Saturday, November 11. 2006
When I started on my way to my Masters degree seriously in January 2006, several questions began to arise in the course of time, and they continue to do so. While there were some very fundamental questions at the beginning (because my last contact to FA and basic analysis had been quite some time ago), they matured to some more advanced questions today. Here, I give a list of them, in their order of appearance. The answers, however, and as far as they have already been provided anyway, did not come up in the same order. Some questions are so general that they are to be understood as a challenge to look into the books and get familiar with it.
This list will continue to grow in separate entries. Also, for the sake of documentation, currently missing answers will only be given separately.

What is the adjoint of an operator?
See Functional Analysis. In detail: For normed spaces and and a bounded linear operator , the adjoint operator is defined by . I.e., in the end a functional had been evaluated on . The functional on the image has been mapped by to a functional on the preimage. For Hilbert spaces and , given , and the isomorphisms , the adjoint operator is defined by . It can now be written as . A remaining question is about the meaning of the mentioned isomorphisms  see 62).

What happens to an operator if it is moved into/out of an inner product?
This question is nonsense. The confusion came from the notation . The remaining question is how comes into the inner product.

What is a unitary operator?
yields and .
 What does it mean when windows are (in)complete in ?
A sequence is complete if it spans the whole space.

To what extent are the dual windows not uniquely defined?
There are several dual sequences possible to reconstruct , but only the canonical dual has minimal norm.

What is a WeylHeisenberg system?
Another name for a Gabor system.

What does “locally compact” mean?

What’s a separable TFlattice and what’s so special about it?

What’s a separable Hilbert space and what’s so special about it?
Separable means that there’s a countable dense subset. Now every element can be approximated by finite linear combinations of elements in that subset. This is true for every Banach space.

What properties do bases of infinitely dimensional spaces have compared to those of spaces with finite dimension?

What’s “Parseval’s formula for orthonormal bases”?
An orthonormal basis of a Hilbert space implies that .

Why are bounds used in the frame definition? Why is not enough?
For the properties of the frame operator, this is enough, but it is often convenient to have estimates for the bounds, as they define the redundancy of the frame and therefore important properties. The remaining question is what exactly that means  see 55).

Why do we always talk about the special case and e.g. not ? The same goes with .
Well, and for are no Hilbert spaces! The inner product is essential for frame theory. What qualifies compared to other Hilbert spaces is that it is a function space. But there are sometimes more suitable function spaces which are dense in , e.g. the Schwartz space , as it is FTinvariant. A similar statement is valid for the sequence space —look at Parseval’s formula or the frame definition!

What does “essentially bounded” mean?
A property is “essentially” valid if it is only hurt on zerosets. is the space of essentially bounded functions.

What does look like for ?
Note that is the inner product on and therefore always real. So is staying the circular function with values on the unit circle. The remaining question is how the inner product behaves as a function , e.g., when is it in such that the circle function has value = 1 (one period finished)?

What is an example of a continuous function which is not uniformly continuous?

What’s so special about continuous functions with compact support? Where’s the connection to partial integration? What actually is partial integration?
Well, partial integration occurs when integrating a product of functions: . (BTW, this follows by integrating the wellknown equation .) For functions with compact support the integral vanishes at the borders (at infinity), because the function equals zero from a certain point on. Therefore you only need to do finite calculations.

What’s the Gaussian integral theorem?

What are commonized (=weak) derivatives?
This is a question of distribution theory.

Why is for ?
The answer has already been given in 17).

Why is and why should this be clear?
Well, look at the definition of the FT: It’s an integral of a product of a function with an exponential . The absolute value of that product only leaves , as . The inequality is finally just the triangular inequality, and on the lefthand side the supreme is taken. Note that this inequality means that the FT is essentially bounded only for functions. The remaining question is how the prefactor is arising, see 56).

When does exist and when is it continuous?

Why are absolutely convergent series so special?
The order of the summands does not matter, i.e., absolute convergence implies unconditional convergence.

What is the functional determinant? Why does yield ?

Why is “nearly constant” on if circles around the origin with frequency in the timeinterval ?
Note: (polar representation).

Find visualizations and interpretations of the convolution of functions.

What are holomorphic functions?

What are Taylor series?

Why are the eigenvalues of a selfadjoint operator real?
Trivial: .

What is a regular operator/matrix? What’s so special about those?

What is a positive (semi)definite operator/matrix? What’s so special about those?

When are FS absolutely convergent?

Recall basic terms of linear algebra: Jordan normal form, determinants, invertibility, bases, SVD, Vandermonde matrix/determinant, symmetric matrices, positive (semi)definite matrices.

What’s the squareroot of operators? Why/when is ?

Why do the coefficients of Gabor systems always have to be in ?
This is essential for frame theory! Look at the frame definition.

Why are operators invertible if they are bounded below?
This implies that the kernel of the operator just consists of the zero, which implies injectivity and therefore invertibility on the image.

What’s the difference between and the dual ?
are the bounded continuous functions on , whereas denotes the bounded linear functionals on , where linearity also implies continuity.

What’s the difference between and when talking about direct sums?

What does being “totally bounded” mean?
Some kind of .

What’s a linear manifold?

What’s a Hamel basis? Why is a Hamel basis for an infinitelydimensional Hilbert space uncountable?

What’s a homeomorphism?

What’s the connection between the FT of an and the FS of ?

What’s ? To what extent is it “half way between and ”? Why is it a limit of polynomials in in the strong operator topology?

What’s the connection between dual frames and dual lattices?

What are Besov spaces?

What’s weak*convergence and what’s so special about that?

What is weak*continuity?

What are traceclass operators?

What are HilbertSchmidt operators?

What are Sech and Hermite functions/windows?

What are Bsplines? What are the orders of Bplines?

What are Wilson bases and what are they good for?

What is the Frobenius norm of a matrix, and what’s its use?

What’s the redundancy of a frame and why is it important?

What are Segal algebras?

What is the canonical tight window?
It seems that this is that dual window such that the Gabor frame using the same lattice is tight. But why is it unique?

Why is the FT of the canonical dual of a “too wide” Gaussian window equal to the canonical dual of a “too narrow” Gaussian?

Why should one want to have the dual frame tight?
Maybe 5.7 in Christensen tells more, and 9.3.2, 9.3.4 and 9.5. You don’t have to compute the inverse of the frame operator.

Why is a Banach algebra?

Where does the factor in some definitions of the FT come from and why does it disappear when using instead of ?

Why are the isomorphisms of 1) only available/usable for Hilbert spaces?

In question 15 I want to make a picture how looks like for . I want to take at least twodimensional vectors, as the inner product is too trivial otherwise. But then is a function on , which is rather difficult to plot. So I want to fix and plot , whic
Tracked: Feb 21, 13:22
What is ? What are singular matrices? What is the condition number of a matrix? (Continued from questions 162)
Tracked: Feb 21, 13:24
(Q15) I’m gonna show you some cool Gabor atoms on . First, the function , as described previously, indeed looks like this: [xx yy] = meshgrid(linspace(2,2,100)); v=[1 1.5]; e1 = exp&#
Tracked: Feb 21, 13:26
What are Fourier multipliers? What are Frechet spaces? Why are and not Banach, but only Frechet? Is there a difference between and in HGFei’s notation? Why is ? (There’s a relation to Q2.)
Tracked: Feb 21, 13:27
(Q26) Meanwhile, I know that a convolution of one function with another function means that will be “smeared”, and vice versa. I visualized some convolutions using Octave. I was prepared to implement for two vectors, but then stepped over
Tracked: Feb 21, 13:29
(Q34+44) If an operator can be written as , then we abbreviate . In the mapping sequence , clearly “lies between” and . We know that , so far existant, and so . So we can abbreviate , and . It is now clear that . It remains to show that i
Tracked: Feb 21, 13:29