%%  function-spaces.tex
%%  Approximation Theory: 
%%  Exercises on Function Spaces (Normed Spaces)
%%  Neal L. Carothers
%%  Bowling Green State University
%%  Bowling Green, Ohio  43403
%%  carother@math.bgsu.edu
%%  http://www.bgsu.edu/~carother/


\input 680-setup.tex

\parskip = 6 true pt minus 2 true pt

\chaptertitle = {Function Spaces}

\centerline{\hfil\tf Function Spaces\hfil}
\vskip-\baselineskip
\line{\sc Math 680 \hfil 6/27/94}

\noindent
The most important collection of
functions for our purposes is the space  $C[\,a,b\,]$,
consisting of all continuous functions  $f : [\,a,b\,]\to \R$.  
It's easy to see that 
$C[\,a,b\,]$ is a vector space under the usual pointwise
operations on functions: $(f+g)(x)=f(x)+g(x)$ and 
$(\alpha f)(x)=\alpha f(x)$ for $\alpha\in\R$.
Actually, we will be most interested in the
finite-dimensional subspaces  ${\cal P}_n$  of  $C[\,a,b\,]$,
consisting of all {\sl algebraic polynomials\/} of degree
at most  $n$.  

\prob{\cmark}{%
Each  ${\cal P}_n$  has dimension exactly  
$n +1$.  Why?
}

\noindent
Another useful subset of  $C[\,a,b\,]$  is the
collection  ${\rm lip}_K\alpha$, consisting of all those 
$f$'s  which satisfy a {\sl Lipschitz condition\/} of order 
$\alpha > 0$  with constant  $0<K<\infty$; i.e., those
$f$'s for which
$|f(x) - f(y)|\le K\,|x - y|^\alpha$  for all 
$x$, $y$  in  $[\,a,b\,]$. 
[Some authors would say that $f$ is {\sl H\"older
continuous\/} with exponent  $\alpha$.]

\prob{\gmark}{%
\vskip-\baselineskip
\vskip-\parskip
\itemitem{(a)}  
Show that
${\rm lip}_K\alpha$ is, indeed, a subset of  $C[\,a,b\,]$. 
\itemitem{(b)}  
If  $\alpha > 1$, show that ${\rm lip}_K\alpha$
contains only the constant functions.
\itemitem{(c)}
Show that  $\sqrt{�x }$ is in  ${\rm lip}_1(1/2)$  and that  
$\sin��x$ is in  ${\rm lip}_1 1$  on  $[\,0,1\,]$.
\itemitem{(d)}
Show that the collection  ${\rm lip}\,\alpha$, consisting of all those 
$f$'s  which are in  ${\rm lip}_K\alpha$  for some  $K$, is a
subspace of  $C[\,a,b\,]$.
\itemitem{(e)}
Show that  ${\rm lip}\,1 $ contains all the polynomials.
\itemitem{(f)}
If $f\in{\rm lip}\,\alpha$ for some $\alpha>0$, show that 
$f\in{\rm lip}\,\beta$ for all $0<\beta<\alpha$.
\itemitem{(g)}
Given $0<\alpha<1$, show that $x^\alpha$ is in ${\rm lip}_1\alpha$ on 
$[\,0,1\,]$ but {\sl not\/} in ${\rm lip}\,\beta$ for any 
$\beta>\alpha$.\par
}

\noindent
We will also be interested in the collection
of all {\sl trigonometric polynomials}.  A (real) trigonometic
polynomial, or trig polynomial for short, is a function
of the form
$$a_0+\sum_{k=1}^n\bigl(\,a_k\cos kx + b_k\sin kx\,\bigr),\eqno{(\ast)}$$
where $a_0,\ldots,a_n$ and $b_1,\ldots,b_n$ are real numbers.
The {\sl degree\/} of a trig polynomial is the highest frequency 
occurring in any representation of the form $(\ast)$; thus, $(\ast)$ 
has degree $n$ provided that one of $a_n$ or $b_n$ is nonzero.  
We will use ${\cal T}_n$ to denote the collection of trig polynomials
of degree at most $n$.
In this case it is convenient to take the space of all 
continuous $2\pi$-periodic functions on $\R$ as the
containing space for ${\cal T}_n$;  a space we denote by $C^{2\pi}$. 
The space $C^{2\pi}$ has several equivalent descriptions.
For one, it's obvious that $C^{2\pi}$ is a subspace of
$C(\R)$, the space of all continuous functions on $\R$.
But we might also consider $C^{2\pi}$ as a subspace of
$C[\,0,2\pi\,]$ in the following way: The $2\pi$-periodic
continuous functions on $\R$ may be identified with the
set of functions $f\in C[\,0,2\pi\,]$ satisfying 
$f(0) =�f(2\pi)$.  Each such $f$ extends to a 
$2\pi$-periodic element of $C(\R)$ in an obvious way,
and it's not hard to see that the condition $f(0) = f(2\pi)$ 
defines a subspace of $C[\,0,2\pi\,]$.
As a third description, it is often convenient to identify  
$C^{2\pi}$ with the collection of continuous real-valued 
functions on  $\T$, where  $\T$  is the unit circle in the 
complex plane $\C$.  That is, we simply make the identifications
$$\theta\longleftrightarrow e^{i\theta}\qquad\hbox{and}\qquad
	f(\theta)\longleftrightarrow f(e^{i\theta}).$$

\prob{\cmark}{%
\vskip-\baselineskip
\vskip-\parskip
\itemitem{(a)}  
By using the recurrence formulas  
$\cos kx+\cos(k-2)x = 2\cos(k-1)x\,\cos x$ and   
$\sin(k+1)x-\sin(k-1)x = 2 \cos kx\,\sin x$, show
that each of the functions  $\cos kx$  and  
$\sin(k+1)x/\sin x$ 
may be written as algebraic polynomials of degree
exactly  $k$  in  $\cos x$.  In each case, what is the  
coefficient of $\cos^kx$\/?
\itemitem{(b)}
Equivalently, use the binomial formula to write the real 
and imaginary parts of $(\cos�x + i\sin�x)^n=\cos nx + i\sin nx$
as algebraic polynomials in $\cos x$ and $\sin x$.  
Again, what are the leading coefficients of these polynomials?
\itemitem{(c)}
If $P(x,y)$ is an algebraic polynomial
(in two variables) of degree at most $n$, show that
$P(\cos x,\sin x)$ may be written as $Q(\cos x)+R(\cos x)\sin x$,
where $Q$ and $R$ are algebraic polynomials (in one variable)
of degrees at most $n$ and $n-1$, respectively.
\itemitem{(d)}
Show that $\cos^nx$ can be written as a linear combination
of the functions $\cos kx$, $k=1,\ldots,n$, and that 
$\cos^{n-1}x\sin x$ can be written as a
linear combinations of the functions $\sin kx$, $k=1,\ldots,n$.
Thus, each polynomial $P(\cos x,\sin x)$
in $\cos x$ and $\sin x$ can be written in the form $(\ast)$.  
\itemitem{(e)}
If $(\ast)$ represents an even function, show that it can
be written using only cosines.  Conversely, if $P(x,y)$ is
an even polynomial, show that $P(\cos x,\sin x)$ can be
written using only cosines.\par
}

\prob{}{%
Show that ${\cal T}_n$ has dimension $2n+1$ (as a vector 
space over $\R$\/).
}

\prob{}{%
We might also consider complex trig polynomials; that is,
functions of the form $(\ast)$, but now allowing the
$a_k$'s and $b_k$'s to be complex.  
\itemitem{(a)}
Show that every trig
polynomial, whether real or complex, may be written as
$$\sum_{k=-n}^nc_ke^{ikx},\eqno(\ast\ast)$$
where the  $c_k$'s  are complex.  Thus, 
complex trig polynomials are just algebraic
polynomials in  $z$  and  $\bar z$,  where  $z =  e^{ix}\in \T$.
\itemitem{(b)}
If $(\ast\ast)$ represents a real-valued function, 
show that it may be written as a real trig polynomial; that is, 
it may be written in the form $(\ast)$ using real coefficients.
\itemitem{(c)}
Show that $(\ast\ast)$ is real-valued 
if and only if $\bar{c}_k=c_{-k}$ for any $k$.\par
}

\noindent
We will also want to consider a norm on the vector
space  $C[\,a,b\,]$;  we typically use the {\sl uniform\/}
or {\sl sup norm\/} (Rivlin calls this the {\sl Chebyshev
norm\/}) defined by  $\Vert f\Vert=\Max_{a\le x\le b}|f(x)|$.
[Some authors write  $\Vert f\Vert_u$ or $\Vert f\Vert_\infty$.]  

\prob{\gmark}{%
Show that ${\cal P}_n$  and  ${\rm lip}_K�\alpha$  
are closed subsets of  $C[\,a,b\,]$ (under the sup norm).  
Is ${\rm lip}\,\alpha$ closed?
A bit harder:  Show that  ${\rm lip}\,1$  is 
both first category and dense in  $C[\,a,b\,]$.  
}

\prob{\cmark\gmark}{%
Fix $n$ and consider the norm $\Vert p\Vert_1=\sum_{k=0}^n|a_k|$
for $p(x)=a_0+a_1x+\cdots+a_nx^n\in{\cal P}_n$.
Show that there are constants $0<A_n,B_n<\infty$ such that 
$A_n\Vert p\Vert_1\le\Vert p\Vert\le B_n\Vert p\Vert_1$.
Do $A_n$ and $B_n$ really depend on $n$\/?
}

\noindent
We will occasionally consider spaces of real-valued functions 
defined on finite sets;  that is, we will 
consider  $\R^n$  under various norms. (Why is this the
same?)  We define a scale of norms on  $\R^n$  by 
$\Vert x\Vert_p=\left(\sum_{i=1}^n|x_i|^p\right)^{1/p}$,
where $x=(x_1,\ldots,x_n)$ and $1\le p<\infty$
(we need $p\ge1$ in 
order for this expression to be a legitimate norm,
but the expression makes perfect sense
for any  $p > 0$, and even for  $p < 0$ provided no  $x_i$  is 
$0$).  Notice, please, that the usual norm on  $\R^n$  is
given by  $\Vert x\Vert_2$.  

\prob{}{%
Show that  $\Lim_{p\to\infty}\Vert x\Vert_p=\Max_{1\le i\le n}|x_i|$.
For this reason we define $\Vert x\Vert_\infty=\Max_{1\le i\le n}|x_i|$.
Thus $\R^n$, under the norm $\Vert \cdot\Vert_\infty$, 
is the same as  $C(\{1,2,\ldots,n\})$, with its usual norm. 
}

\prob{}{%
Assuming $x_i\ne0$ for $i=1,\ldots,n$, compute
$\Lim_{p\to0+}\Vert x\Vert_p$ and $\Lim_{p\to-\infty}\Vert x\Vert_p$.
}

\prob{}{%
Consider $\R^2$ under the norm $\Vert x\Vert_p$.  Draw the graph 
of the unit sphere  $\{ x :  \Vert x\Vert_p= 1 \}$ for various 
values of $p$ (especially $p = 1$, $2$, $\infty$).
}

\prob{\gmark}{%
(Young's inequality):
Let  $1<p<\infty$  and let $q$ satisfy ${1\over p}+{1\over q}=1$.
Show that $ab\le{1\over p}\,a^p+{1\over q}\,b^q$ for
all $a$, $b\ge0$ with equality if and only if $a^p=b^q$.
}

{\baselineskip=26truept\prob{\gmark}{%
(H\"older's inequality):
Let  $1<p<\infty$  and let $q$ satisfy ${1\over p}+{1\over q}=1$.
Show that
\itemitem{(a)}
$\sum_{i=1}^n|a_i\,b_i|\le\left(\sum_{i=1}^n|a_i|^p\right)^{1/p} 
	\left(\sum_{i=1}^n|b_i|^q\right)^{1/q}$, and 
\itemitem{(b)}
$\int_a^b|f(x)\,g(x)|\,dx\le\left(\int_a^b|f(x)|\,dx\right)^{1/p}
	\left(\int_a^b|g(x)|\,dx\right)^{1/p}$.
\vskip0pt
Describe the case for equality in each inequality.
What happens if $p=1$ (and $q=\infty$)?\par}}

{\baselineskip=26truept\prob{\gmark}{%
(Minkowski's inequality):
For  $1\le p<\infty$, show that  
\itemitem{(a)}
$\left(\sum_{i=1}^n|a_i+b_i|^p\right)^{1/p}\le 
	\left(\sum_{i=1}^n|a_i|^p\right)^{1/p}+\left(\sum_{i=1}^n|b_i|^p\right)^{1/p}$
and that
\itemitem{(b)}
$\left(\int_a^b|f(x)+g(x)|\,dx\right)^{1/p}\le
	\left(\int_a^b|f(x)|\,dx\right)^{1/p}+\left(\int_a^b|g(x)|\,dx\right)^{1/p}$.
\vskip0pt
Describe the case for equality in each inequality.
What happens if $p=\infty$?\par}}

\noindent
Exercise {\bf 13} shows that  $\Vert\cdot\Vert_p$
is indeed a norm for  $1\le p < \infty$.  We write 
$L_p[\,a,b\,]$  to mean the vector space of functions on  $[\,a,b\,]$ 
for which the integral norm is defined and finite, we
write  $\ell_p^n$ to mean the vector space of sequences
of length  $n$; that is, $\R^n$ supplied with the norm
$\Vert\cdot\Vert_p$, and we write $\ell_p$ to mean
the vector space of infinite sequences $x=(x_n)_{n=1}^\infty$
for which $\Vert x\Vert_p<\infty$.  In each space, the usual 
algebraic operations are defined pointwise (or coordinatewise)
and the norm is understood to be $\Vert\cdot\Vert_p$.

\noindent
A normed space $(X,\Vert\cdot\Vert)$ is said to be {\sl strictly convex\/}
if $\Vert x+y\Vert=\Vert x\Vert+\Vert y\Vert$ always implies that
$x$ and $y$ are lie in the same direction; that is, either $x=\alpha y$ or 
$y=\alpha x$ for some nonnegative scalar $\alpha$.  Equivalently, the triangle 
inequality is strict on nonparallel vectors.

\prob{}{%
Prove that the following are equivalent:
\itemitem{(a)}
$(X,\Vert\cdot\Vert)$ is strictly convex.
\itemitem{(b)}
If $x$, $y\in X$ are nonparallel, then
$\displaystyle\left\Vert {{x+y}\over2}\right\Vert<{{\Vert x\Vert+\Vert y\Vert}\over2}$.
\itemitem{(c)}
If $x\ne y\in X$ with $\Vert x\Vert=1=\Vert y\Vert$,
then
$\displaystyle\left\Vert {{x+y}\over2}\right\Vert<1$.
}

\prob{}{%
Show that $L_p$ and $\ell_p$ are 
strictly convex for $1 < p < \infty$.  Show also that this fails
in case $p = 1$.  [Hint: This is actually a statement about the function
$|t|^p$, $1<p<\infty$.]
}

\noindent
Strictly convex spaces are of interest when considering
the problem of {\sl nearest points\/}:  Given a nonempty subset
$K$ of a normed space $X$ and a point
$x\notin K$, we ask whether there is a best approximation
to $x$ from elements of $K$; that is, we want to know if
there exist one or more points $y_0\in K$ satisfying
$$\Vert x-y_0\Vert = \inf_{y\in K}\Vert x-y\Vert = {\rm dist}\,(x,K).$$
It's not hard to see that a satisfactory answer to the
question will require that we take $K$ to be a {\sl closed\/} 
set in $X$ (for otherwise the points in $\,\overline{\!K}\setminus K$ 
wouldn't have nearest points).  Less easy to see is that we typically also
want to assume that $K$ is a {\sl convex\/} set.  Recall
that a subset $K$ of a vector space $X$ is said to be convex
if it contains the line segment joining any pair of its points;
that is, $K$ is convex if
$$x,y\in K, \ 0\le\lambda\le1\implies \lambda x+(1-\lambda)y\in K.$$
Obviously, any subspace of $X$ is a convex set and, for our
purposes at least, this is the most important example.  

\prob{\gmark}{%
Let $X$ be a normed space and let $B=\{x\in X:\Vert x\Vert\le1\}$.
Show that $B$ is a closed convex set.  
}

\prob{}{%
Consider  $\R^2$  under the norm  $\Vert\cdot\Vert_\infty$.  Let 
$B = \{ y\in\R^2 : \Vert y\Vert_\infty\le 1 \}$  and let  
$x = �(2,0)$.  Show that there are infinitely many
points in  $B$  nearest to  $x$.
}

\prob{}{%
\vskip-\parskip
\vskip-\baselineskip
\itemitem{(a)}
Let  $K = \{f\in L_1[\,0,1\,]:f\ge 0  \hbox{ and } \Vert f\Vert_1 = 1\}$.  
Show that $K$  is a closed convex set in $L_1[\,0,1\,]$, that $0 \in K$, 
and that every point in  $K$ is a nearest point to $0$.
\itemitem{(b)}
Let  $K = \{f\in C[\,0,1\,]:f(0) = 0 \hbox{ and } \int_0^1f = 1 \}$.  
Again, show that $K$  is a closed convex
set in $C[\,0,1\,]$, that $0\in K$, but that no point in  $K$  is
nearest to  $0$.\par
}

\prob{}{%
Let $K$ be a {\sl compact\/} convex set in a strictly convex space $X$
and let $x\in X$.  Show that $x$ has a unique nearest point
$y_0\in K$.
}

\prob{}{%
Let $K$ be a closed subset of a complete normed space $X$.  Prove 
that $K$ is convex if and only if $K$ is {\sl midpoint convex\/}:  
that is, if and only if $(x+y)/2\in K$ whenever $x$, $y\in K$.  
Is this result true in more general settings?  For example, can you
prove it without assuming completeness?  Or, for that matter, is
it true for arbitrary sets in any vector space (i.e., without even 
assuming the presence of a norm)?
}


\bye

%%  end of function-spaces.tex


