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

Orthogonal Bases

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

Definition 1

A collection of vectors BB in an inner product space VV is called an orthogonal basis if

1. span ( B ) = V span ( B ) = V
2. vivjvivj (i.e., vi,vj=0vi,vj=0) ijij

If, in addition, the vectors are normalized under the induced norm, i.e., vi=1ivi=1i , then we call VV an orthonormal basis (or “orthobasis”). If VV is infinite dimensional, we need to be a bit more careful with 1. Specifically, we really only need the closure of span (B) span (B) to equal VV. In this case any xVxV can be written as

x = i = 1 c i v i x = i = 1 c i v i
(1)

for some sequence of coefficients {ci}i=1.{ci}i=1.

(This last point is a technical one since the span is typically defined as the set of linear combinations of a finite number of vectors. See Young Ch 3 and 4 for the details. This won't affect too much so we will gloss over the details.)

Example 1

• V=R2V=R2, standard basis
v1=10v2=01v1=10v2=01
(2)

Example 2

• Suppose V={ piecewise constant functions on [0,14),[14,12),[12,34),[34,1]}V={ piecewise constant functions on [0,14),[14,12),[12,34),[34,1]}. An example of such a function is illustrated below. Consider the set The vectors {v1,v2,v3,v4}{v1,v2,v3,v4} form an orthobasis for VV.
• Suppose V=L2[-π,π]V=L2[-π,π]. B={12πejkt}k=-B={12πejkt}k=-, i.e, the Fourier series basis vectors, form an orthobasis for VV. To verify the orthogonality of the vectors, note that:
12πejkt,12πejkt=12π-ππej(k1-k2)t=12πej(k1-k2)tj(k1-k2)-ππ=12π·-1+1j(k1-k2)=0(k1k2)12πejkt,12πejkt=12π-ππej(k1-k2)t=12πej(k1-k2)tj(k1-k2)-ππ=12π·-1+1j(k1-k2)=0(k1k2)
(3)
See Young for proof that the closure of BB is L2[-π,π]L2[-π,π], i.e., the fact that anyfL2[-π,π]fL2[-π,π] has a Fourier series representation.

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