A collection of vectors BB in an inner product space VV is called an
orthogonal basis if
-
span
(
B
)
=
V
span
(
B
)
=
V
- vi⊥vjvi⊥vj (i.e., 〈vi,vj〉=0〈vi,vj〉=0) ∀i≠j∀i≠j
If, in addition, the vectors are normalized under the induced norm, i.e., ∥vi∥=1∀i∥vi∥=1∀i , 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 x∈Vx∈V 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.)
- 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(k1≠k2)12πejkt,12πejkt=12π∫-ππej(k1-k2)t=12πej(k1-k2)tj(k1-k2)-ππ=12π·-1+1j(k1-k2)=0(k1≠k2)
(3)
![Illustrations of the Haar basis functions. v_1 is the all constant function, i.e., it is 1 on [0,1].](07_2a.png)
![Illustrations of the Haar basis functions. v_2 is 1 on [0,0.5) and -1 on [0.5,1].](07_2b.png)
![Illustrations of the Haar basis functions. v_3 is sqrt(2) on [0,0.25), -sqrt(2) on [0.25,0.5), and 0 on [0.5,1].](07_2c.png)
![Illustrations of the Haar basis functions. v_4 is 0 on [0,0.5), sqrt(2) on [0.5,0.75), and -sqrt(2) on [0.75,1].](07_2d.png)