UDC 517.946

MATHEMATICS

E. V. VORONOVSKAYA

ON THE CONSTRUCTION OF EXTREMAL TRIGONOMETRIC POLYNOMIALS OF MIXED TYPE

(Presented by Academician S. L. Sobolev on 24 V 1967)

Put

[
T_{l,m}(\theta)=\sum_0^l a_k\cos k\theta+\sum_1^m b_k\sin k\theta
=c_l(\theta)+s_m(\theta)
]

under the condition

[
\max |T_{l,m}(\theta)|=1
]

(a normalized polynomial). In the special cases (T=c_l(\theta)) or (T=s_m(\theta)) it was proved ((^1)) that all of them are obtained analytically from the corresponding algebraic polynomials. Our problem is to show that every (T_{l,m}(\theta)) of mixed type can also be constructed with the aid of extremal algebraic polynomials.

Theorem 1. To every normalized (T_{l,m}(\theta)) there corresponds a completely determined pair of normalized analytic functions

[
\Phi_{1,2}(y)=Q_l(y)\pm P_{m-1}(y)\sqrt{1-y^2},
]

each normalized on ([-1,+1]); conversely, to every such pair (\Phi_{1,2}(y)) there corresponds a unique (T_{l,m}(\theta)).

1) Let (T_{l,m}(\theta)) be given. Without loss of generality one may suppose that

[
|T_{l,m}(\pm\pi)|<1
\quad\text{and}\quad
|T_{l,m}(0)|<1
]

(otherwise we take (T_{l,m}(\theta+\varphi_0))). We have

[
c_l(\theta)=\sum_0^l A_k\cos^k\theta;\qquad
s_m(\theta)=\sum_0^{m-1} B_k\cos^k\theta\,\sin\theta,
]

where ((A_k)) and ((B_k)) are determined uniquely.

Splitting the interval into ([0,\pi]) and ([-\pi,0]), put (\cos\theta=y). Then

[
T_{l,m}(\theta)=
\left{
\begin{array}{c}
\displaystyle \sum_0^l A_k y^k+\sum_0^{m-1} B_k y^k\sqrt{1-y^2}\[6pt]
\displaystyle \sum_0^l A_k y^k-\sum_0^{m-1} B_k y^k\sqrt{1-y^2}
\end{array}
\right}
=
]

[
=Q_l(y)\pm P_{m-1}(y)\sqrt{1-y^2}=\Phi_{1,2}(y).
]

2) If

[
\Phi_{1,2}(y)=Q_l(y)\pm P_{m-1}(y)\sqrt{1-y^2}
]

are normalized on ([-1,+1]), then the substitution (y=\cos\theta) in each of them ((0\le \theta\le \pi)) gives on ([-\pi,+\pi]) a single normalized trigonometric polynomial.

Thus the question of constructing all (T_{l,m}(\theta)) is equivalent to the question of finding all pairs (\Phi_{1,2}(y)).

Remark 1. Under the indicated substitution (direct and inverse), the degrees (l) and (m) are preserved.

Remark 2. By the nodes of the polynomial (T_{l,m}(\theta)) we shall mean the points (\theta_1^+), if

[
T_{l,m}(\theta_1^+)=+1,
]

and (\bar{\theta}_2), if

[
T_{l,m}(\bar{\theta}_2)=-1.
]

If ((\bar{\theta}k^{\pm(1)})) are the nodes of (T)) are, respectively, the nodes of (\Phi_1(y)) and (\Phi_2(y)) on ([-1,+1]) (for (\Phi_1) the nodes are arranged in the reverse order).}) on ([0,\pi]), and ((\bar{\theta}_k^{\pm(2)})) are its nodes on ([-\pi,0]), then ((\bar{\sigma}_k^{\pm(1)})=(\cos \bar{\theta}_k^{\pm(1)})) and ((\bar{\sigma}_k^{\pm(2)})=(\cos \bar{\theta}_k^{\pm(2)

Extremal analytic functions and their properties

Remark on divisibility. If (f(y)) is represented by its Maclaurin series (\sum_0^\infty a_i y^i) on the whole interval ([-1,+1]), then the “remainder upon division” of (f(y)) by

[
R_s^2(y)=\prod_1^s (y-\sigma_i)^2,
]

where (-1<\sigma_i<+1), is the uniquely determined polynomial (r(y)) of degree lower than (2s) (see (2)). Since

[
\sqrt{1-y^2}=\sum_0^\infty (-1)^n {^{1/2}\choose n} y^{2n}
]

converges absolutely for (-1\le y\le 1), (P(y)\sqrt{1-y^2}) also gives a completely determined remainder upon division by (R_s^2(y)).

By a distribution ((\stackrel{\pm}{\sigma_i})_1^s) of a polynomial (or function) we shall mean the complete set of its nodes, taking into account the sign assumed by the polynomial at this node (the polynomial or function is reduced).

Let some distribution ((\stackrel{\pm}{\sigma_i})_1^s) on ((-1,+1)) be given ((^2)), and let

[
R_1(x)=\prod_1^{s_1}(x-\stackrel{+}{\sigma_i});\qquad
R_2(x)=\prod_1^{s_2}(x-\stackrel{-}{\sigma_i}) \quad (s_1+s_2=s).
]

In the identity

[
R_1^2(x)\varphi(x)+R_2^2(x)\psi(x)\equiv 2
]

the polynomials (\varphi(x)) and (\psi(x)) are found uniquely by successive division, and their degrees are not higher than (2s_2-1) and (2s_1-1), respectively. A polynomial (of Hermite) of the form

[
Q_n(x)=1-R_1^2(x)\varphi(x)\equiv -1+R_2^2(x)\psi(x)
]

will be the principal extremal polynomial of the distribution ((\sigma_i)_1^s) (i.e. of the least degree) if and only if on ([0,1]) one has (\varphi(x)\ge 0) and (\psi(x)\ge 0). Otherwise we shall call it quasi-extremal and denote it by (\widetilde Q_n(x)) (the same on the interval ((-1,+1)) after the substitution (y=2x-1)).

Remark 3. We shall call any analytic function (\Phi(y)) extremal for the distribution ((\stackrel{\pm}{\sigma_i})1^s) on ([-1,+1]) if (\Phi(\sigma)=\pm 1) and (\max|\Phi(y)|=1). (\Phi(y)) is obtained from (\widetilde Q_n(y)) by the formula

[
\Phi(y)=\widetilde Q_n(y)+R_s^2(y)\Omega(y),
]

where (\Omega(y)) is an analytic function and

[
-\psi(y)/R_1^2(y)\le \Omega(y)\le \varphi(y)/R_2^2(y).
\tag{1}
]

The width of the “reducedness strip” is equal to (2/R_s^2(y)) ((R_s=R_1R_2)). In general, every extremal (\Psi(y)) can be obtained from any other extremal (\Phi(y)) by the formula (\Psi(y)=\Phi(y)+R_s^2(y)\overline\Omega(y)), with the corresponding reducedness strip for (\overline\Omega(y)).

Theorem 2. If the given function reduced on ([-1,+1]), with distribution ((\stackrel{\pm}{\sigma_1})_1^s), has the form

[
\Phi(y)=\bigl(Q_l(y)+P_{m-1}(y)\sqrt{1-y^2}\bigr),
]

where (l<2s), then the identity

[
Q_l(y)\equiv \widetilde Q_n(y)-\rho(y),
\tag{2}
]

holds, where (\rho(y)) is the remainder upon division of (P_{m-1}(y)\sqrt{1-y^2}) by (R_s^2(y)).

Indeed,

[
P(y)\sqrt{1-y^2}\equiv \rho(y)+R_s^2(y)\Omega(y),
]

and, since (\Phi(y)) is an extremal function, we have

[
\Phi(y)=\widetilde Q_n(y)+R_s^2(y)\Omega'(y)=Q_l(y)+\rho(y)+R_s^2(y)\Omega(y).
]

Hence, from the uniqueness both of the remainder and of the incomplete quotient, we obtain

[
\widetilde Q_n(y)=Q_l(y)+\rho(y),\qquad \Omega'(y)=\Omega(y).
]

Remark 4. In finding (\rho_{2s-1}(y)) in the identity

[
P_{m-1}(y)\sqrt{1-y^2}=\rho_{2s-1}(y)+R_s^2(y)\Omega(y)
\tag{3}
]

one may assume (m-1\le 2s-1), since otherwise one can first find

[
P_{m-1}(y)=\rho(y)+R_s^2(y)\omega(y).
]

Remark 5. In formula (3) the polynomials (P_{m-1}(y)) and (\rho_{2s-1}(y)) are determined uniquely from one another, being respectively Hermite polynomials: (\rho_{2s-1}(y)) for the function (P_{m-1}(y)\sqrt{1-y^2}), and (P_{m-1}(y)) for the function (\rho_{2s-1}(y)/\sqrt{1-y^2}).

Theorem 3. If (T_{l,m}(\theta)=C_l(\theta)+S_m(\theta)\sqrt{1-y^2}) is reduced to ([0,\pi]) with distribution (\binom{\pm}{\theta_i}_1^s) on ((0,\pi)), where (0<\theta_i<\pi), then there exists an infinite set of trigonometric polynomials reduced to ([0,\pi]) with the same distribution, for which: 1) (C_l(\theta)) is preserved while the degree of (S_m(\theta)) is raised; 2) (S_m(\theta)) is preserved while the degree of (C_l(\theta)) is raised.

Indeed, pass from (T_{l,m}(\theta)) to the corresponding
[
\Phi_1(y)=Q_l(y)+P_{m-1}(y)\sqrt{1-y^2}
]
with distribution (\binom{\pm}{\sigma}_1^s) on ((-1,+1)).

1) Construct, according to Remark 3, the extremal function
[
\Psi(y)=\Phi_1(y)+R_s^2(y)L_k(y),
]
where (L_k(y)) is a polynomial (preferably of lowest possible degree). The reducedness requirement consists in the inequalities
[
-\Omega_2(y)/R_1^2(y)\leq L_k(y)\leq \Omega_1(y)/R_2^2(y), \tag{4}
]
where
[
\Phi_1(y)=1-R_1^2(y)\Omega_1(y)\equiv -1+R_2^2(y)-\Omega_2(y).
]
Then
[
\Psi(y)=[Q_l(y)+R_s^2(y)L_k(y)]+P_{m-1}(y)\sqrt{1-y^2}.
]
Returning to trigonometric form, we have
[
\Psi(\cos\theta)=C_{2s+k}(\theta)+S_m(\theta).
]

2) If the chosen (L_k(y)) lies in the strip of reducedness (4), then
(L_k(y)\sqrt{1-y^2}) also lies in it, and we have
[
\Psi(y)=\Phi_1(y)+R_s^2(y)L_k(y)\sqrt{1-y^2}
=Q_l(y)+[P_{m-1}(y)+R_s^2(y)L_k(y)]\sqrt{1-y^2}
]
— a function reduced on ([-1,+1]) with distribution (\binom{\pm}{\sigma_i}1^s), after which we obtain
[
\Psi(\cos\theta)=C_l(\theta)+S(\theta).
]
The theorem is proved.

In what follows we shall be interested in trigonometric polynomials of possibly low degrees (l) and (m) with possibly large (s).

Let (\binom{\pm}{\sigma_i}_1^s), for (-1<\sigma_i<1), be a complete distribution of a polynomial (Q_n(y)) of class II, i.e. (n<2s). Then the quasi-extremal (\widetilde Q_n(y)) coincides with (Q_n(y)), and in the identity we have
[
\varphi(y)>0,\qquad \psi(y)>0
]
for (-1\leq y\leq +1).

Theorem 4. One can always construct an infinite set of functions of the form
[
\Phi_1(y)=Q_l(y)+P_{m-1}(y)\sqrt{1-y^2},
]
reduced to ([-1,+1]), having the complete distribution (\binom{\pm}{\sigma_i}_1^s) of a given polynomial of class II (Q_n(y)), where (l<2s) and (m-1<2s); moreover, functions of the form
[
\Phi_2(y)=Q_l(y)-P_{m-1}(y)\sqrt{1-y^2}
]
are also reduced to ([-1,+1]) (the distribution of (\Phi_2) remains unknown).

Let us first write the conditions of double reducedness for (\Phi_1) and (\Phi_2) under the given distribution for (\Phi_1(y)). By Theorem 2,
[
\Phi_{1,2}(y)=Q_n(y)-\rho(y)\pm P(y)\sqrt{1-y^2}.
]
It is required that
[
-1\leq Q_n(y)-\rho(y)\pm P(y)\sqrt{1-y^2}\leq 1,
]
or
[
Q_n(y)-1\leq \rho(y)\mp P(y)\sqrt{1-y^2}\leq Q_n(y)+1. \tag{5}
]
The inequalities (5) show that
[
Q_n(y)-1\leq \rho(y)\leq Q_n(y)+1,
]
i.e. (\rho(y)) must have no fewer sign changes than (Q_n(y)). But since
[
P(\sigma_i)\sqrt{1-\sigma_i^2}=\rho(\sigma_i),
]
it follows that (P(y)) also has no fewer sign changes.

If an arbitrarily chosen (\rho(y)) of degree (<2s) inside the strip (5) and the (P(y)) computed from it yield
[
\rho(y)\mp P(y)\sqrt{1-y^2}
]
outside the strip (5), replace (\rho(y)) by (\lambda\rho(y)) ((\lambda>0)); then (P(y)) is replaced by (\lambda P(y)), and we have the same requirement for
[
\lambda\rho(y)\mp \lambda P(y)\sqrt{1-y^2}.
]

Write the conditions separately for (\Phi_1) (sign (-)) and (\Phi_2) (sign (+)):

1)
[
Q_n(y)-1\leq \lambda[\rho(y)-P(y)\sqrt{1-y^2}]\leq Q_n(y)+1
]
or
[
-R_1^2(y)\varphi(y)\leq -\lambda R_s^2(y)\Omega(y)\leq R_2^2(y)\psi(y),
]
which will always be satisfied for some
[
0<\lambda\leq \lambda_0;
]

2)
[
Q_n(y)-1\leq \lambda[\rho(y)-P(y)\sqrt{1-y^2}]+
]

[
+2\lambda P(y)\sqrt{1-y^2}\leq Q_n(y)+1.
]
If 1) is satisfied, then (\lambda) can be decreased further so that 2) is also satisfied, since (P(y)\sqrt{1-y^2}) is also inside the strip (5); the theorem is proved.

Corollary 1. Passing to trigonometry, we have: to each extremal polynomial (Q_n(y)) of class II with distribution ((\bar{\sigma}i^{\pm})_1^s) on ((-1,+1)) there corresponds an infinite set of trigonometric polynomials (T)=\cos\theta_i).}(\theta)) possessing the following properties: (l<2s,\ m<2s;\ T_{l,m}(\theta)) reduced on ([-\pi,+\pi]); on ([0,\pi]) it has ((\bar{\theta}_i^{\pm})_1^s) nodes corresponding to the substitution ((\bar{\sigma}_i^{\pm

Corollary 2. Each (Q_n(y)) of class II generates a family ({\Phi_{1,2}(y)}{Q_n(y)}), containing (2s) parameters—the coefficients of (\rho(y)), independent but bounded by some (\lambda).

Corollary 3. Any analytic pair (\Phi_{1,2}(y)) (generating (T_{l,m}(\theta))) with the distribution of at least one of them, say (\Phi_1(y)), of class II belongs to ({\Phi_{1,2}(y)}{Q_n(y)}), where (Q_n(y)) is the principal polynomial of this distribution. In particular, this includes all such pairs for which (\Phi_2(y)) also has on ([-1,+1]) some distribution of class II ((\bar{\sigma}_i^{\pm})_1^{s'}); we denote such pairs by (\Phi^*(y)) and call them analytic pairs of class II.

Corollary 4. If there is a pair (\Phi^{1,2}(y)) with corresponding distributions ((\bar{\sigma}_i^{\pm})_1^{s_1}) and ((\bar{\sigma}_i^{\pm})_1^{s_2}) and extremal polynomials (Q(y)}}(y)) and (Q_{n_2}(y)), then this pair is contained in both families ({\Phi_{1,2{Q(y)}}}) and ({\Phi_{1,2{Q). Conversely: if two such families have a common pair, then it is (\Phi^}_{1,2}(y)).

Theorem 5 (criterion for analytic pairing of class II). Necessary and sufficient conditions for given (Q_{n_1}(y)) and (Q_{n_2}(y)) with their complete distributions ((\bar{\sigma}i^{\pm})_1^{s_1}) and ((\bar{\sigma}_i^{\pm})_1^{s_2}) to generate a pair of class II
[
\Phi^*}(y)=Q_1(y)\pm P_{m-1}(y)\sqrt{1-y^2
]
are the following: there exists a polynomial (P(y)) of degree (<2s',\,2s''), possessing the following properties: if (\rho_1(y)) and (\rho_2(y)) are the remainders, respectively, from division of (P(y)\sqrt{1-y^2}) by (R_{s'}^2(y)) and of (-P(y)\sqrt{1-y^2}) by (R_{s''}^2(y)), then the conditions
[
Q_{n_2}(y)-Q_{n_1}(y)\equiv \rho_2(y)-\rho_1(y);
\tag{6}
]
[
Q_{n_1}(y)-1\leq \rho_1(y)\mp P(y)\sqrt{1-y^2}\leq Q_n(y)+1.
\tag{7}
]
are satisfied.

On the basis of what has been set forth above, these conditions require no proof, being evident. The inequalities (7) are preserved, in consequence of (6), when (n_1) is replaced by (n_2) and (\rho_1(y)) by (\rho_2(y)).

Remark 6. The problem of actually checking the suitability of (Q_{n_1}(y)) and (Q_{n_2}(y)) for forming a pair (\Phi^*_{1,2}(y)) is a linear problem. Indeed, taking
[
P(y)=\sum_0^{2s-1} a_i y^i
]
with undetermined coefficients ((s=\max s'\ \text{and}\ s'')), we find unique
[
\rho_1(y)=\sum_1^{2s_1-1} b_i^{(1)}y^i
\quad\text{and}\quad
\rho_2(y)=\sum_1^{2s_2-1} b_i^{(2)}y^i,
]
where ((b_i^{(1)})) and ((b_i^{(2)})) are linear homogeneous functions of ((a_i)). The identity (6) decomposes into (2s) equations with unknowns ((a_i)_0^{2s-1}), which, generally speaking, have unique solutions, after which (7) is checked.

Leningrad Electrotechnical
Institute of Communications
named after M. A. Bonch-Bruevich

Received
24 V 1967

CITED LITERATURE

1. E. V. Voronovskaya, DAN, 129, No. 1 (1959).
2. E. V. Voronovskaya, The Method of Functionals and Its Applications, L., 1963.