T. A. TIMAN

ON THE GROWTH OF FUNCTIONS CONJUGATE TO ENTIRE FUNCTIONS OF FINITE DEGREE

(Presented by Academician S. N. Bernstein, 1 IX 1964)

Let \(W^2\) denote the Wiener space of functions \(f(x)\), measurable in the sense of Lebesgue, defined on the real axis and satisfying the condition

\[ \int_{-\infty}^{\infty}\frac{|f(x)|^2}{1+x^2}\,dx<\infty . \tag{1} \]

If, for a function \(f(x)\in W^2\), for some values of the numbers (in general, complex) \(a,b\),

\[ \int_{-\infty}^{\infty}\frac{|f(x)-a|^2}{|x-b|^2}\,dx<\infty, \]

then, relying on Plancherel’s theorem and using the Fourier transform \(F^*(x)\) of the fraction \(F_1(x)=f(x)-a/x-b\), one can define the function \(\widetilde f(x)\) conjugate to \(f(x)\) by the formula (see \((^2)\), p. 69)

\[ \widetilde f(x)=\frac{x-b}{\sqrt{2\pi}}\frac{d}{dx}\int_{-\infty}^{\infty}F^*(t)\,\operatorname{sign}t\cdot\frac{1-e^{ixt}}{t}\,dt. \]

In particular, such a definition of the conjugate function with \(a=f(0)\), \(b=0\) is applicable to all functions belonging to \(W^2\).

In the present note we consider functions, defined in this way, that are conjugate to entire functions of exponential type from \(W^2\).

Let, for some \(\sigma>0\), as is customary, \(B_\sigma\) denote the class, introduced by S. N. Bernstein \((^1)\), of entire functions of degree \(\leq\sigma\), bounded on the real axis. For a function \(f(x)\in B_\sigma\) the following representations, resulting from the Wiener–Paley theorem, are known (see \((^2)\), p. 73):

\[ f(x)=f(0)+\frac{x}{\sqrt{2\pi}}\int_{-\sigma}^{\sigma}\Psi(u)e^{iux}\,dx, \tag{2} \]

\[ \widetilde f(x)=\frac{x}{\sqrt{2\pi}}\int_{-\sigma}^{\sigma} i\,\operatorname{sign}u\cdot e^{iux}\Psi(u)\,du, \tag{3} \]

in which \(\Psi(u)\) is an arbitrary measurable function satisfying the condition

\[ \int_{-\sigma}^{\sigma}|\Psi(u)|^2\,du<\infty . \]

These representations, which are valid for all entire functions of degree \(\leq\sigma\) belonging to \(W^2\), show that if an entire function \(f(x)\) of degree \(\leq\sigma\) belongs to \(W^2\), then \(\widetilde f(x)\) is also an entire function of degree \(\leq\sigma\), belonging to the space \(W^2\).

Consequently, for any entire function \(f(x)\) of exponential type satisfying condition (1), the function \(\widetilde f(x)\) conjugate to it grows on the real axis, as \(|x|\to\infty\), more slowly than \(|x|\), and in the general case this conclusion admits no refinement.

In connection with this, the question arises of the possibility of obtaining a more precise estimate for the modulus of the conjugate function \(\widetilde f(x)\) in the case when \(f(x)\in B_\sigma\). The answer to this question is given by the following

Theorem 1. If an entire function \(f(x)\) of degree not exceeding \(\sigma>0\) satisfies everywhere on the real axis the condition \(|f(x)|\leqslant 1\), then for the function \(\widetilde f(x)\) conjugate to it the inequality
\[ |\widetilde f(x)|\leqslant {4\over \pi}\ln (|x|+1)+O(1) \tag{4} \]
holds.

Inequality (4) shows that, instead of the general estimate \(\widetilde f(x)=o(|x|)\) as \(|x|\to\infty\), which cannot be improved on the class of all entire functions of degree \(\leqslant\sigma\) from \(W^2\), in the case when \(f(x)\in B_\sigma\) a more precise estimate is valid, in which the majorant for \(|\widetilde f(x)|\) already has order of growth \(\ln |x|\). This result on the whole class \(B_\sigma\), in a certain sense, cannot be further refined.

Theorem 2. Whatever function \(\varepsilon(x)>0\), defined for \(-\infty<x<\infty\), is such that
\[ \lim_{|x|\to\infty}\varepsilon(x)=0, \]
there exists an entire function \(f(x)\in B_\sigma\) such that
\[ \varlimsup_{|x|\to\infty} {\bigl|\widetilde f(x)\bigr|\over \varepsilon(x)\ln(|x|+1)}=\infty . \tag{5} \]

Theorem 2 confirms the circumstance that the function \(\widetilde f(x)\) conjugate to a function \(f(x)\in B_\sigma\) may turn out to be unbounded on the real axis, and also shows that the estimate of growth given above,
\[ |\widetilde f(x)|=O\{\ln(|x|+1)\}, \]
on the whole class \(B_\sigma\) cannot be replaced by a better, in order, estimate
\[ O\{\varepsilon(x)\ln(|x|+1)\}, \]
where \(\varepsilon(x)>0\) and \(\lim_{|x|\to\infty}\varepsilon(x)=0\).

Consider the class \(W_\sigma^{(1)}\) of entire functions \(f(x)\) of degree not exceeding \(\sigma\), satisfying condition (1) and such that
\[ \left|f\left\{(2k+1){\pi\over \sigma}\right\}\right|\leqslant 1 \quad (k=0,\pm1,\pm2,\pm3,\ldots), \tag{6} \]
and also the class \(W_\sigma^{(2)}\) of entire functions \(f(x)\) of degree \(\leqslant\sigma\), satisfying condition (1) and the condition
\[ \left|f\left({2k\pi\over \sigma}\right)\right|\leqslant 1 \quad (k=0,\pm1,\pm2,\pm3,\ldots). \tag{7} \]

Along with the theorems stated above for the class \(B_\sigma\), we note the following proposition concerning conjugate functions for the classes \(W_\sigma^{(1)}\) and \(W_\sigma^{(2)}\).

Theorem 3. For functions conjugate to functions of the classes \(W_\sigma^{(1)}\) and \(W_\sigma^{(2)}\), the equalities
\[ \sup_{f\in W_\sigma^{(1)}}\widetilde f\left({2k\pi\over \sigma}\right) = {4\over \pi}\ln(|k|+1)+O\{1\}, \tag{8} \]
\[ \sup_{f\in W_\sigma^{(2)}}\widetilde f\left({2k+1\over 2\sigma}\pi\right) = {4\over \pi}\ln(|k|+1)+O(1). \tag{9} \]
are valid.

For the classes \(W_\sigma^{(1)}\) and \(W_\sigma^{(2)}\) propositions analogous to Theorem 3 are valid.

Received
10 VII 1964

REFERENCES

1. S. N. Bernstein, S. R., 176, 1603 (1923).
2. N. I. Akhiezer, Lectures on the Theory of Approximation, 1947.