Lobachevskii Journal of Mathematics http://ljm.ksu.ru Vol. 25, 2007, 197–216

© M. S. Martirosyan and S.V. Samarchyan

Abstract. We have developed a new common method to investigate geometrically fast approximation problems. Fisher-Micchelli’s, Bernstein-Walsh’s and Batirov-Varga’s well known results are obtained as applications.

Introduction: Fisher-Micchelli’s & Bernstein-Walsh’s type problems

Let $K$ be a compact subset of the open unit disk $D$, $H^{\infty }\left(D\right)$ be the set of bounded analytic functions on $D$ and $C\left(K\right)$ be the set of continuous functions on $K$.

Then each function $f\in H^{\infty }\left(D\right)$ is approximable by finite linear combinations of the system of powers ${\left\{z^k\right\}}_0^{\infty }$ uniformly on $K$ at a rate of geometrical progression and as approximants one can take Taylor polynomials, i.e.

where $P_n\left(f\right)\left(z\right):=\sum _{k=0}^n\frac{f^{\left(k\right)}\left(0\right)}{k!}{z}^k$.

If $f\in H\left(\mathbb{ℂ}\right)$ then one can obtain faster approximation

From now on approximation which is faster than geometrical progression will be called fast approximation.

In case when logarithmic capacity of $K$ is positive, $\gamma \left(K\right)>0$, Bernstein and Walsh [BW] obtained the following result.

BW type result (Bernstein-Walsh): The class of functions $f:K\to \mathbb{ℂ}$ permitting the fast approximation by finite linear combinations of the system ${\left\{z^k\right\}}_0^{\infty }$ in $C\left(K\right)$, coincides with $H\left(\mathbb{ℂ}\right)$.

We have seen that to make the fast approximation of the class $H^{\infty }\left(D\right)$ we have to miniaturize it up to the class of entire functions. The natural question arises if we can fast approximate whole class $H^{\infty }\left(D\right)$ using some other system instead of ${\left\{z^k\right\}}_0^{\infty }$. Due to Fisher and Micchelli [FM] the answer is negative.

FM type result (Fisher-Micchelli) There is no system of functions

such that every function $f\in H^{\infty }\left(D\right)$ admits the fast approximation by polynomials

in $C\left(K\right)$.

Reasoning from these results, let us consider the following problems named Fisher-Micchelli’s and Bernstein-Walsh’s type problem, respectively.

Problems:

FM)For a given space $H$, find a system $e_k^{\left(n\right)}$ such that each element of $H$ admits the fast approximation by linear combinations $\sum _{k=1}^n{a}_k^{\left(n\right)}{e}_k^{\left(n\right)}$.

BW) For a given system $e_k^{\left(n\right)},$ find the class of elements permitting the fast approximation by linear combinations $\sum _{k=1}^n{a}_k^{\left(n\right)}{e}_k^{\left(n\right)}$.

We have developed a common method to investigate both problems based on the notion of $q$-bounded systems.

1. $q$-bounded systems

Definition 1.1 Let $X$ be a Banach space, ${\left(e_k^{\left(n\right)}\right)}_{k\le n}\subset X$, $n=1,2,\dots$ be a triangle matrix, and $L_n$ be the linear span of the finite system ${\left\{e_k^{\left(n\right)}\right\}}_{k=1}^n$. For $q>0$ the matrix ${\left(e_k^{\left(n\right)}\right)}_{k\le n}$ is called

i. $q$-lower bounded in $X$, if for each sequence $P_n=\sum _{k=1}^n{a}_k^{\left(n\right)}{e}_k^{\left(n\right)}\in L_n$ one has

ii. $q$-upper bounded in $X$, if

Definition 1.2 The matrix ${\left(e_k^{\left(n\right)}\right)}_{k\le n}$ is called $0$-lower bounded ($\infty$-upper bounded), if it is $q$-lower (upper) bounded for some $q\in \left(0,\infty \right)$.

Definition 1.3 The system ${\left\{e_k\right\}}_{k=1}^{\infty }\subset X$ is called $q$-lower (upper) bounded, if the matrix ${\left(e_k^{\left(n\right)}\right)}_{k\le n}$ is $q$-lower (upper) bounded for $e_k^{\left(n\right)}:=e_k$, $k\le n$.

Checking of $q$-upper boundedness is usually easy. As regards checking of $q$-lower boundedness, it seems difficult. The following lemma shows that checking of $q$-lower boundedness can be reduced to checking of $q$-upper boundedness of biorthogonal system.

Lemma 1.1 (Checking lower boundedness, [F]) If the finite systems ${\left\{{\varphi }_k^{\left(n\right)}\right\}}_{k=1}^n$ and ${\left\{e_k^{\left(n\right)}\right\}}_{k=1}^n$ are biorthogonal for all $n\in N$ and the matrix ${\left({\varphi }_k^{\left(n\right)}\right)}_{k\le n}$ is $q$-upper bounded in a normed space $Y$ then the matrix ${\left(e_k^{\left(n\right)}\right)}_{k\le n}$ is $\frac{1}{q}$-lower bounded in the dual space $Y^{\ast }$.

2. Basic lemma

There is a close relation between Kolmogorov $n$-widths and $q$-bounded systems.

Definition 2.1 Let $K$ be a subset of a normed linear space $X$. The quantity

where the leftmost infimum is taken over all subspaces $X_n\subset X$ of dimension $n$, is called the Kolmogorov $n$-width of $K$ in $X$.

Lemma 2.1($q$-bounded systems and Kolmogorov $n$-widths)

Let $H,X$ be Banach spaces and $H\subset X$. If there exists a matrix ${\left({\varphi }_k^{\left(n\right)}\right)}_{k\le n}\subset H$ which is $q_1$-lower bounded in $X$ and $q_2$-upper bounded in $H$, then

for the unit ball $B_H=\left\{x\in H:{∥x∥}_H\le 1\right\}$ of $H$.

Lemma 2.1 could be established by Tikhomirov’s well known result.

Lemma (Tikhomirov, [T]) If $B_{n+1}$ is a unit ball of some $n+1$-dimensional subspace of a Banach space $X$, then $d_n\left(B_{n+1},X\right)=1$.

Instead, we shall give here an elementary proof in the sense that it does not depend on Borsuk’s theorem.

Proof. Assuming the converse, there is a positive number $\delta$ and the $n$-dimensional $\left(n\ge n_0\right)$ subspaces $Y_n\subset X$ such that

Let $\left\{e_1^{\left(n\right)},\dots ,e_n^{\left(n\right)}\right\}$ be a basis of $Y_n$. Denoting

we have

For each $x\in B_H$ take coefficients $a_k^{\left(n\right)}\left(x\right)$ such that

Since ${\left({\varphi }_k^{\left(n\right)}\right)}_{k\le n}$ is $q_2$-upper bounded in $H$, it follows that

for each positive $\varepsilon$ beginning from some $N\left(\varepsilon \right)$.

The linear dependence of the system

implies the existence of coefficients $c_i^{\left(n+1\right)}$, $i=1,\dots ,n+1$ such that $\sum _{i=1}^{n+1}{c}_i^{\left(n+1\right)}{a}_k^{\left(n\right)}\left({\psi }_i^{\left(n+1\right)}\right)=0$, $k=1,2,\dots ,n$ and

Denote $P_{n+1}=\sum _{i=1}^{n+1}{c}_i^{\left(n+1\right)}{\psi }_i^{\left(n+1\right)}$. Combining (1) – (4), we obtain

This yields that $\frac{q_1}{q_2}-\delta \ge \frac{q_1}{q_2}$ and the contradiction proves the lemma.

The following basic lemma shows that checking of $q$-boundedness of even one system leads to solution of both problems at once.

Lemma 2.2 (Basic lemma)

FM) Suppose $H\subset X$ are Banach spaces and ${∥x∥}_X\le C{∥x∥}_H,x\in H.$ If exists a matrix ${\left({\varphi }_k^{\left(n\right)}\right)}_{k\le n}\subset H$, which is $q_1$-lower bounded in $X$ and $q_2$-upper bounded in $H$, then for every matrix ${\left(e_k^{\left(n\right)}\right)}_{k\le n}\subset X$ there is an element $x\in H$ such that

for all numerical matrices ${\left(a_k^{\left(n\right)}\right)}_{k\le n}$.

BW) Let ${\left\{e_k\right\}}_{k=1}^{\infty }$ be a $0$-lower and $\infty$-upper bounded system in a Banach space $X$. For $x\in X$ there are polynomials $P_n=\sum _{k=1}^n{a}_k^{\left(n\right)}{e}_k$ satisfying $\sqrt[n]{{∥x-P_n∥}_X}\underset{n\to \infty }{\to }0$ if and only if $x=\sum _{k=1}^{\infty }{x}_k{e}_k,\sqrt[k]{\left|x_k\right|}\underset{k\to \infty }{\to }0$.

Proof. One can find BW) proof in [F] under even more general conditions.

FM) Assuming the converse, it is possible to take ${\left(e_k^{\left(n\right)}\right)}_{k\le n}\subset X$ such that $\underset{n\to \infty }{lim\; sup}{r}_n\left(x\right)<\frac{q_1}{q_2}\left(\forall x\in H\right)$ for $r_n\left(x\right):=\underset{a_k^{\left(n\right)}}{inf}\sqrt[n]{{∥x-\sum _{k=1}^n{a}_k^{\left(n\right)}{e}_k^{\left(n\right)}∥}_X}$. Then let $H=\bigcup _{i=1}^{\infty }{H}_i$, where $H_i:=\left\{x\in H:\underset{n\to \infty }{lim\; sup}{r}_n\left(x\right)<\frac{q_1}{q_2}-\frac{1}{i}\right\}$. Since $H$ is a Banach space, one can take some $H_{i_0}$ being a set of the second category there.

Denote $q_0=\frac{q_1}{q_2}-\frac{1}{i_0}$ and

As ${∥\ast ∥}_H$ is stronger than ${∥\ast ∥}_X$, the sets $E_k$ are closed in $H$. Since $H_{i_0}\subset \bigcup _{k=1}^{\infty }{E}_k$, then some $E_{k_0}$ contains a ball in $H$, that is the estimates

hold for some positive number $\mu$, for some $x_0\in X$ and for each $x$ chosen from the unit ball $B_H=\left\{x\in H:{∥x∥}_H\le 1\right\}$. As

then

Therefore

that contradicts lemma 2.1.

To each pair $\left(X,{\varphi }_k^{\left(n\right)}\right)$, where ${\left({\varphi }_k^{\left(n\right)}\right)}_{k\le n}$ is some matrix of elements from a Banach space $X$, let’s put in correspondence the set ${\Im }_X\left({\varphi }_k^{\left(n\right)}\right)$, consisting of elements $x\in X$ that

Corollary. Let ${\left({\varphi }_k^{\left(n\right)}\right)}_{k\le n}$ be $q$-lower bounded matrix in the Banach space $X$ and ${∥{\varphi }_k^{\left(n\right)}∥}_X\le C,k\le n,n=1,2,\dots$. Then for each matrix ${\left(e_k^{\left(n\right)}\right)}_{k\le n}\subset X$ there is an element $x\in {\Im }_X\left({\varphi }_k^{\left(n\right)}\right)$ such that

for all numerical matrices ${\left(a_k^{\left(n\right)}\right)}_{k\le n}$.

Proof. Let $M_n$ be the Banach space of all $n\times n$ triangular numerical matrices $a=\left(a_{ij}\right)$, $1\le i\le j\le n$, equipped with the norm ${\left|a\right|}_n=\sum _{j=1}^n\sum _{i=1}^j\left|a_{ij}\right|$. For $a=\left(a_{ij}\right)\in M_n$ denote

Consider the set $M$ of sequences $A={\left\{a^{\left(n\right)}\right\}}_{n=1}^{\infty },a^{\left(n\right)}\in M_n$ that converges in $X$,

and $\underset{n}{sup}{\left|a^{\left(n\right)}\right|}_n<\infty$. Then $M$ is a Banach space, with the norm

Therefore the set ${\Im }_X\left({\varphi }_k^{\left(n\right)}\right)$ that coincide with $\left\{A\varphi :A\in M\right\}$ turns to a Banach space, equipped with the norm

Indeed, ${\Im }_X\left({\varphi }_k^{\left(n\right)}\right)$ is isometrically isomorphic to the factor space $M\mathrm{<mfenced\; separators=""\; open="/"\; close=""><mrow>\; </mrow></mfenced>}{M}_0$, which is a Banach space as $M_0=\left\{A:A\varphi =0\right\}$ is a closed subspace of $M$.

To complete the proof it is enough to note that

and use basic lemma FM).

3. Application 1

Let $K$ be a compact subset of the open unit disk $D$ with positive logarithmic capacity $\gamma \left(K\right)$, $C\left(K\right)$ be the set of continuous functions on $K$ and $H^{\infty }\left(D\right)$ be the set of bounded analytic functions on the unit disk $D$.

Take $H=H^{\infty }\left(D\right),X=C\left(K\right)$ then $H\subset X$ and ${∥\ast ∥}_X\le {∥\ast ∥}_H$.

It is obvious that the system ${\varphi }_k^{\left(n\right)}\left(z\right)={\varphi }_k\left(z\right)=z^k$ is $1$-upper bounded in $H$. On the other hand, one can easily establish $\frac{\gamma \left(K\right)}{4}$-lower boundedness of ${\left\{z^k\right\}}_0^{\infty }$ in $X$ using the following

Proposition ([F]). For each polynomial $P_n\left(z\right)=\sum _{k=0}^n{a}_k{z}^{n-k},a_0\ne 0$ there is a polynomial $Q_n\left(z\right)=\sum _{k=0}^n{b}_k{z}^{n-k},b_0=1$ satisfying

Now applying basic lemma we obtain Fisher-Micchelli’s and Bernstein-Walsh’s results.

(FM) Let ${\left(e_k^{\left(n\right)}\right)}_{k\le n}$ be a matrix of continuous functions on $K$. Then there is a function $f\in H^{\infty }\left(D\right)$ such that

for all numerical matrices ${\left(a_k^{\left(n\right)}\right)}_{k\le n}$.

(BW) The class of functions $f:K\to \mathbb{ℂ}$, permitting the fast approximation by finite linear combinations of the system ${\left\{z^k\right\}}_0^{\infty }$ in $C\left(K\right)$, coincide with $H\left(\mathbb{ℂ}\right)$.

As you could see we have obtained FM and BW results by checking q-boundedness of just one system ${\left\{z^k\right\}}_{k=0}^{\infty }$.

4. Application 2

It was mentioned above that to make the fast approximation of $H^{\infty }\left(D\right)$ we have to miniaturize it up to the class of entire functions. Now let’s consider another miniaturization.

Suppose $0\le n_1<n_2<\dots <n_k<\dots$ are integers with density $\tau$, i.e. $\underset{k\to \infty }{lim}\frac{k}{n_k}=\tau$.

Denote

Let $K$ be a compact subset of the unit disk $D=\left\{z\in \mathbb{ℂ}:\left|z\right|<1\right\}$ with positive logarithmic capacity $\gamma \left(K\right)$.

Theorem 4.1

FM 1) If $\tau =0$ then $\underset{n\to \infty }{lim}\sqrt[n]{d_n\left(B,C\left(K\right)\right)}=0$, for the unit ball $B$ of $H_{\left\{n_k\right\}}^{\infty }\left(D\right)$.

FM 2) If $\tau >0$ then for each matrix ${\left(e_k^{\left(n\right)}\right)}_{k\le n}$ of continuous functions on $K$ there is a function $f\in H_{\left\{n_k\right\}}^{\infty }\left(D\right)$ satisfying

for all numerical matrices ${\left(a_k^{\left(n\right)}\right)}_{k\le n}$.

BW) If $\tau >0$ then the class of functions $f:K\to \mathbb{ℂ}$, permitting the fast approximation by the finite linear combinations of the system ${\left\{z^{n_k}\right\}}_{k=1}^{\infty }$ in $C\left(K\right)$, coincide with $H_{\left\{n_k\right\}}^{\infty }\left(D\right)\cap H\left(\mathbb{ℂ}\right)$.

Proof. To prove FM1) one can take the partial sums of corresponding lacunary series as approximants. Proofs of FM2) and BW) can be established by the basic lemma.

Indeed, take $H=H_{\left\{n_k\right\}}^{\infty }\left(D\right),X=C\left(K\right)$ and ${\varphi }_k\left(z\right)=z^{n_k}\in H_{\left\{n_k\right\}}^{\infty }\left(D\right)$. It can be easily checked that ${\left\{z^{n_k}\right\}}_{k=1}^{\infty }$ is $1$-upper bounded in $H$ and ${\left(\frac{\gamma \left(K\right)}{4}\right)}^{\frac{1}{\tau }}$-lower bounded in $X$.

5. Application 3

Consider the system of exponents

where ${\lambda }_k$ are disjoint numbers, which satisfy

To every function $f\in L^2\left(0,\infty \right)$ we put in correspondence its approximation error by finite linear combinations of first $n$ elements of (5), i.e. $E_n\left(f\right)=\underset{a_k^{\left(n\right)}}{inf}{∥f\left(x\right)-\sum _{k=1}^n{a}_k^{\left(n\right)}{e}^{-{\lambda }_{k}x}∥}_{L^2\left(0,\infty \right)}$. There is an analogue of Bernstein theorem for exponents.

Theorem (Musoyan, [M1]) Let $f\in L^2\left(0,\infty \right)$. Then the estimate $\underset{n\to \infty }{lim\; sup}\sqrt[n]{E_n\left(f\right)}<1$ holds if and only if there exists an entire function of exponential type with indicator diagram located in the open left half-plane coinciding with $f$ on $\left(0,\infty \right)$ almost everywhere.

In [Z] it has been shown that in a sense such rate of approximation can’t be improved.

Theorem ([Z]) Let $D$ be a set of positive measure $\mu \left(D\right)$ and $D\subset \left\{z:\left|z\right|\le M_1,Rez<0\right\}$. If (6) takes place then

where $f_{\lambda }\left(z\right)=e^{-\lambda z}$ for some $\lambda$ chosen from $-\overline{D}=\left\{z:-\overline{z}\in D\right\}$.

For entire function $f$ of exponential type introduce its Borel transform ${\beta }_f\left(z\right)=\sum _{n=0}^{\infty }{f}^{\left(n\right)}\left(0\right)z^{-n-1}$. Let $K$ be a compact subset of the open right half-plane with positive logarithmic capacity $\gamma \left(K\right)$ and $-K=\left\{z:-z\in K\right\}$. For $1\le p\le \infty$ we denote by $L_K^p$ the class of functions $f\in L^p\left(0,\infty \right)$ admitting the extension up to entire function of exponential type with ${\beta }_f$ holomorphic on the complement of $-K$.

For $f\in L^p\left(0,\infty \right)$ and any matrix ${\left(e_k^{\left(n\right)}\right)}_{k\le n}\subset L^p\left(0,\infty \right)$ consider

Theorem 5.1 (FM) For each matrix ${\left(e_k^{\left(n\right)}\right)}_{k\le n}\subset L^p\left(0,\infty \right),1\le p\le \infty$, there is a function $f\in L_K^p$ satisfying

where $d\left(K\right)=inf\left\{2M:K\subset \left\{z:\left|z\right|\le M\right\}\right\}$.

Proof. There exists a matrix of numbers ${\left({\lambda }_{kn}\right)}_{k\le n}\subset K$ such that the matrix of exponents ${\left(e^{-{\lambda }_{kn}x}\right)}_{k\le n}$ is $\frac{\gamma \left(K\right)}{d\left(K\right)}$-lower bounded in $L^p\left(0,\infty \right)$. To prove this, we introduce Fekete’s $n$-th transfinite diameter and Chebyshev $n$-th constant of $K$ ([L], p. 606) that are ${\tau }_n=\underset{z_1,\dots ,z_n\in K}{max}\prod _{j\ne k}{\left|z_k-z_j\right|}^{1\mathrm{<mfenced\; separators=""\; open="/"\; close=""><mrow></mrow></mfenced>}n\left(n-1\right)}$ and $c_n=\underset{z_1,\dots ,z_n\in \mathbb{ℂ}}{min}\underset{z\in K}{max}\prod _{k=1}^n{\left|z_k-z\right|}^{1∕n}$ respectively. The theorem of Fekete – Szego states that both ${\tau }_n$ and $c_n$ tend to $\gamma \left(K\right)$ as $n\to \infty$.

We take ${\left({\lambda }_{kn}\right)}_{k\le n}$ in such a way that

Consider the finite system of exponents

and Blashke product $B_n\left(\lambda \right)=\prod _{k=1}^n\frac{\lambda -{\lambda }_{kn}}{\lambda +\overline{{\lambda }_{kn}}}$ for ${\left\{{\lambda }_{kn}\right\}}_{k=1}^n$. The biorthogonal system generated by (8) is the system of functions [M2]

that is, $\underset{o}{\overset{\infty }{\int \; <!--nolimits-->}}{e}^{-{\lambda }_{pn}x}\overline{{\varphi }_{qn}\left(x\right)}dx={\delta }_{pq}$ (${\delta }_{pq}$ is Kronecker’s delta) and the linear spans of (8) and (9) coincide. According to lemma 1.1, we just need $\frac{d\left(K\right)}{\gamma \left(K\right)}$-upper boundedness of ${\left({\varphi }_k^{\left(n\right)}\right)}_{k\le n}$ in all spaces $L^p\left(0,\infty \right),1\le p\le \infty$.

The function ${\varphi }_k^{\left(n\right)}\left(x\right)$ can be written in the integral form

where ${\Gamma }_{r,R}$ is the contour consisting of the segment $\left[r+iR,r-iR\right]$ and semicircle $\zeta =r+R{e}^{i\varphi },-\frac{\pi }{2}\le \varphi \le \frac{\pi }{2}$ running in positive direction. Besides, interior of ${\Gamma }_{r,R}$ contains the compact set $K$. For any positive number $\varepsilon$ one can fix $r$ such small and $R$ such large that $\left|\frac{\zeta -\lambda }{\zeta +\overline{\lambda }}\right|>1-\varepsilon$ when $\left(\zeta ,\lambda \right)\in {\Gamma }_{r,R}\times K$. Consequently, (10) implies

where the constant $C$ does not depend on $k$ and $n$. As $\varepsilon$ was arbitrary, the last estimate leads to

Indeed, according to (7) we get

for all $m=1,2,\dots ,n$, therefore

Now let’s prove ${\Im }_{L^p\left(0,\infty \right)}\left(e^{-{\lambda }_{kn}x}\right)\subset L_K^p$.

For $f\in {\Im }_{L^p\left(0,\infty \right)}\left(e^{-{\lambda }_{kn}x}\right)$ consider the exponential polynomials $f_n\left(z\right)=\sum _{j=1}^n\sum _{i=1}^j{a}_{ij}^{\left(n\right)}{e}^{-{\lambda }_{ij}z}\left(z\in \mathbb{ℂ}\right)$ such that

The chosen sequence of polynomials is a normal family of entire functions because of its uniformly boundedness inside of $\mathbb{ℂ}$. Similarly, the sequence of Borel transforms

is a normal family on the complement of $-K$. So $f_n\left(z\right)\underset{n\to \infty }{\to }\tilde{f}\left(z\right)$ uniformly on each compact subset of the complex plane and ${\beta }_{f_n}\left(z\right)\underset{n\to \infty }{\to }\beta \left(z\right)$ uniformly on each compact subset of the complement of $-K$.

As entire function $\tilde{f}$ is of exponential type and $\tilde{f}=f$ almost everywhere on $\left(0,\infty \right)$, it remains to prove ${\beta }_{\tilde{f}}\left(z\right)=\beta \left(z\right),z\in \mathbb{ℂ}\setminus \left(-K\right)$. Indeed, if $Rez\ge \delta >0$ then

Therefore