1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
lozanna [386]
2 years ago
5

State and prove bessel inequality​

Physics
2 answers:
maria [59]2 years ago
5 0

Statement :- We assume the orthagonal sequence {{\{\phi\}}_{1}^{\infty}} in Hilbert space, now {\forall \sf \:v\in \mathbb{V}}, the Fourier coefficients are given by:

{\quad \qquad \longrightarrow \sf a_{i}=(v,{\phi}_{i})}

Then Bessel's inequality give us:

{\boxed{\displaystyle \bf \sum_{1}^{\infty}\vert a_{i}\vert^{2}\leqslant \Vert v\Vert^{2}}}

Proof :- We assume the following equation is true

{\quad \qquad \longrightarrow \displaystyle \sf v_{n}=\sum_{i=1}^{n}a_{i}{\phi}_{i}}

So that, {\bf v_n} is projection of {\bf v} onto the surface by the first {\bf n} of the {\bf \phi_{i}} . For any event, {\sf (v-v_{n})\perp v_{n}}

Now, by Pythagoras theorem:

{:\implies \quad \sf \Vert v\Vert^{2}=\Vert v-v_{n}\Vert^{2}+\Vert v_{n}\Vert^{2}}

{:\implies \quad \displaystyle \sf ||v||^{2}=\Vert v-v_{n}\Vert^{2}+\sum_{i=1}^{n}\vert a_{i}\vert^{2}}

Now, we can deduce that from the above equation that;

{:\implies \quad \displaystyle \sf \sum_{i=1}^{n}\vert a_{i}  \vert^{2}\leqslant \Vert v\Vert^{2}}

For {\sf n\to \infty}, we have

{:\implies \quad \boxed{\displaystyle \bf \sum_{1}^{\infty}\vert a_{i}\vert^{2}\leqslant \Vert v\Vert^{2}}}

Hence, Proved

tensa zangetsu [6.8K]2 years ago
3 0

Answer:

hope this helps and if you have any questions please feel free to ask me any time

You might be interested in
Jupiter is 5.2 AU from the sun on average. How long is one year on Jupiter?
velikii [3]

Answer:

12 years

Explanation:

12 years is correct because how long is Jupiter one year is 12 years

3 0
2 years ago
The direction of the magnetic field is parallel to the field line at any point in space. A small compass will point in a directi
Slav-nsk [51]
I think the answer is false
6 0
3 years ago
Weightlessness is experienced by an astronaut in space. This means that the astronaut's muscles have to be stronger to move his
just olya [345]
The answer is false. The speed of the astronaut cancels out the force of gravity, causing a 'stationary freefall'. While under these effects, it is not required for an astronaut to 'strengthen' his body.
4 0
3 years ago
Read 2 more answers
Sics and Chemistry A (2nd Six Weeks) - SC3200SFA/01
Andreyy89
Force = Mass * Acceleration therefore the red ball with the higher mass will have more force and greater acceleration
3 0
3 years ago
Early experiments with light were understood by explaining the behavior of light in terms of waves. Which experimental result re
STALIN [3.7K]
The answer to what experimental result required considering the particle nature of light is A. The ultraviolet catastrophe of blackbody radiation.
8 0
3 years ago
Other questions:
  • What is an example of electrons moving?
    6·1 answer
  • If a snail travels at 5 m/s, how far will it travel in 90 seconds?
    11·2 answers
  • A spring of force constant 285.0 N/m and unstretched length 0.230 m is stretched by two forces, pulling in opposite directions a
    14·1 answer
  • Using Hooke's law, Fspring=kΔx, find the elastic constant of a spring that stretches 2 cm. when a 4 Newton force is applied to i
    7·1 answer
  • What color is mercury
    5·2 answers
  • Fire trucks have the word FIRE written two ways, as seen here, on the front of the truck. Why is that?
    10·2 answers
  • What is a black hole
    6·1 answer
  • A 5.0 kg block is pushed 2.0 m at a constant velocity up a vertical wall by a constant force applied at an angle of 26
    9·1 answer
  • You can make a solute dissolve more quickly in a solvent by
    9·1 answer
  • A 60 kg person runs up a set of stairs in B-Building 6 meters high, in a time of 8 seconds.
    14·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!