=\int_0^1 (5s^2t^2+2)(f(t))dt$$ and I wish to calculate $||T||$, how do I go about doing this:
I know that in $L^2(0,1)$ we have that relation:$$||T||\leq \left ( \int_0^1\int_0^1 |(5s^2t^2+2)|^2dtds\right )...
Hi all,
I have a basic optimisation question. I keep reading that L2 norm is easier to optimise than L1 norm. I can see why L2 norm is easy as it will have a closed form solution as it has a derivative everywhere.
For the L1 norm, there is derivatiev everywhere except 0, right? Why is this...
Having bounded set ##U\subset\mathbb{R}^n## with ##C^1## boundary ##\partial U## and a function ##g\in C^2(\partial U)##, does one automatically have ##g\in L^2(\partial U)##?
I don't need a proof/explanation, yes/no answer is sufficient.
Homework Statement
Let H= C[-1,1] with L^2 norm and consider G={f belongs to H| f(1) = 0}. Show that G is a closed subspace of H.
Homework Equations
L^2 inner product: <f,g>\to \int_{-1}^{1}f(t)\overline{g(t)} dt
The Attempt at a Solution
I've been trying to prove this for a...
Homework Statement
Suppose we have a sequence {x} = {x_1, x_2, ...} and we know that \{x\}\in\ell^2, i.e. \sum^\infty x^2_n<\infty. Does it follow that there exists a K>0 such that x_n<K/n for all n?
Homework Equations
The converse is easy, \sum 1/n^2 = \pi^2/6, so there would be a finite...
Hello!
According to many sites L1 is internal cache
and Level 2 cache, cache memory that is external to the microprocessor.
but according to this intel core2duo box (photo) , intel is selling this L2 cache on processor not on another chip.
http://i40.tinypic.com/i4fu4n.jpg
why is...
I got this open problem from my advisor: find a time function which belongs to L2 and L_infinity, but is not vanishing as t -> infinity. Lp here is the p-norm space of functions http://en.wikipedia.org/wiki/Lp_space#Lp_spaces_2
I've worked on this for two weeks now, and still not found the way...
Does anyone know where I can find one? I've written a program to test the stability of the lunar L2 point by placing a massless test object in orbit there and I've found an optimised value based on the approximation
r_{m}(\frac{M_{M}}{3M_{E}})^{1/3}
where r_{m} is the radius of the...
Homework Statement
1. Prove that if a metric space (X,d) is separable, then
(X,d) is second countable.2. Prove that \ell^2 is separable.
Homework Equations
The Attempt at a Solution
1. \{ x_1,\ldots,x_k,\ldots \} is countable dense subset. Index the
basis with rational numbers, \{ B(x,r) | x...
Let S be a subspace of L^{2}(\left[0,1\right]) and suppose \left|f(x)\right|\leq K \left\| f \right\| for all f in S.
Show that the dimension of S is at most K^{2}
---------
The prof hinted us to use Bessel's inequality.
Namely, let \left\{ u_1,\dots, u_m \right\} be a set of...
Homework Statement
\|x\|_2\le\|x\|_1\le\sqrt{n}\|x\|_2
where |x|1 is the l1 norm and |x|2 is the l2 normHomework Equations
See aboveThe Attempt at a Solution
I have \|\mathbf{x}\|_1 := \sum_{i=1}^{n} |x_i|
and \|x\|_2 = \left(\sum_{i\in\mathbb N}|x_i|^2\right)^{\frac12}
I have tried to...
Hello,
I have a (infinite dimensional) vector space and defined an inner product on it.
The vectors element are infinite sequence of real numbers (x_1, x_2,\ldots).
The inner product has the common form: x_iy_i
The problem now is that the vectors have an infinite number of elements, so the...
Let x and y be n-tuples of non-negative integers.
Furthermore,
sum x_i = sum y_i
and,
sum x_i^2 = sum y_i^2
Is it true that x must be a permutation of y?
Cheers!
Homework Statement
Prove that f(x)=\sin(\pi x)/(\pi x) is in L^2(R) but not in L^1(R)
This is in a chapter of the book dealing with Inverse Fourier Transform
f is in L^1 if \int|f|<\infty
f is in L^2 if \sqrt{\int|f|^2}<\infty
Homework Equations
I just have no idea how to do it
The Attempt...
I'm trying to show that if f(x) is analytic, then for large enough n,
|| f^{(n)} (x) || \leq c n! || f(x) ||,
where
|| f ||^2=\int_a^b{|f|^2}dx
and f^{(n)} denotes the nth derivative.
I tried to use the Taylor series, and then manipulated some inequalities, but I wasn't getting...
It is clear that there are functions in L2 that are not in L1, but what about the other way? And what effect does considering L2(R) versus L2([a,b]) have?
Thanks.
Homework Statement
Construct a function u of the space H'(B), where B is a unit sphere in R^3, which does not belong to L∞(B).
Homework Equations
(relevant facts)
all L2 are hilbert, therefore the problem reduces for us finding an L2 function which is not L-infinity.
The Attempt...
Hi this is my first time posting on this forum. I have an question about Lagragian points.
I was trying to find L2 lagrangian point, a point that lies on the line defined by the two large masses, beyond the smaller of the two. Here, the gravitational forces of the two large masses balance...
Homework Statement
(I'm posting this because my proofs seem to be lousy. I want to see if I'm missing anything.)
Show that if f_n \in L^2(a,b) and f_n \rightarrow f in norm, then <f_n,g> \rightarrow <f,g> for all g \in L^2(a,b)
Homework Equations
L^2(a,b) is the space of...
L1-, L2-, Linfty-Norm Proofs - Please Help!
Homework Statement
Show that ||x||1 < or = n||x||infinity and ||x||2 < or = sqrt(n)*||x||infinity for x exists in the set of all real numbers.
Homework Equations
||x||2 is defined here: http://mathworld.wolfram.com/L2-Norm.html
||x||1 is...
Hello.
I am having trouble answering the following question:
"Show that the L2 transformation reduces to the L1 transformation when the two reference frames are in standard configuration."
Am I wrong to assume that r = xi + yj + zk
Any help would be beautiful!
Thanx much