Raising questions, 1-62

Saturday, November 11. 2006

Raising questions, 1-62

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 $X$ and $Y$ and a bounded linear operator $T\in \mathscr{B}(X,Y)$ , the adjoint operator $T^\prime\colon Y^\prime\to X^\prime$ is defined by $T^\prime y^\prime(x)=y^\prime(Tx)$ . I.e., in the end a functional had been evaluated on $x$ . The functional $y^\prime$ on the image has been mapped by $T^\prime$ to a functional on the preimage. For Hilbert spaces $\mathscr{H}$ and $\mathscr{K}$ , given $T\in \mathscr{B}(\mathscr{H},\mathscr{K})$ , $T^\prime\in \mathscr{B}(\mathscr{K}^\prime,\mathscr{H}^\prime)$ and the isomorphisms $\Phi_{\mathscr X}\colon\mathscr{X}\to\mathscr{X}^\prime$ , the adjoint operator $T^*\in\mathscr{B}(\mathscr{K}, \mathscr{H})$ is defined by $T^*:=\Phi_{\mathscr H}^{-1}T^\prime\Phi_{\mathscr K}$ . It can now be written as $\langle Tx,y\rangle_{\mathscr K}=\langle x,T^*y\rangle_{\mathscr H}$ . A remaining question is about the meaning of the mentioned isomorphisms $\Phi_{\mathscr X}$ - 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 $f=\sum_k\langle f,S^{-1}f_k\rangle f_k=\sum_k\langle f,f_k\rangle S^{-1}f_k$ . The remaining question is how $S^{-1}$ comes into the inner product.
What is a unitary operator?
$T\in\mathscr{B}(\mathscr{H},\mathscr{K})$ yields $TT^*=\operatorname{Id}_{\mathscr K}$ and $T^*T=\operatorname{Id}_{\mathscr H}$ .
What does it mean when windows $g_{m,n}$ are (in-)complete in $\Lsp^2(\mathbb R)$ ?
A sequence is complete if it spans the whole space.
To what extent are the dual windows $\gamma_{m,n}$ not uniquely defined?
There are several dual sequences possible to reconstruct $f=\sum_{m,n}\langle f,g_{m,n}\rangle\gamma_{m,n}$ , but only the canonical dual $\{S^{-1}g_{m,n}\}_{m,n}$ has minimal $\ell^2$ -norm.
What is a Weyl-Heisenberg system?
Another name for a Gabor system.
What does “locally compact” mean?
What’s a separable TF-lattice 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 $\mathscr E$ of a Hilbert space $\mathscr H$ implies that $\|x\|^2=\sum_{e\in\mathscr E}|\langle x,e\rangle|^2 \quad\forall x\in\mathscr H$ .
Why are bounds used in the frame definition? Why is $0<\sum_k|\langle f,f_k\rangle|^2<\infty$ 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 $\Lsp^2$ and e.g. not $\Lsp^3$ ? The same goes with $\ell^2$ .
Well, $\Lsp^p$ and $\ell^p$ for $p\neq2$ are no Hilbert spaces! The inner product is essential for frame theory. What qualifies $\Lsp^2$ compared to other Hilbert spaces is that it is a function space. But there are sometimes more suitable function spaces which are dense in $\Lsp^2$ , e.g. the Schwartz space $\mathscr S$ , as it is FT-invariant. A similar statement is valid for the sequence space $\ell^2$ —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 zero-sets. $\Lsp^\infty$ is the space of essentially bounded functions.
What does $e^{2\pi\i x\cdot\omega}$ look like for $x, \omega\in\bbR^d$ ?
Note that $x\cdot\omega$ is the inner product on $\bbR^d$ and therefore always real. So $e^{2\pi\i x\cdot\omega}$ is staying the circular function with values on the unit circle. The remaining question is how the inner product $u(x,\omega):=x\cdot\omega$ behaves as a function $\bbR^{2d}\to\bbR$ , e.g., when is it in $\bbZ$ 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: $\int fg = Fg-\int Fg^\prime$ . (BTW, this follows by integrating the well-known equation $(fg)^\prime=f^\prime g+fg^\prime$ .) 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 $\langle f^\prime,\varphi\rangle=-\langle f,\varphi^\prime\rangle$ for $\varphi\in\Csp_c(\Omega)$ ?
The answer has already been given in 17).
Why is $\|\mathscr{F}f\|_\infty\leq\frac{1}{(2\pi)^{d/2}}\|f\|_{\Lsp^1}$ and why should this be clear?
Well, look at the definition of the FT: It’s an integral of a product of a function $f(x)$ with an exponential $e^{\i z}$ . The absolute value of that product only leaves $|f(x)|$ , as $|e^{\i z}|=1\quad\forall z\in\bbC$ . The inequality is finally just the triangular inequality, and on the left-hand side the supreme is taken. Note that this inequality means that the FT is essentially bounded only for $\Lsp^1$ -functions. The remaining question is how the prefactor is arising, see 56).
When does $f^{-1}$ 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 $y=Ax$ yield $dy=|\det A|dx$ ?
Why is $\arg(f(t)e^{-\i\xi t})$ “nearly constant” on $I$ if $f$ circles around the origin with frequency $\xi$ in the time-interval $I$ ?
Note: $z=|z|e^{\i\arg z} \quad\forall z\in\bbC$ (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 self-adjoint operator real?
Trivial: $\lambda\langle x,x\rangle=\ldots=\overline{\lambda}\langle x,x\rangle$ .
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 square-root of operators? Why/when is $\operatorname{Id}=S^{-1/2}SS^{-1/2}$ ?
Why do the coefficients of Gabor systems always have to be in $\ell^2$ ?
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 $\Csp_b(X)$ and the dual $X^*$ ?
$\Csp_b(X)$ are the bounded continuous functions on $X$ , whereas $X^*=\mathscr{B}(X,\bbF)$ denotes the bounded linear functionals on $X$ , where linearity also implies continuity.
What’s the difference between $A+B$ and $A\oplus B$ when talking about direct sums?
What does $X$ being “totally bounded” mean?
Some kind of $X\subseteq\bigcup^n B_j$ .
What’s a linear manifold?
What’s a Hamel basis? Why is a Hamel basis for an infinitely-dimensional Hilbert space uncountable?
What’s a homeomorphism?
What’s the connection between the FT of an $f\in\Lsp^1(\bbR)$ and the FS of $f\in\Lsp^2(\bbT), \,\bbT=\bbR/\bbZ$ ?
What’s $S^{-1/2}$ ? To what extent is it “half way between $S^0$ and $S^{-1}$ ”? Why is it a limit of polynomials in $S^{-1}$ 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 trace-class operators?
What are Hilbert-Schmidt operators?
What are Sech and Hermite functions/windows?
What are B-splines? What are the orders of B-plines?
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 $\Lsp^1$ a Banach algebra?
Where does the factor $\frac{1}{\sqrt{2\pi}}$ in some definitions of the FT come from and why does it disappear when using $e^{-2\pi\i x\cdot\omega}$ instead of $e^{-\i x\cdot\omega}$ ?
Why are the isomorphisms $\Phi_{\mathscr X}\colon\mathscr{X}\to\mathscr{X}^\prime$ of 1) only available/usable for Hilbert spaces?

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