Function Inner Product

Source: A linear algebra problem book

Problem:

Find the function f  ( member of span {1, sin x, cos x} ) that minimizes norm of ( sin 2x − f(x) ), where the norm comes from the inner product

< f , g > = integral over x from -pi to pi [ f(x)g(x) ]

 Update (Oct 30, 2011):
Solution posted by Harsh Pareek (Graduate Student at UT Austin, CSE IITB 2011 Alumnus) in comments.

Comments

  1. Harsh Pareek12:40 PM, October 11, 2011

    This is one of your easier ones, but it's so nice I couldn't help commenting.

    First, you're asking us to find the point in P=span{1,sin x, cos x} closest to sin 2x.
    Usually in vector spaces, the answer would be the projection of sin 2x on P. sin 2x is orthogonal to {1,sin x, cos x} so in this case we expect the element 0, and as we'll see the intuition goes through.

    Let m = = norm(sin 2x)^2 + norm(f)^2 - 2

    f(x) = A + B sin x + C cos x => = 0

    So m is minimum when norm(f) is minimum, i.e. f=0.

    In general, if you substitute sin 2x by g(x), our answer will be +sin x + cos x

    ReplyDelete
  2. Pratik Poddar11:34 AM, October 30, 2011

    @Harsh.. you probably tried to use latex.. Hence your comment is not well formed..

    Let me try to decipher what you wrote:

    First, you're asking us to find the point in P=span{1,sin x, cos x} closest to sin 2x.
    Usually in vector spaces, the answer would be the projection of sin 2x on P. sin 2x is orthogonal to {1,sin x, cos x} so in this case we expect the element 0, and as we'll see the intuition goes through.

    Let m = norm(sin2x - f(x))^2 = norm(sin 2x)^2 + norm(f)^2 - 2

    f(x) = A + B sin x + C cos x

    Hence, = 0 (Since, sin2x is orthogonal to all 1, sinx and cosx)

    So m is minimum when norm(f) is minimum, i.e. f=0.

    In general, if you substitute sin 2x by g(x), our answer will be the projection of g(x) onto P.

    Is this what you meant? I think that is correct.

    ReplyDelete

Post a Comment

Popular posts from this blog

Fraction Brainteaser

Source: Sent to me by Gaurav Sinha Problem: Siddhant writes a Maths test and correctly answers 5 out of 6 Arithmetic questions and 20 out of 28 Geometry questions.  In total, Siddhant scores 25 out of 34.  Vaibhav writes another Maths test and correctly answers 20 out of 25 Arithmetic questions and 6 out of 9 Geometry questions. in total, Vaibhav scores 26 out of 34. Note that a) Vaibhav scores more than Siddhant b) Siddhant score better than Vaibhav in both individual topics -  5/6 > 20/25 and 20/28 > 6/9 How is it possible? 
Read more

Buying Dimsums

Source: Alok Goyal (Stellaris VP, Ex-Helion VC) puzzle blog Problem: A fast food restaurant sells dimsums in boxes of 7 and 3. What’s the greatest number of dimsums a person cannot buy. Generalize it for p and q where p and q are relatively prime. I loved the puzzle. Hope you enjoy it too.
Read more

Consecutive Heads

Let's say A keep tossing a fair coin, until he get 2 consecutive heads, define X to be the number of tosses for this process; B keep tossing another fair coin, until he get 3 consecutive heads, define Y to be the number of the tosses for this process. 1) Calculate P{X>Y} 2) What's the expected value of X 3) What's the expected value of Y This is probably the hardest puzzle I have ever put on my blog. Hence, I will post its solution in the post directly rather than on comment. Solution: 1) (Solved by me finally after 13 months :)) Make a state diagram. Let the state be (a,b) where a is the number of consecutive heads streak "A" is on currently and b is the number of consecutive heads streak "B" is on currently. So, (0,3) (1,3) are final accepted states and (2,0) (2,1) (2,2) (2,3) are failure states. If you get tails, your contribution to the state reaches to "0" f(State) = P(X>Y | "State" configuration initially)
Read more