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Hilbert Spaces and Orthogonal Expansions

Module by: Roy Ha. E-mail the author

Summary: (Blank Abstract)

Much of the language in this section will be familiar to you - you've been exposed to the concepts of

in the context of Rn n . We're going to take what we know about vectors and apply it to functions (continuous time signals).

Vector Spaces

Definition 1: Vector space
A linear vector space 𝕊 𝕊 is a collection of "vectors" such that
  1. if f 1 𝕊α f 1 𝕊 f 1 𝕊 α f 1 𝕊 for all scalars αα ( αR α or αC α ) and
  2. if f 1 𝕊 f 1 𝕊 , f 2 𝕊 f 2 𝕊 , then ( f 1 + f 2 )𝕊 f 1 f 2 𝕊

If the scalars αα are real, 𝕊𝕊 is called a real vector space.

If the scalars αα are complex, 𝕊𝕊 is called a complex vector space.

If the "vectors" in 𝕊𝕊 are functions of a continuous variable, we sometimes call 𝕊𝕊 a linear function space

Example 1

  • Rn=real vector space n real vector space
  • Cn=complex vector space n complex vector space
  • L 1 ( ) = ft ft |ft|dt< L 1 ( ) f t t f t f t is a vector space
  • L ( ) = ft ft f ( t )  is bounded L ( ) f t f ( t )  is bounded f t is a vector space
  • L 2 ( ) = ft ft |ft|2dt< =finite energy signals L 2 ( ) f t t f t 2 f t finite energy signals is a vector space
  • L 2 ( [ 0 , T ] ) =finite energy functions on interval [0,T] L 2 ( [ 0 , T ] ) finite energy functions on interval [0,T]
  • 1 ( ) 1 ( ) , 2 ( ) 2 ( ) , ( ) ( ) are vector spaces
  • the collection of functions piecewise constant between the integers is a vector space
Figure 1
Figure 1 (fig1.png)

More examples:

  • + 2= x 0 x 1 x 0 x 1 ( x 0 >0)( x 1 >0) + 2 x 0 x 1 x 0 0 x 1 0 x 0 x 1 is not a vector space. 11 + 2 1 1 + 2 , but α,α<0:α11 + 2 α α 0 α 1 1 + 2
  • D=zCzC |z|1 D z z 1 z is not a vector space. ( z 1 =1)D z 1 1 D , ( z 2 =i)D z 2 D , but ( z 1 + z 2 )D z 1 z 2 D , | z 1 + z 2 |=2>1 z 1 z 2 2 1
Vector spaces can be collections of functions, collections of sequences, as well as collections of traditional vectors (i.e. finite lists of numbers)

Norms

The norm of a vector is a real number that represents the "size" of the vector.

Example 2

In R2 2 , we can define a norm to be a vectors geometric length.

Figure 2
Figure 2 (fig2.png)

x= x 0 x 1 T x x 0 x 1 , norm x= x 0 2+ x 1 2 x x 0 2 x 1 2

Mathematically, a norm · · is just a function (taking a vector and returning a real number) that satisfies three rules.

To be a norm, · · must satisfy:

  • the norm of every vector is positive x,x𝕊:x>0 x x 𝕊 x 0
  • scaling a vector scales the norm by the same amount αx=|α|x α x α x for all vectors x x and scalars α α
  • triangle property x+yx+y x y x y for all vectors x x, y y. "The "size" of the sum of two vectors is less than or equal to the sum of their sizes"

A vector space with a well defined norm is called a normed vector space or normed linear space.

Example 3

Rn n (or Cn n ), x= x 0 x 1 x n - 1 x x 0 x 1 x n - 1 , x1 1 x i 0 n 1 x i , Rn n with this norm is called 1 ( [ 0 , n - 1 ] ) 1 ( [ 0 , n - 1 ] ) .

Figure 3: Collection of all xR2 x 2 with x1 1 x
Figure 3 (fig3.png)

Rn n (or Cn n ), with norm x2=i=0n1| x i |212 2 x i 0 n 1 x i 2 1 2 , Rn n is called 2 ( [ 0 , n - 1 ] ) 2 ( [ 0 , n - 1 ] ) (the usual "Euclidean"norm).

Figure 4: Collection of all xR2 x 2 with x2=1 2 x 1
Figure 4 (fig4.png)

Rn n (or Cn n , with norm x=maxii| x i | x i x i is called ( [ 0 , n - 1 ] ) ( [ 0 , n - 1 ] )

Figure 5: xR2 x 2 with x=1 x 1
Figure 5 (fig5.png)

We can define similar norms for spaces of sequences and functions.

Discrete time signals = sequences of numbers xn= x -2 x -1 x 0 x 1 x 2 x n x -2 x -1 x 0 x 1 x 2

  • xn2=i=|xi| 2 x n i x i , xn1Z(x1<) x n 1 1 x
  • xn2=i=|xi|212 2 x n i x i 2 1 2 , xn2Z(x2<) x n 2 2 x
  • xnp=i=|xi|p1p p x n i x i p 1 p , xnpZ(xp<) x n p p x
  • xn= sup i | x [ i ] | x n sup i | x [ i ] | , xnZ(x<) x n x

For continuous time functions:

  • ftp=|ft|pdt1p p f t t f t p 1 p , ftLpR(ftp<) f t L p p f t
  • (On the interval) ftp=0T|ft|pdt1p p f t t 0 T f t p 1 p , ftLp 0 T (ftp<) f t L 0 T p p f t

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