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Orthobasis Expansions

Module by: Mark A. Davenport. E-mail the author

Suppose that the vjj=1Nvjj=1N are a finite-dimensional orthobasis. In this case we have

x ^ = ∑ j = 1 N x , v j v j . x ^ = ∑ j = 1 N x , v j v j .
(1)

But what if x∈ span (vj)=Vx∈ span (vj)=V already? Then we simply have

x = ∑ j = 1 N x , v j v j x = ∑ j = 1 N x , v j v j
(2)

for all x∈Vx∈V. This is often called the “reproducing formula”. In infinite dimensions, if VV has an orthobasis vjj=1∞vjj=1∞ and x∈Vx∈V has

∑ j = 1 ∞ x , v j 2 < ∞ ∑ j = 1 ∞ x , v j 2 < ∞
(3)

then we can write

x = ∑ j = 1 ∞ x , v j v j . x = ∑ j = 1 ∞ x , v j v j .
(4)

In other words, xx is perfectly captured by the list of numbers x,v1,x,v2,...x,v1,x,v2,...

Sound familiar?

Example 1

  • V=CnV=Cn, vkvk is the standard basis.
    xk=x,vkvk.xk=x,vkvk.
    (5)
  • V=L2-π,πV=L2-π,π, vkt=12πejktvkt=12πejkt For any f∈Vf∈V we have
    ft= ∑ k=-∞ ∞ cxvxft= ∑ k=-∞ ∞ cxvx
    (6)
    where
    ck=f,vk=12π∫-ππfte-jktdt.ck=f,vk=12π∫-ππfte-jktdt.
    (7)

The general lesson is that we can recreate a vector xx in an inner product space from the coefficients x,vkx,vk. We can think of x,vkx,vk as “transform coefficients.”

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