Mathematics

Interpolation on Lebesgue Spaces

ca.classical-analysis, Expository, fa.functional-analysis No Responses »

Dec 172011

This is the set of notes I wrote up for the talk I gave at the student analysis seminar on December 1 and December 8.

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Let $T:X \to Y$ be an operator between vector spaces satisfying some linearity condition. We fix Banach subspaces $A_0$ and $A_1$ fo $X$ and $B_0$ and $B_1$ of $Y$ . If the restrictions $T|_{A_0}$ and $T|_{A_1}$ are operators into $B_0$ and $B_1$ , respectively, satisfying some norm estimates, then an interpolation theorem tells us that we can often find “intermediate” Banach spaces $(A_t,B_t)$ between $(A_0,B_0)$ and $(A_1,B_1)$ such that $T|_{A_t}$ , for each $t \in [0,1]$ , is a bounded operator into $B_t$ with its norm depending canonically on the norm estimates at the endpoints.

Contents

Lecture 1 – Strong Endpoint Estimates: The theorems of Riesz-Thorin and Stein. If a linear map is bounded as an operator from to and from to , respectively, the the Riesz-Thorin interpolation theorem establishes the boundedness of as an operator from to , where and depend on the exponents and the parameter . As an immediate application, we shall prove the Hausdorff-Young inequality for the Fourier transform and the generalized Young’s inequality. We also mention the variable-parameter version of Riesz-Throin, proven by E. Stein in his 1955 thesis.
Lecture 2 – Weak Endpoint Estimates: The theorems of Marcinkiewicz and Fefferman-Stein. What if an operator just fails to be bounded at the endpoints? In the second part of the talk, we shall consider the weak-type operators, which are bounded as an operator from a Lebesgue space to a weak Lebesgue space. The prototypical examples of such operators are the Hardy-Littlewood maximal function in the theory of differentiation and the Hilbert transform in the theory of singular integrals. We also mention an extension of Stein’s interpolation theorem by C. Fefferman, which allows the endpoint-operator to be bounded from a Lebesgue space to a Hardy space.

Triangular Fourier Series and Physical Reality

ca.classical-analysis, ds.dynamical-systems, la.linear-algebra No Responses »

Oct 192011

This is a transcription of the September 15 talk by Prof. Camil Muscalu at the Courant Institute. Any errors in this post are due to my interpretation of the talk.

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Part 1. Triangular Fourier Series

What is a triangular Fourier series? Let us suppose that $f$ is a $2\pi$ -periodic function on $\mathbb{R}$ . If $f \in L^p([0,2\pi])$ for some $1 < p < \infty$ , we know that the classical Fourier series

$\displaystyle \sum_{n=-\infty}^{\infty} \hat{f}(n) e^{i n x}$

converges to $f(x)$ , both in the $L^p$ -sense (M. Riesz) and in the pointwise almost-everywhere sense (L. Carleson and R. Hunt). These results can be phrased in terms of boundedness of certain operators. For one, the $L^p$ -convergence happens if and only if the operator

$\displaystyle f \mapsto \sum_{-N \leq n \leq N} \hat{f}(n) e^{inx}$

is bounded on $L^p$ for all $N$ . Similarly, the almost-everywhere convergence happens if and only if the operator

$\displaystyle f \mapsto \sup_N \left| \sum_{-N \leq n \leq N} \hat{f}(n) e^{inx} \right|$

is bounded on $L^p$ .

So then, we have a clear notion of the convergence

$\displaystyle \sum_{-N \leq n \leq N} \hat{f}(n) e^{inx} \xrightarrow{N \to \infty} f(x)$ .

Why not take the square, and obtain:

$\displaystyle \left( \sum_{-N \leq n \leq N} \hat{f}(n) e^{inx} \right)^2 \xrightarrow{N \to \infty} f(x)^2$ .

We expand the left-hand side as follows:

$\begin{array}{rcl} \displaystyle \left( \sum_{-N \leq n \leq N} \hat{f}(n) e^{inx} \right)^2 &=& \displaystyle \sum_{-N \leq n_1,n_2 \leq N} \hat{f}(n_1) \hat{f}(n_2) e^{in_1x} e^{in_2x} \\ &=& \displaystyle \sum_{-N \leq n_1 < n_2 \leq N} \hat{f}(n_1)\hat{f}(n_2) e^{in_1x}e^{in_2x} \\ & & + \displaystyle \sum_{-N \leq n_2 < n_1 \leq N} \hat{f}(n_1)\hat{f}(n_2) e^{in_1x}e^{in_2x} \\ & & + \displaystyle \sum_{-N \leq n_1 = n_2 \leq N} \hat{f}(n_1)\hat{f}(n_2) e^{in_1x}e^{in_2x}\end{array}$ .

The last term is the convolution $(f*f)(2x)$ .

We can now ask ourselves the following questions: (1) Does the following convergence happen?

$\displaystyle \sum_{-N \leq n_1 < n_2 \leq N} \hat{f}(n_1) \hat{f}(n_2) e^{in_1x} e^{in_2x} \to \frac{1}{2} (f(x)^2 - (f*f)(2x));$

(2) Similarly, does the following convergence happen?

$\textbf{(1) }\displaystyle \sum_{-N \leq n_1 < n_2 \leq N} \hat{f}(n_1) \hat{g}(n_2) e^{in_1x} e^{in_2x} \to \frac{1}{2}(f(x)g(x) - (f*g)(2x)),$

where $f,g \in L^2([0,2\pi])$ ?

The Hahn-Banach Theorem and Symmetry Invariance II

fa.functional-analysis No Responses »

Oct 112011

This is the set of notes I wrote up for the talk I gave at the student analysis seminar on September 14.

* * * * *

Recall that we adopt the notation of Agnew and Morse [AM38] and write $[\mathscr{L},\rho,f,X]$ to denote the following:

$X$ is a real vector space;
$\rho$ is a sublinear functional on $X$ ;
$f$ is a linear functional defined on a subspace $Y$ of $X$ ;
$f$ is dominated by $\rho$ , viz., $f(x) \leq \rho(x)$ for all $x \in Y$ ;
$\Lambda$ is a collection of endomorphisms on $X$ ;
$\rho$ is invariant under each $\Lambda$ , viz., $\rho ( \Lambda x ) = \rho(x)$ for each $\Lambda \in \mathscr{L}$ ;
$f$ is invariant under each $\Lambda$ , viz., $f(\Lambda x) = f(x)$ for each $\Lambda \in \mathscr{L}$ ;
$Y$ is invariant under each $\Lambda$ , viz., $\Lambda(Y) \subseteq Y$ for each $\Lambda \in \mathscr{L}$ .

Furthermore, if $[\mathscr{L},\rho,f,X]$ is satisfied, we write $\{\mathscr{L},\rho,f,X\}$ to denote the collection of extensions $F$ of $f$ on $X$ such that $F$ is still dominated by $\rho$ , and that

$F ( \Lambda x ) = F(x)$

for all $\Lambda \in \mathscr{L}$ . In this notation, the classical Hahn-Banach theorem can be phrased as follows:

Theorem 1 (Hahn-Banach). Let $X$ be a real vector space, and $\mathscr{L}=\{I\}$ , where $I:X \to X$ is the identity mapping. If $[\mathscr{L},\rho,f,X]$ , then $\{\mathscr{L},\rho,f,X\}$ is nonempty.

Recall that a semigroup is a set with an associative binary operation. In the last talk, we have proved the following symmetry-invariant extension of Hahn-Banach:

Theorem 2 (Agnew-Morse). Let $\mathscr{L}$ be a semigroup of commuting endomorphisms on a real vector space $X$ . If $[\mathscr{L},\rho,f,X]$ , then $\{\mathscr{L},\rho,f,X]$ is nonempty.

For a proof, see pages 25-27 of Lax [Lax02].

We also recall the key example from the last talk. We write $X^*$ to denote the space of all real-valued sequences, and $X_b$ to denote the space $l^\infty(\mathbb{R})$ of all bounded real-valued sequences. The classical Banach limit extend the usual limit in the following manner:

Theorem 3. There exists an operator $L:X_b \to X_b$ , called a generalized limit or a Banach limit, such that

The generalized limit agrees with the usual limit if the usual limit exists;
$L ((x_n)_{n=1}^\infty + (y_n)_{n=1}^\infty) = L( (x_n)_{n=1}^\infty ) + L( (y_n)_{n=1}^\infty )$ ;
$L( (x_n)_{n=1}^\infty ) = L( (x_n)_{n=k}^\infty)$ for any $k \in \mathbb{N}$ ;
$\displaystyle \liminf_{n \to \infty} x_n \leq L( (x_n)_{n=1}^\infty ) \leq \limsup_{n \to \infty} x_n$ .

We now see that the above theorem is an easy consequence of the Agnew-Morse theorem. Indeed, the Agnew-Morse theorem furnishes an extension of the regular limit operator that is invariant under the translation operator. A slight modification of the Agnew-Morse theorem allows us to extend the limit operator even further. Continue reading »

The Hahn-Banach Theorem and Symmetry Invariance I

fa.functional-analysis No Responses »

Sep 232011

This is the set of notes I wrote up for the talk I gave at the student analysis seminar on September 13.

* * * * *

Definition 1. Let $X$ be a real vector space. A sublinear functional on $X$ is a real-valued function $\rho$ on $X$ satisfying

Positive homogeneity. $\rho(a x) = a \rho(x)$ for all $a > 0$ and $x \in X$ ;
Subadditivity. $\rho(x+y) \leq \rho(x)+\rho(y)$ for all $x,y \in X$ .

A linear functional is a sublinear functional that is additive, viz.,

$\rho(x+y) = \rho(x)+\rho(y)$

for all $x,y \in X$ .

Theorem 2 (Hahn, 1927; Banach, 1932). If a linear functional $f$ on a subspace $Y$ of a real vector space $X$ is
dominated by a sublinear functional on $X$ , then there exists an extension of $f$ on $X$ still dominated by the same sublinear functional.

Proof. Extend the linear functional by one dimension and use transfinite induction.

We now discuss a well-known application of the Hahn-Banach theorem, known as Banach limits, which extends the notion of limit to a more general class of sequences.

Definition 3. $X_b = l^\infty(\mathbb{R})$ is the normed linear space of all bounded real-valued sequences with the norm

$\|(x_n)_{n=1}^\infty\| = \sup_{n \in \mathbb{N}} |x_n|$ .

Of course, not every bounded sequence converges. The classical Banach limit extends the notion of limit as follows:

Theorem 4. There exists an operator $L:X_b \to X_b$ , called a generalized limit or a Banach limit, such that

The generalized limit agrees with the usual limit if the usual limit exists;
$L ((x_n)_{n=1}^\infty + (y_n)_{n=1}^\infty) = L( (x_n)_{n=1}^\infty ) + L( (y_n)_{n=1}^\infty )$ ;
$L( (x_n)_{n=1}^\infty ) = L( (x_n)_{n=k}^\infty)$ for any $k \in \mathbb{N}$ ;
$\displaystyle \liminf_{n \to \infty} x_n \leq L( (x_n)_{n=1}^\infty ) \leq \limsup_{n \to \infty} x_n$ .

Proof. We define $p:X_b \to \mathbb{R}$ by setting

$\displaystyle p((x_n)_{n=1}^\infty) = \limsup_{n \to \infty} x_n$

This is a sublinear functional on $X_b$ . If $S$ is the left-translation map

$S((x_n)_{n=1}^\infty) = (x_n)_{n=2}^\infty$ ,

then $p(Sx)=p(x)$ for all $x \in X_b$ . Let $X_c$ be the space of convergent real-valued sequences, which is a linear subspace of $X_b$ . We define a linear functional $l:Y \to \mathbb{R}$ by

$\displaystyle l((y_n)_{n=1}^\infty) = \lim_{n \to \infty} y_n$ .

Then $l(y)=p(y)$ for all $y \in X_c$ , and $l(Sy)=l(y)$ for all $y \in X_c$ . We invoke the Hahn-Banach theorem to furnish an extension $L:X_b \to \mathbb{R}$ of $l$ such that $L$ is dominated by $p$ and that $L$ is invariant under left translation. In particular,

$\displaystyle \liminf_{n \to \infty} x_n \leq L((x_n)_{n=1}^\infty) \leq \limsup_{n \to \infty} x_n$ ,

for each $x \in X_b$ satisfies the inequality $-p(-x) \leq L(x) \leq p(x)$ . It thus follows that $L$ is the desired map. $\square$

Note that one of the key properties of the limit operator was “invariance under the translation operator,” viz.,

$L((x_n)_{n=k}^\infty) = L((x_n)_{n=1}^\infty)$ .

Keeping this in mind, we introduce a new notation, which will bring out the key elements of the Hahn-Banach theorem. The following is a slightly modified version of the notation employed in Agnew and Morse [AM38].

We shall write $[\mathfrak{L},\rho,f,X]$ to denote the following:

$X$ is a real vector space;
$\rho$ is a sublinear functional on $X$ ;
$f$ is a linear functional defined on a subspace $Y$ of $X$ ;
$f$ is dominated by $\rho$ , viz., $f(x) \leq \rho(x)$ for all $x \in Y$ ;
$\Lambda$ is a collection of endomorphisms on $X$ ;
$\rho$ is invariant under each $\Lambda$ , viz., $\rho(\Lambda x) =\rho(x)$ for each $\Lambda \in \mathfrak{L}$ .
$f$ is invariant under each $\Lambda$ , viz., $f(\Lambda x) = f(x)$ for each $\Lambda \in \mathfrak{L}$ ;
$Y$ is invariant under each $\Lambda$ , viz., $\Lambda(Y) \subseteq Y$ for each $\Lambda \in \mathfrak{L}$ .

If $[\mathfrak{L},\rho,f,X]$ , then we write $\{\mathfrak{L},\rho,f,X\}$ to denote the set of extensions $F$ of $f$ on $X$ such that $F$ is still dominated by $\rho$ , and that

$F(\Lambda x) = F(x)$

for all $\Lambda \in \mathfrak{L}$ . With the new shorthand, we can rephrase the classical Hahn-Banach theorem as follows:

Theorem 5 (Hahn-Banach, as stated in [AM38]). Let $X$ be a real vector space, and $\mathfrak{L}=\{I\}$ , where $I:X \to X$ is the identity mapping. If $[\mathfrak{L},\rho,f,X]$ , then $\{\mathfrak{L},\rho,f,X\}$ is nonempty.

The main theorem of the talk is the symmetry-invariant version of the Hahn-Banach theorem known as the Agnew-Morse theorem. The theorem was first proven by A. G. Agnew and A. P. Morse in 1938 [AM38], and was subsequently generalized by E. J. McShane, R. B. Warfield, and V. M. Warfield in 1969 [MWW69]. We state here the minor generalization of the McShane-Warfield-Warfield formulation of the theorem, as stated in [Lax02]:

Theorem 6 (Agnew-Morse, 1937; McShane-Warfield-Warfield, 1969; Lax, 2002). Let $X$ be a real vector space, and $\mathfrak{L}$ be a collection of endomorphisms on $X$ that commute, viz.,

$\Lambda_t \Lambda_s = \Lambda_s \Lambda_t$

for any pair of operators $\Lambda_t$ and $\Lambda_s$ in $\mathfrak{L}$ . If $[\mathfrak{L},\rho,f,X]$ , then $\{\mathfrak{L},\rho,f,X\}$ is nonempty.

Connectedness and Discrete-Valued Maps

gn.general-topology 1 Response »

Aug 032011

I found a neat characterization of connectedness in Bredon’s Topology and Geometry: so nice, in fact, that I am compelled to write up a quick note about it.

Let us recall that a topological space $X$ is disconnected if there exists a disjoint pair of open subsets $U$ and $V$ of $X$ whose union is $X$ , and that $X$ is connected otherwise. A discrete-valued map is a continuous map $d:X \to D$ from a topological space $X$ to a discrete space $D$ , which is a topological space in which every subset is open. An alternate characterization of connectedness is as follows:

Proposition 1 (Alternate characterization of connectedness). A topological space $X$ is connected if and only if every discrete-valued map $d:X \to D$ on $X$ is constant.

The proof is quite simple. If $X$ is connected, then the preimage $d^{-1}(y)$ of an element $y$ in the image of $d$ is nonempty, open, and closed: therefore, $d^{-1}(y)$ must be the whole space. Conversely, if $X$ is not connected, then we can find a disjoint pair of open subsets $U$ and $V$ of $X$ whose union is $X$ , whence the map $d:X \to \{0,1\}$ which is 0 on $U$ and 1 on $V$ is a nonconstant discrete-valued map on $X$ .

Why would anyone think of such an alternate definition? Glen Wilson offered the following perspective, which I very much like. First off, following the “categorical” way of thinking, we submit that describing a collection of object in terms of maps between them—as opposed to the objects themselves—is a good thing. But why constant maps? We can consider “labeling” each connected component in a topological space $X$ by quotienting $X$ out by its connected components. The resulting space $Y$ is a discrete space, and the quotient map $\pi:X \to Y$ is a surjective continuous map that is constant if and only if $Y$ is a singleton. Of course, this happens precisely when $X$ has one connected component, i.e., if $X$ is connected.

This alternate characterization leads us to swift proofs of the key properties of connected space. Let us first consider the following

Proposition 2. The continuous image of a connected space is connected.

Here is a one-liner for the proof: If $X$ is a connected space, $f:X \to Y$ a continuous map, and $d:f(X) \to D$ any discrete-valued map, then the composition $d \circ f:X \to D$ must be constant, and so $f(X)$ is connected. The following triumph of intuition also admits a devastatingly simple proof:

Proposition 3. If a collection of connected sets share a point, then the union is connected.

Here, any discrete-valued map $d$ must be constant on each connected set, and the value of $d$ must be the same because they all share a point: it follows that $d$ is constant on the union. Another illustrative example is as follows:

Proposition 4. If $A$ is a connected subset of a topological space $X$ , and if $B$ is a subset of $X$ such that $A \subseteq B \subseteq \bar{A}$ , then $B$ is connected.

Again, the proof is very short. Any discrete-valued map $d$ on $X$ is a continuous, hence sequentially continuous, map that is constant on $A$ , whence we conclude that the value of $d$ on any limit point of $A$ is the same as the value $d$ takes on $A$ . It follows that $d$ is constant on $\bar{A}$ , hence on $B$ .

Mark H. Kim

Interpolation on Lebesgue Spaces

Triangular Fourier Series and Physical Reality

Part 1. Triangular Fourier Series

The Hahn-Banach Theorem and Symmetry Invariance II

The Hahn-Banach Theorem and Symmetry Invariance I

Connectedness and Discrete-Valued Maps

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Interpolation on Lebesgue Spaces

Triangular Fourier Series and Physical Reality

Part 1. Triangular Fourier Series

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The Hahn-Banach Theorem and Symmetry Invariance I

Connectedness and Discrete-Valued Maps

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