Abstract
We prove that the order of \(L^1\)-approximation by elements of the disc algebra given by Khavinson, Pérez-González and Shapiro is precise.
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Let \(\Delta \) be the unit disc, \(\mathbf{T} \) its boundary and consider the disk algebra \(\mathcal{A}\) of those continuous functions on \(\overline{\Delta }\) that are holomorphic in \(\Delta \). In \(\mathcal{A}\) the norm is the supremum norm
but we shall also consider the \(L^1\) norm given by
In connection with approximation in \(L^1\)-norm by elements of a uniform algebra D. Khavinson, F. Pérez-González and H. Shapiro proved the following theorem (see [2, Theorem 3.3]).
Let f be a continuous function on \(\mathbf{T} \) with \(\Vert f\Vert _\infty =1\). Assume there exists an \(H^1\)-function G such that
Then there exists a function \(G^*\) in the disk algebra \(\mathcal{A}\) such that \(\Vert G^*\Vert _\infty \le 1\) and
where C is a constant independent of f.
See [2] for the motivation of this result and its connection to a theorem of Hoffman and Wermer on homomorphisms of uniform algebras.
The authors of [2] also verified that in (1) the bound \(C\varepsilon \log \frac{1}{\varepsilon }\) cannot be replaced by \(C\varepsilon \) ( [2, Theorem 3.4]), but the problem if the order \(O(\varepsilon \log \frac{1}{\varepsilon })\) in (1) can be improved at all, i.e., if it is precise or not, remained open and stated explicitly in Remark (i) in [2]. That problem was communicated to us by D. Khavinson [1]. In this note we prove that the stated order is, indeed, precise.
There is a constant \(c>0\) with the property that for every sufficiently small \(\varepsilon >0\) there is a continuous function \(f=f_\varepsilon \), \(\Vert f\Vert _\infty =1\), such that
for some \(G\in \mathcal{A}\), but for any \(G^*\) in \(\mathcal{A}\) with \(\Vert G^*\Vert _\infty \le 1\) we have
It will be convenient to verify the claim with \(\varepsilon \) replaced by \(\varepsilon ^2\).
Let
where u(z) and v(z) are real. Using that
for \(z=e^{it}\) we haveFootnote 1
while
Indeed, these are easy consequences of the inequality
i.e. of
For example, for \(z=e^{it}\), \(\varepsilon \le |t|\le \pi \) (\(0<\varepsilon \le 1\)), we obtain
as was claimed above.
The preceding relations show that
while
Therefore, if we set \(F_\varepsilon =F(z)=\exp (\varepsilon ^2/(1+\varepsilon -z))\), \(|z|\le 1\), then for small \(\varepsilon \)
This is so, because we subtract from a term \(\sim \varepsilon \) (\(|t|\le \varepsilon \)) resp \(\succeq \varepsilon ^3/t^2\) (\(|t|\ge \varepsilon \)) a term that is at most \(\preceq \varepsilon ^2\) resp. \(\preceq \varepsilon ^4/t^2\). Now the preceding inequality implies in view of the maximum principle that \(\mathfrak {R}F(z)-1\) is positive in the unit disk.
Let \(f_\varepsilon =f=F/|F|\), for which we have for small \(\varepsilon \)
where we used that \(u\le e^u-1\le 2u\) provided \(0\le u\le 1/2\) (cf. (2)).
Note that F is in the disk algebra and f is a continuous function with \(\Vert f\Vert _\infty = 1\). Now let \(G^*\in \mathcal{A}\), \(\Vert G^*\Vert _\infty \le 1\), be any function. We are going to show that
with some \(c>0\) independent of \(\varepsilon \), and that will prove the theorem (with \(\varepsilon \) replaced by \(\varepsilon ^2\) and \(f_{\alpha \varepsilon }\) resp. \(F_{\alpha \varepsilon }\) replacing f resp. G in it, where \(\alpha \) is a constant for which \(\Vert f_{\alpha \varepsilon }-F_{\alpha \varepsilon }\Vert _1\le \varepsilon ^2\); see (4)).
For the \(L^1\) distance of F and \(G^*\) we have
The real part of
is clearly nonnegative. Now to the pairs \(g_1(z)\) and \(g_2(z):=F(z)-1\) with nonnegative real part in \(\Delta \) and with imaginary part \(=0\) at the origin we can apply the “reverse triangle inequality"
proved in [2, Lemma 3.5], where \(C_0\) is an absolute constant. This yields
On the right
so, in view of (6),
follows. Since on the left
(where, for the \(\sim \) relation we used (3)), the inequality (5) follows from (7) for all sufficiently small \(\varepsilon \). \(\square \)
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.Footnote 2
Notes
In what follows \(A\sim B\) means that A/B lies in between two positive absolute constants, and \(A\preceq B\) and \(B\succeq A\) stand for A/B being bounded.
This statement was requested to be included during the submission process.
References
Khavinson, D.: Personal communication
Khavinson, D., Pérez-González, P., Shapiro, H.: Approximation in \(L^1\)-norm by elements of a uniform algebra. Constr. Approx. 14, 401–410 (1998)
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Totik, V. The order of \(L^1\)-approximation by elements of the disc algebra. Anal.Math.Phys. 12, 4 (2022). https://doi.org/10.1007/s13324-021-00606-0
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DOI: https://doi.org/10.1007/s13324-021-00606-0