Abstract Generated abstract
The paper studies conditional extremal problems for real entire functions of finite exponential degree that are nonnegative on the real axis and belong to an Lp class, subject to admissible linear constraints. Using a Nikolskii type inequality and comparison with extremal problems in L1 and uniform metrics, it establishes upper and lower bounds for the minimal Lp norm and the existence of an extremal function for 1 < p < infinity. For at most two constraints, the L1 extremal is represented through a Paley Wiener form and Legendre coefficients, yielding more explicit estimates. The results are illustrated for normalization conditions at the origin, where the bounds become sharp in particular ranges of p.
Full Text
UDC 517.512.6
MATHEMATICS
I. I. IBRAGIMOV, B. A. RYMARENKO
ON SOME CONDITIONAL EXTREMAL PROBLEMS IN THE CLASS OF ENTIRE FUNCTIONS OF FINITE DEGREE
(Presented by Academician S. N. Bernstein on 7 V 1965)
Denote by \(\overset{+}{W}{}_\sigma^{(p)}\) \((p \ge 1)\) the class of real entire functions \(\varphi_\sigma(z)\) of degree \(\sigma\), nonnegative on the real axis, with norm
\[ \|\varphi_\sigma\|_{L_p} = \left( \int_{-\infty}^{\infty} [\varphi_\sigma(x)]^p\,dx \right)^{1/p} <\infty . \tag{1} \]
Let \(F_j(\varphi_\sigma)\) \((j=1,\ldots,\nu)\) be linear functionals defined on the set \(\overset{+}{W}{}_\sigma^{(p)}\).
We shall consider functions \(\varphi_\sigma(x)\) subject to the conditions
\[ F_j(\varphi_\sigma)=A_j \qquad (j=1,\ldots,\nu), \tag{2} \]
where \(A_j\) are given real numbers, with not all \(A_j\) equal to zero; the conditions (2) are assumed admissible, i.e. compatible and not contradicting the membership of the functions \(\varphi_\sigma(x)\) in the class \(\overset{+}{W}{}_\sigma^{(p)}\) \((p\ge 1)\).
In the present article upper and lower estimates are indicated for the quantity
\[ \|\varphi_\sigma^*\|_{L_p} = \inf_{\varphi_\sigma\in \overset{+}{W}{}_\sigma^{(p)}} \|\varphi_\sigma\|_{L_p}. \tag{3} \]
The function \(\varphi_\sigma^*(x)\in \overset{+}{W}{}_\sigma^{(p)}\) will be called the extremal function of our problem in the class \(\overset{+}{W}{}_\sigma^{(p)}\). The founder of this type of problem in the metric \(C(-\infty,\infty)\) is S. N. Bernstein \((^1)\).
It is obvious that \(\varphi_\sigma^*\ne \mathrm{const}\), since otherwise one of the conditions (1) or (2) would not be satisfied.
It is known \((^{1,2})\) that if \(\varphi_\sigma(x)\in \overset{+}{W}{}_\sigma^{(p)}\) and \(p_1>p\ge 1\), then an inequality of S. M. Nikolskii type holds:
\[ \|\varphi_\sigma\|_{L_{p_1}} \le \left(\frac{s\sigma}{2\pi}\right)^{1/p-1/p_1} \|\varphi_\sigma\|_{L_p}, \tag{4} \]
where \(s=|[-p/2]|\) is the least integer not less than \(p/2\). Therefore, denoting by \(\Phi_\sigma(x)\) a function belonging to \(\overset{+}{W}{}_\sigma^{(1)}\), and taking into account that \(\overset{+}{W}{}_\sigma^{(1)}\subset \overset{+}{W}{}_\sigma^{(p)}\subset \overset{+}{W}{}_\sigma^{(\infty)}\), we have
\[ \|\varphi_\sigma^*\|_{L_p} \le \|\Phi_\sigma^*\|_{L_p} \le \left(\|\Phi_\sigma^*\|_{C}\right)^{(p-1)/p} \left(\|\Phi_\sigma^*\|_{L}\right)^{1/p}, \tag{5} \]
where a function \(f_\sigma(z)\in \overset{+}{W}{}_\sigma^{(\infty)}\) is considered bounded on \((-\infty,\infty)\).
Consequently, the following is valid.
Theorem. If \(f_\sigma^*(x)\) and \(\Phi_\sigma^*(x)\) are the extremal functions of our problem, respectively, in the classes \(\overset{+}{W}{}_\sigma^{(\infty)}\) and \(\overset{+}{W}{}_\sigma^{(1)}\), then in the class \(\overset{+}{W}{}_\sigma^{(p)}\) \((1<p<\infty)\) there exists a function \(\varphi_\sigma^*(x)\), subject to the conditions (2),
with the least norm \(\|\varphi_\sigma^*\|_{L_p}\), for which the inequalities
\[ \left(\frac{2\pi}{s\sigma}\right)^{1/p}|f_\sigma^*|_C \leq \|\varphi_\sigma^*\|_{L_p} \leq \|\Phi_\sigma^*\|_{L_p} \leq \left(\|\Phi_\sigma^*\|_C\right)^{(p-1)/p}\left(\|\Phi_\sigma^*\|_L\right)^{1/p}. \tag{6} \]
As is known \((^4)\), if \(\Phi_\sigma(x)\in \overset{+}{W}{}_\sigma^{(1)}\), then for \(\nu \leq 2\)
\[ \Phi_\sigma^*(x)=[\Psi_{\sigma/2}(x)]^2, \tag{7} \]
where \(\Psi_{\sigma/2}(x)\) is a certain real entire function of degree \(\sigma/2\) from the class \(W_\sigma^{(2)}\) (see \((^2)\), p. 38). Then, by the well-known Wiener–Paley theorem, we have
\[ \Psi_{\sigma/2}(x)=\int_{-\sigma/2}^{\sigma/2}\gamma(t)e^{ixt}\,dt, \tag{8} \]
where \(\gamma(t)\in L_2[-\sigma/2,\sigma/2]\), and
\[ \int_{-\infty}^{\infty}[\Psi_{\sigma/2}(x)]^2\,dx = 2\pi\int_{-\sigma/2}^{\sigma/2}|\gamma(t)|^2\,dt. \tag{9} \]
Writing the formal expansion of the function \(\gamma(t)\) in the Legendre polynomials \(\{\hat P_n(t)\}_0^\infty\) normalized on \([-\sigma/2,\sigma/2]\),
\[ \gamma(t)\sim \sum_{k=0}^{\infty} b_k \hat P_k(t), \tag{10} \]
where
\[ b_k=\int_{-\sigma/2}^{\sigma/2}\gamma(t)\hat P_k(t)\,dt \qquad (k=0,1,2,\ldots), \tag{11} \]
we obtain
\[ \|\Phi_\sigma^*\|_L = \int_{-\infty}^{\infty}[\Psi_{\sigma/2}(x)]^2\,dx = 2\pi\int_{-\sigma/2}^{\sigma/2}|\gamma(t)|^2\,dt = 2\pi\sum_{k=0}^{\infty} b_k^2. \tag{12} \]
Moreover,
\[ \|\Phi_\sigma^*\|_C = \left[\sup_{-\infty<x<\infty}|\Psi_{\sigma/2}(x)|\right]^2 \leq \left(\int_{-\sigma/2}^{\sigma/2}|\gamma(t)|\,dt\right)^2 \leq \sigma\int_{-\sigma/2}^{\sigma/2}|\gamma(t)|^2\,dt = \sigma\sum_{k=0}^{\infty} b_k^2. \tag{13} \]
Finally, denote by \(\widetilde B_\sigma\) the class of entire functions \(g_\sigma(z)\) belonging to the class \(B_\sigma\) (\(B_\sigma\) is the class of entire functions of degree \(\sigma\), bounded on the entire real axis \((^1)\), and satisfying the conditions (2)). Let \(g_\sigma^*(x)\) be an entire function from the class \(\widetilde B_\sigma\) with the least norm,
\[ \|g_\sigma^*\|_C=\inf_{g_\sigma\in \widetilde B_\sigma}\|g_\sigma\|_C. \]
It is not difficult to observe that \(\|f_\sigma^*\|_C\geq |f_\sigma^*(0)|\) and, moreover,
\[ |g_\sigma^*(0)|\leq \|g_\sigma^*\|_C \leq \|f_\sigma^*\|_C. \tag{14} \]
Thus, inequality (6) (taking into account (12), (13), and (14), for \(\nu \leq 2\)) takes the form
\[ \left(\frac{2\pi}{s\sigma}\right)^{1/p}\|g_\sigma^*\|_C \leq \|\varphi_\sigma^*\|_{L_p} \leq \sigma\left(\frac{2\pi}{\sigma}\right)^{1/p}\sum_{k=0}^{\infty} b_k^2. \tag{15} \]
Let us now consider some particular cases of inequalities (6) or (15).
- Let \(\nu=1\) and let relation (2) be
\[ \varphi_\sigma(0)=1. \tag{16} \]
S. N. Bernstein 1 proved that in this case \(g_\sigma^*(x)=\sin \sigma x/\sigma x\) is an extremal function in the class \(\widetilde B_\sigma\) and \(\|g^*\|_C=1\). On the other hand, as shown in 4, in this case
\[ b_0=\frac{1}{\sqrt{\sigma}},\qquad b_k=0\quad (k=1,2,\ldots);\qquad \Phi_\sigma^*(x)=\left(\frac{\sin \sigma x/2}{\sigma x/2}\right)^2 . \tag{17} \]
Therefore inequality (15) takes the form
\[ (2\pi/s\sigma)^{1/p}\leq \|\varphi_\sigma^*\|_{L_p}\leq (2\pi/\sigma)^{1/p}. \tag{18} \]
2. Let now \(\nu=2\), and let the relations (2) have the form
\[ \varphi_\sigma(0)=1,\qquad \varphi_\sigma'(0)=0. \tag{19} \]
Then for \(\Phi_\sigma^*(x)\in \overset{+}{W}{}_\sigma^{(1)}\) (under conditions (19)) the coefficients \(b_k\) are determined by the same formulas (17).
On the other hand, it is known 5 that in the class \(\widetilde B_\sigma\), under conditions (19), the extremal function is \(\cos \sigma x\), with norm equal to 1. Therefore, as in the preceding case, we have
\[ (2\pi/s\sigma)^{1/p}\leq \|\varphi_\sigma^*\|_{L_p}\leq (2\pi/\sigma)^{1/p}. \tag{18′} \]
It is interesting to note that for \(p=2\) (then \(s=1\)) inequalities (18) and (18′) become the equality
\[ \|\varphi_\sigma^*\|_{L_2}=(2\pi/\sigma)^{1/2}; \]
moreover, for all \(p\in[1,2]\) the equality
\[ \|\varphi_\sigma^*\|_{L_p}=(2\pi/\sigma)^{1/p} \]
holds.
Received
20 IV 1965