The standard orthonormal basis (ONB) in the Hilbert space consists of the vectors
(1, 0, 0, 0, …)
(0, 1, 0, 0, …)
(0, 0, 1, 0, …)
Let S be the forward shift operator: . The aforementioned ONB is precisely the orbit of the first basis vector under the iteration of S. Are there any other vectors x whose orbit under S is an ONB?
If one tries to construct such x by hand, taking some finite linear combination of , this is not going to work. Indeed, if the coefficient sequence has finitely many nonzero terms, then one of them, say, , is the last one. Then is not orthogonal to because the inner product is and that is not zero.
However, such vectors x exist, and arise naturally in complex analysis. Indeed, to a sequence we can associate a function . The series converges in the open unit disk to a holomorphic function which, being in the Hardy space , has boundary values represented by an square-integrable function on the unit circle . Forward shift corresponds to multiplication by . Thus, the orthogonality requires that for every the function be orthogonal to in . This means that is orthogonal to all such ; and since it’s real, it is orthogonal to for all by virtue of conjugation. Conclusion: |f| has to be constant on the boundary; specifically we need |f|=1 a.e. to have a normalized basis. All the steps can be reversed: |f|=1 is also sufficient for orthogonality.
So, all we need is a holomorphic function f on the unit disk such that almost all boundary values are unimodular and f(0) is nonzero; the latter requirement comes from having to span the entire space. In addition to the constant 1, which yields the canonical ONB, one can use
- A Möbius transformation where .
- A product of those (a Blaschke product), which can be infinite if the numbers converge to the boundary at a sufficient rate to ensure the convergence of the series.
- The function which is not a Blaschke product (indeed, it has no zeros) yet satisfies for all .
- Most generally, an inner function which is a Blaschke product multiplied by an integral of rotated versions of the aforementioned exponential function.
Arguably the simplest of these is the Möbius transformation with ; expanding it into the Taylor series we get
Thus, the second simplest ONB-by-translations after the canonical one consists of
(-1/2, 3/4, 3/8, 3/16, 3/32, 3/64, …)
(0, -1/2, 3/4, 3/8, 3/16, 3/32, …)
(0, 0, -1/2, 3/4, 3/8, 3/16, …)
and so on. Direct verification of the ONB properties is an exercise in summing geometric series.
What about the exponential one? The Taylor series of begins with
I don’t know if these coefficients in parentheses have any significance. Well perhaps they do because the sum of their squares is . But I don’t know anything else about them. For example, are there infinitely many terms of either sign?
Geometrically, a Möbius transform corresponds to traversing the boundary circle once, a Blaschke product of degree n means doing it n times, while the exponential function, as well as infinite Blaschke products, manage to map a circle onto itself so that it winds around infinitely many times.
Finally, is there anything like that for the backward shift ? The vector is orthogonal to if and only if is orthogonal to , so the condition for orthogonality is the same as above. But the orbit of any vector under tends to zero, thus cannot be an orthonormal basis.