ISSN 0001-4346, Mathematical Notes, 2017, Vol. 102, No. 4, pp. 480–491. © Pleiades Publishing, Ltd., 2017.

Some Extremal Problems for the Fourier Transform on the Hyperboloid* D. V. Gorbachev** , V. I. Ivanov*** , and O. I. Smirnov**** Tula State University, Tula, Russia Received July 21, 2017

´ ´ Delsarte, Logan, and Bohman extremal Abstract—We give the solution of the Turan, Fejer, problems for the Fourier transform on the hyperboloid Hd or Lobachevsky space. We apply the averaging function method over the sphere and the solution of these problems for the Jacobi transform on the half-line. DOI: 10.1134/S0001434617090206 Keywords: hyperboloid, Lobachevsky space, Fourier transform, Turan ´ extremal problem, Fejer ´ extremal problem, Delsarte extremal problem, Logan extremal problem, Bohman extremal problem.

1. INTRODUCTION ´ Fejer, ´ Delsarte, Logan and Bohman extremal The paper is devoted to the solution of the Turan, problems for the Fourier transform on the hyperboloid Hd or Lobachevsky space. First, these problems were investigated for the Fourier transform on the Euclidean space Rd and another locally compact groups (see [1]–[16]). ´ Fejer ´ and Delsarte problems were studied in periodical case as well (see [10], [17]–[23]). The Turan, The conditions in these extremal problems are imposed on both the values of the function and those of its Fourier transform. The nonnegativity condition for the Fourier transform is equivalent to the ´ Delsarte, Logan, and Bohman problems condition that the function is positive definite. The Fejer, are posed for entire functions of exponential type, which are the Fourier transforms of nonnegative ´ problem is posed for continue functions with compact functions with compact support. The Turan ´ problem from functions to there support and nonnegative Fourier transform. If we pass in the Turan ´ extremal problem. Fourier transforms, we get the Fejer In these problems extremizers are radial and by averaging functions over the sphere they are reduced to the similar problems for the Hankel transform on the half-line (see [8], [9], [10], [24], [25]). Common estimates in these problems are obtained by means of the Gauss and Markov quadrature formulae for entire functions of exponential type at zeros of the Bessel function. The extremal functions are obtained by analysis of equality conditions in these estimates. The extremal problems for the Dunkl transform on Rd with Dunkl weight are reduced to the extremal problems for the Hankel transform on the half-line as well (see [26], [27]). Many well-known transforms on the half-line are defined by kernels that are eigenfunctions of the Sturm–Liouville problem    ∂ ∂ w(t) uλ (t) + λ2 + λ20 w(t)uλ (t) = 0, ∂t ∂t (1.1) ∂uλ (0) = 0, λ, λ0 ∈ R+ = [0, ∞), t ∈ R+ . uλ (0) = 1, ∂t ∗

The article was submitted by the authors for the English version of the journal. E-mail: [email protected] *** E-mail: [email protected] **** E-mail: [email protected] **

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For the Hankel transform, when α ≥ −1/2,

w(t) = t2α+1 ,

λ0 = 0,

the Gauss and Markov quadrature formulae for entire functions of exponential type at zeros of the Bessel function were proved in [28], [29]. We proved the similar quadrature formulae at zeros of an eigenfunction ´ Fejer, ´ Delsarte, and of the Sturm–Liouville problem (1.1) [30]. This allowed us to solve the Turan, Bohman extremal problems for the Jacobi transform (see [31]–[33]), whose kernel is an eigenfunction of the Sturm–Liouville problem with weight function w(t) = 22ρ (sinh t)2α+1 (cosh t)2β+1 , where α ≥ β ≥ −1/2, α > −1/2, ρ = λ0 = α + β + 1. In the article we solve the Logan extremal problem for the Jacobi transform on the half-line and using the solution of the extremal problems for the Jacobi transform and the averaging function method over ´ Fejer, ´ Delsarte, Logan, and Bohman extremal problems the sphere we obtain the solution of the Turan, d for the Fourier transform on the hyperboloid H . The extremal functions in these problems are unique and spherical. 2. ELEMENTS OF HARMONIC ANALYSIS ON THE HYPERBOLOID We will use some facts of harmonic analysis on hyperboloid Hd and Lobachevsky space from [34, Chapt. 9]. is d-dimensional real Euclidean space with inner product Let d ∈ N, d ≥ 2, and suppose that Rd  (x, y) = x1 y1 + . . . + xd yd and norm |x| = (x, x), Sd−1 = {x ∈ Rd : |x| = 1} is the Euclidean sphere, Rd,1 is (d + 1)-dimensional real pseudoeuclidean space with bilinear form [x, y] = −x1 y1 − . . . − xd yd + xd+1 yd+1 , Hd = {x ∈ Rd,1 : [x, x] = 1, xd+1 > 0} is the upper sheet of the hyperboloid of two sheets, and d(x, y) = arccosh [x, y] = ln([x, y] +



[x, y]2 − 1)

is the distance between x, y ∈ Hd .   The pair Hd , d(·, ·) is known as the Lobachevsky space. Let o = (0, . . . , 0, 1) ∈ Hd , d(x) = d(x, o), and let Br = {x ∈ Hd−1 : d(x) ≤ r}, r > 0, be the ball in the Lobachevsky space. Let t > 0, η ∈ Sd−1 , x = (sinh t η, cosh t) ∈ Hd , and let dμ(t) = Δ(t) dt = 2d−1 sinhd−1 t dt,

dω(η) =

1 |Sd−1 |

dη,

dν(x) = dμ(t) dω(η)

be the Lebesgue measures on R+ , Sd−1 and Hd , respectively. Note that dω is the probability measure on the sphere, invariant under the rotation group SO(d) and the measure dν is invariant under the hyperbolic rotation group SO0 (d, 1). Let λ ∈ R+ , ξ ∈ Sd−1 , y = (λ, ξ) ∈ R+ × Sd−1 = Ωd , and let     d−1 2 3−2d −2 d  Γ 2 + iλ  Γ dσ(λ) = s(λ) dλ = 2   dλ, dτ (y) = dσ(λ) dω(ξ) 2 Γ(iλ) be the Lebesgue measures on R+ and Ωd . Note that s(λ) is an even function. Let X = R+ , Sd−1 , Hd , Ωd , let dρ be a measure on X, let Lp (X, dρ), 1 ≤ p < ∞, be the space of Lebesgue measurable complex functions f on X with finite norm ˆ 1/p |f |p dρ < ∞, f p,dρ = X

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let Cb (X) be the space of continuous bounded functions f on X with norm f ∞ = supX |f |, and let supp f be the support of the function f . Harmonic analysis in L2 (Hd , dν) and L2 (Ωd , dτ ) is carried out by using the direct and inverse Fourier transforms ˆ f (x)[x, ξ  ]−

Ff (y) = F −1 g(x) =

d−1 −iλ 2

dν(x),

(2.1)

d−1 +iλ 2

dτ (y),

(2.2)

Hd

ˆ

g(y)[x, ξ  ]−

Ωd

where ξ  = (ξ, 1), ξ ∈ Sd−1 . If f ∈ L2 (Hd , dν), g ∈ L2 (Ωd , dτ ), then Ff ∈ L2 (Ωd , dτ ),

F −1 g ∈ L2 (Hd , dν)

and f (x) = F −1 (Ff )(x), g(y) = F(F −1 g)(y) in the mean-square sense. Here the following Plancherel relations hold: ˆ ˆ ˆ ˆ 2 2 2 |f (x)| dν(x) = |Ff (y)| dτ (y), |g(y)| dτ (y) = |F −1 g(x)|2 dν(x). Hd

Ωd

Hd

Ωd

Let F (a, b; c; z) be the Gauss hypergeometric function, and let  (d − 1)/2 + iλ (d − 1)/2 − iλ d ((d−2)/2,−1/2) 2 , ; ; −(sinh t) (t) = F ϕλ (t) = ϕλ 2 2 2 be the Jacobi function. The Jacobi function is an eigenfunction of the Sturm–Liouville problem    d − 1 2  ∂ ∂ Δ(t) ϕλ (t) + λ2 + Δ(t)ϕλ (t) = 0, ∂t ∂t 2 ∂ϕλ (0) = 0. ϕλ (0) = 1, ∂t The Jacobi function is an even and analytic function of t on R and it is an even entire function of exponential type |t| with respect to λ. For it, the following conditions hold: |ϕλ (t)| ≤ 1,

ϕ0 (t) > 0,

λ, t ∈ R.

Using [34, Chapt. 9, 9.3.2] and the parity of ϕλ (t), we obtain ˆ ˆ d−1  − d−1 +iλ 2 [x, ξ ] dω(ξ) = (cosh t − sinh t (η, ξ))− 2 +iλ dω(ξ) = ϕλ (t) d−1 d−1 S S ˆ ˆ d−1 − d−1 −iλ (cosh t − sinh t (η, ξ)) 2 dω(η) = [x, ξ  ]− 2 −iλ dω(η). = Sd−1

Sd−1

(2.3)

If f (x) = f0 (d(x)) = f0 (t), x ∈ Hd , and g(y) = g0 (λ), y = (λ, ξ) ∈ Ωd , are spherical functions, then using (2.1)–(2.3), we obtain ˆ ˆ ∞ d−1 f0 (t) dμ(t) (cosh t − sinh t (η, ξ))− 2 −iλ dω(η) Ff (y) = Sd−1 ˆ0 ∞ f0 (t)ϕλ (t) dμ(t) = J f0 (λ), (2.4) = 0

F

−1

ˆ

ˆ

∞

g(x) =

g0 (λ) dσ(λ) ˆ0 ∞

=

Sd−1

(cosh t − sinh t (η, ξ))−

d−1 +iλ 2

g0 (λ)ϕλ (t) dσ(λ) = J −1 g0 (t),

dω(ξ) (2.5)

0

where J f and J −1 g are the direct and inverse Jacobi transforms on the half-line. Elements of Jacobi harmonic analysis can be found in [30], [31], [32], [35]. MATHEMATICAL NOTES

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3. SOME AUXILIARY STATEMENTS be the class of even entire functions of exponential type not higher than r whose restrictions Let to R+ belongs to L1 (R+ , dσ). For functions from the class B1r , the following Paley–Wiener theorem is valid (see [31], [32], and the references there). B1r

Theorem 3.1. A function f belongs to B1r if and only if f ∈ L1 (R+ , dσ) ∩ Cb (R+ ) here

ˆ

∞

f (λ) =

and

supp J −1 f ⊂ [0, r];

J −1 f (t)ϕλ (t) dμ(t).

0

Let 0 < λ1 (t) < . . . < λk (t) < . . . be the positive zeros of the Jacobi function ϕλ (t) of λ. Let us write the Gauss quadrature formula for entire functions of exponential type with nodes at zeros of the Jacobi function [30]. Theorem 3.2. For an arbitrary function f ∈ B1r the Gauss quadrature formula with positive weights holds: ˆ ∞ ∞

f (λ) dσ(λ) = γk (r/2)f (λk (r/2)). (3.1) 0

k=1

The series in (3.1) converges absolutely. Let us consider two averaging operators ⎧ˆ ⎨ f (x) dω(η), x = (sinh t η, cosh t) ∈ Hd , P f (t) = Sd−1 ⎩ f (o), t = 0, ˆ g(y) dω(ξ), y = (λ, ξ) ∈ Ωd . Qg(λ) =

t > 0,

Sd−1

Note that, If f ∈ Cb (Hd ), then limt→0+0 P f (t) = P f (0) = f (o). Lemma 3.3. If r > 0, a function f defined on Hd satisfies the conditions f ∈ Cb (Hd ),

f (o) = 1,

supp f ⊂ Br ,

Ff (y) ≥ 0,

then P f satisfies the conditions P f (0) = 1, supp P f ⊂ [0, r], P f ∈ Cb (R+ ), J P f (λ) ≥ 0, J P f (0) = Q(Ff )(0). Proof. We have P f ∈ Cb (R+ ) and P f (0) = f (o) = 1. Since f (sinh t η, cosh t) = 0 for t > r, η ∈ Sd−1 , it follows that P f (t) = 0 for t > r; hence supp P f ⊂ [0, r]. Further, using (2.1), (2.3), (2.4) we obtain ˆ ˆ ˆ d−1 Ff (y) dω(ξ) = f (x) [x, ξ  ]− 2 −iλ dω(ξ) dν(x) d Sd−1 Sd−1 ˆH∞ ˆ f (sinh t η, cosh t) dω(η)ϕλ (t) dμ(t) = 0 Sd−1 ˆ ∞ P f (t)ϕλ (t) dμ(t) = J P f (λ); = 0

therefore, J P f (λ) ≥ 0. Similarly, ˆ ˆ ˆ  − d−1 f (x)[x, ξ ] 2 dν(x) dω(ξ) = Q(Ff )(0) = Sd−1

Hd

Lemma 3.3 is proved. MATHEMATICAL NOTES

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∞ 0

P f (t)ϕ0 (t) dμ(t) = J P f (0).

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Lemma 3.4. Let r, s > 0, let y = (λ, ξ) ∈ Ωd , g(y) = Ff (y) ∈ L1 (Ωd , dτ ), where f ∈ Cb (Hd ) and supp f ⊂ Br . Then Qg ∈ L1 (R+ , dσ) ∩ Cb (R+ ), supp J −1 Qg ⊂ [0, r], ˆ g(y) dτ (y) = f (o) = P f (0) = J −1 Qg(0). F −1 g(o) = Ωd

In addition, (1) if g(y) ≥ 0, then Qg(λ) ≥ 0; (2) if f (x) ≥ 0, then J −1 Qg(t) ≥ 0; (3) if g(y) ≤ 0, λ ≥ s, then Qg(λ) ≤ 0, λ ≥ s. Proof. Let

ˆ

f (x)[x, ξ  ]−

g(y) =

d−2 −iλ 2

dν(x).

Br

We have

ˆ g ∈ Cb (Ωd ),

Further,

ˆ

∞ ˆ

 

Qg1 dσ = 0

and

Sd−1

Qg(λ) = Sd−1

ˆ   g(λ, ξ) dω(ξ) dσ(λ) ≤

∞ˆ 0

ˆ

ˆ

Sd−1

|g(λ, ξ)| dω(ξ) dσ(λ) = |g1 dτ < ∞ ˆ

∞ˆ

g(y) dτ (y) = Ωd

g(λ, ξ) dω(ξ) ∈ Cb (R+ ).

g(λ, ξ) d omega(ξ) dσ(λ) = 0

Sd−1

∞

Qg(λ) dσ(λ). 0

Applying (2.1), (2.3), we obtain ˆ ˆ d−1 f (x) [x, ξ  ]− 2 −iλ dω(ξ) dν(x) Qg(λ) = B Sd−1 ˆ ˆ r ˆ rr ϕλ (t) f (x) dω(η) dσ(λ) = P f (t)ϕλ (t) dμ(t); = 0

Sd−1

0

hence, by (2.5), the Paley–Wiener theorem 3.1, and Lemma 3.3, we have J −1 Qg(t) = P f (t) ∈ Cb (R+ ), ˆ ˆ g(y) dτ (y) = F −1 g(o) = Ωd

∞

supp J −1 Qg ⊂ [0, r], Qg(λ) dσ(λ) = J −1 Qg(0) = P f (0) = f (o).

0

If f (x) ≥ 0, then from Lemma 3.3 we have J −1 Qg(t) = P f (t) ≥ 0. The properties (1), (3) are evidently. Lemma 3.4 is proved. Lemma 3.5 ([35]). For d ≥ 2, there exists an even entire function ωd (z) of exponential type 2 such that x > 0, ωd (x) > 0, (3.2) |ωd (iy)| y d−1 e2y , y → +∞. ωd (x) xd−1 , x → +∞, Lemma 3.6 ([36, Appl. VII, Akhiezer’s lemma]). Suppose that m ∈ Z+ , F is an even entire function of exponential type τ > 0, bounded on R, G is an even entire function of finite exponential type all of whose roots are contained in the set of roots F , and lim inf e−τ y y 2m |G(iy)| > 0. y→+∞

Then the function ψ(z) = F (z)/G(z) is a polynomial of degree not greater than 2m. MATHEMATICAL NOTES

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We shall need the asymptotics of the Jacobi function (see [30]). If t > 0, λ ∈ C, Re λ ≥ 0, |λ| → ∞, then     −1  (2/π)1/2 π(d − 1)  t|Im λ| ϕλ (t) = +e O |λ| cos λt − , (3.3) 4 (Δ(t)s(λ))1/2 and

  s(λ) = 23−2d Γ−2 (d/2)λd−1 1 + O(|λ|−1 ) .

(3.4)

´ AND FEJER ´ PROBLEMS 4. THE TURAN ´ problem on the half-line. Calculate the quantity The Turan ˆ ∞ f0 (t)ϕ0 (t) dμ(t) T (r, R+ ) = sup J f0 (0) = sup 0

if f0 ∈ Cb (R+ ),

f0 (0) = 1,

supp f0 ⊂ [0, r],

J f0 (λ) ≥ 0.

(4.1)

´ problem on the half-line. Calculate the quantity F (r, R+ ) = sup g0 (0) if The Fejer g0 ∈ L1 (R+ , dσ) ∩ Cb (R+ ),

J −1 g0 (0) = 1,

g0 (λ) ≥ 0,

supp J −1 g0 ⊂ [0, r].

(4.2)

Let w(t) = ϕ20 (t)Δ(t) be the modified weight function, let uλ (t) = ϕλ (t)/ϕ0 (t), and let χr (t) be the characteristic function of the closed interval [0, r]. The generalized translation operator in the space L2 (R+ , dμ) is defined by the equality ˆ ∞ t ϕλ (t)ϕλ (s)J f (λ) dσ(λ), t, s ∈ R+ . T f (s) = 0

It has the following integral representation: ˆ

t+s

t

T f (s) =

f (x)K(t, s, x) dμ(x), |t−s|

where the kernel K is nonnegative and symmetric with respect to all the variables. Using this representation, we can extend the generalized translation operator to the spaces Lp (R+ , dμ), 1 ≤ p ≤ ∞, and, for any t ∈ R+ , we have T t p→p = 1 (see [31], [32], and the references there). With the help of the generalized translation operator, we can define the convolution ˆ ∞ ˆ ∞ t T f (x)g(t) dμ(t) = f (t)T t g(x) dμ(t); (f ∗ g)(x) = 0

0

for its properties, see [31], [32]. Theorem 4.1 ([33]). Let d ≥ 2, r > 0. Then, in the Turan ´ and Fejer ´ problems on the half-line ˆ r/2 w(t) dt, T (r, R+ ) = F (r, R+ ) = 0

the extremal functions can be uniquely written as f0∗ (t)

(ϕ0 χr/2 ∗ ϕ0 χr/2 )(t) , = ´ r/2 w(t) dt 0

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g0∗ (λ)

 ∂uλ (r/2) 2 = a(r) ∂t 2 , λ

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where

w2 (τ /2) . a(r) = ´ r/2 w(t) dt 0

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´ problem on the hyperboloid. Calculate the quantity The Turan T (r, Hd ) = sup Q(Ff )(0) if f ∈ Cb (Hd ),

f (o) = 1,

supp f ⊂ Br ,

Ff (y) ≥ 0.

(4.3)

´ problem on the hyperboloid. Calculate the quantity The Fejer F (r, Hd ) = sup Qg(0) if g = Ff ∈ L1 (Ωd , dτ ),

g(y) ≥ 0,

f ∈ Cb (Hd ),

supp f ⊂ Br .

f (o) = 1,

(4.4)

If a function f defined on Hd satisfies conditions (4.3), then, by Lemma 3.3, P f satisfies conditions (4.1) and Q(Ff )(0) = J P f (0); therefore, T (r, Hd ) = T (r, R+ ). If a function g defined on Ωd satisfies conditions (4.4), then, by Lemma 3.4, Qg satisfies conditions (4.2); therefore, F (r, Hd ) = F (r, R+ ). Using Theorem 4.1, we obtain the following theorem. Theorem 4.2. Let d ≥ 2, r > 0. Then, in the Turan ´ and Fejer ´ problems on the hyperboloid, ˆ r/2 w(t) dt T (r, Hd ) = F (r, Hd ) = 0

and the extremal functions can be uniquely written as f ∗ (x) =

 ∂uλ (r/2) 2 g∗ (y) = a(r) ∂t 2 , λ

(ϕ0 χr/2 ∗ ϕ0 χr/2 )(d(x)) , ´ r/2 w(t) dt 0

where x ∈ Hd , y = (λ, ξ) ∈ Ωd , and w2 (r/2) . a(r) = ´ r/2 w(t) dt 0 5. THE DELSARTE PROBLEM The Delsarte problem on the half-line. Calculate the quantity D(r, s, R+ ) = sup J −1 g(0) if J −1 g ≥ 0. (5.1) Let uλ (t) = ϕλ (t)/ϕ0 (t), let λ1 (t) be the minimal positive zero of ∂uλ (t)/∂t with respect to λ. The Delsarte problem on the half-line can only be solved under an additional relation between the parameters r and s = λ1 (r/2). g ∈ L1 (R+ , dσ) ∩ C(R+ ),

g(0) = 1,

g(λ) ≤ 0,

λ ≥ s,

supp J −1 g ⊂ [0, τ ],

Theorem 5.1 ([31]). Let d ≥ 2, r > 0, λ1 = λ1 (r/2). Then, in the Delsarte problem on the half-line, ˆ r/2 −1  w(t) dt , D(r, λ1 (r/2), R+ ) = 0

the unique extremal function is  λ 2 b(r) ∂u (r/2) ∂t , g0 (λ) = 4  λ 1 − (λ/λ1 )2

 where

b(r) =

1 w(r/2)

ˆ

r/2

w(t) dt

−2

.

0

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The Delsarte problem on the hyperboloid. Calculate the quantity D(r, s, Hd ) = sup F −1 g(o) = sup f (o) if g = Ff ∈ L1 (Ωd , dτ ), f ∈ Cb (Hd ),

Qg(0) = 1,

g(λ, ξ) ≤ 0,

supp f ⊂ Br ,

f (x) ≥ 0.

λ ≥ s,

(5.2)

If a function g defined on Ωd satisfies conditions (5.2), then, by Lemma 3.4, Qg satisfies conditions (5.1) and F −1 g(o) = J −1 Qg(0); therefore, D(r, s, Hd ) = D(r, s, R+ ). Using Theorem 5.1, we obtain the following theorem. Theorem 5.2. Let d ≥ 2, r > 0, λ1 = λ1 (r/2). Then, in the Delsarte problem on the hyperboloid, ˆ r/2 −1 w(t) dt , D(r, λ1 (r/2), Hd ) = 0

the unique extremal function is  λ 2 b(r) ∂u (r/2) ∗ ∂t , g (y) = 4  λ 1 − (λ/λ1 )2

where

y = (λ, ξ) ∈ Ω , d



1 b(r) = w(r/2)

ˆ

r/2

w(t) dt

−2

.

0

6. THE BOHMAN PROBLEM The Bohman Problem on the Half-Line. Calculate the quantity ˆ ∞  d − 1 2  λ2 + g0 (λ) dσ(λ) B(r, R+ ) = inf 2 0 if the function g0 satisfies conditions (4.2). We can assume that λ2 g0 ∈ L1 (R+ , dσ); otherwise, ˆ ∞  d − 2 2  λ2 + g0 (λ) dσ(λ) = +∞ 2 0 and it is not the minimal value of B(r, R+ ). Let λ1 (r/2) and γ1 (r/2) be the first node and the first weight in the Gauss quadrature formula (2.1), respectively. Theorem 6.1 ([32]). Let d ≥ 2, r > 0, λ1 = λ1 (r/2), γ1 = γ1 (r/2). Then, in the Bohman problem on the half-line,  d − 1 2 , B(r, R+ ) = λ21 + 2 the unique extremal function is g0∗ (λ) =

1  ϕλ (r/2) 2 , c(r) λ21 − λ2

where

∂ϕλ1  ϕ (r/2) 2 λ ∂λ (r/2) . = 2 ∂ϕ 2 λ→λ1 λ1 − λ 2λ1 Δ(r/2) ∂tλ1 (r/2)

c(r) = γ1 lim

The Bohman Problem on the Hyperboloid. Calculate the quantity ˆ   d − 1 2  d 2 λ + g(y) dτ (y) B(r, H ) = inf 2 Ωd if y = (λ, ξ) ∈ Ωd and a function g satisfies conditions (4.4). MATHEMATICAL NOTES

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´ problem, if a function g satisfies conditions (4.4), then Qg As we have noted in the case of the Fejer satisfies conditions (4.2) and ˆ  ˆ ∞  d − 1 2   d − 1 2  ˆ 2 2 g(λ, ξ) dω(ξ) dσ(λ) λ + g(y) dτ (y) = λ + 2 2 0 Ωd Sd−1 ˆ ∞  d − 1 2  2 λ + Qg(λ) dσ(λ); = 2 0 therefore, B(r, Hd ) = B(r, R+ ). Using Theorem 6.1, we obtain the following theorem. Theorem 6.2. Let d ≥ 2, r > 0, λ1 = λ1 (r/2), γ1 = γ1 (r/2). Then, in the Bohman problem on the hyperboloid,  d − 1 2 , B(r, Hd ) = λ21 + 2 the unique extremal function is 1  ϕλ (r/2) 2 , g ∗ (y) = c(r) λ21 − λ2 where y = (λ, ξ) ∈ Ωd and ∂ϕλ1  ϕ (r/2) 2 λ ∂λ (r/2) . = 2 ∂ϕ λ→λ1 λ1 − λ2 2λ1 Δ(r/2) ∂tλ1 (r/2)

c(r) = γ1 lim

7. THE LOGAN PROBLEM Let g0 (λ) be a real, continuous function on the half-line, λ(g0 ) = sup{λ > 0 : g0 (λ) > 0}. The Logan problem on the half-line. Calculate the quantity L(r, R+ ) = inf λ(g0 ) if g0 ∈ L1 (R+ , dσ) ∩ Cb (R+ ),

J −1 g0 (t) ≥ 0,

g0 (λ) ≡ 0,

supp J −1 g0 ⊂ [0, r].

(7.1)

Since, in the Logan problem, any admissible function g belongs to L1 (R+ , dσ) ∩ Cb (R+ ), and ⊂ [0, r], it follows, by the Paley–Wiener theorem 3.1, that it belongs to B1r .

J −1 g0

Theorem 7.1. Let d ≥ 2, r > 0, λ1 = λ1 (r/2). Then, in the Logan problem on the half-line, L(r, R+ ) = λ1 , the unique extremal function up to positive factor is g0∗ (λ) =

ϕ2λ (r/2) . 1 − (λ/λ1 )2

(7.2)

Proof. Let the function ωd (z) be from Lemma 3.5. We assume that L(r, R+ ) = λ0 < λ1 . Then, for some admissible function g0 (λ), we have g0 (λ) ≤ 0 for λ ≥ λ1 − ε, ε > 0. It follows from the Paley–Wiener theorem 3.1 that the function g0 belongs to B τ ∩ L1 (R+ , dσ). Applying the Gaussian quadrature formula (3.1), we obtain ˆ ∞ ∞

g0 (λ) dσ(λ) = γk (r/2)g0 (λk (r/2)) ≤ 0; (7.3) 0≤ 0

k=1

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therefore, at the points λk (r/2), k ≥ 1, the function g0 (λ) has double zeros. Let us consider the functions F (z) = ωd (z)g0 (z),

G(z) = ωd (z)ϕ2z (r/2),

From the asymptotics of the Jacobi function (3.3), (3.4), the properties (3.2) of the entire function ωd (z), we have |G(iy)| e(r+2)y ,

y → +∞.

Using Lemma 3.6, we obtain g0 (λ) = cϕ2λ (r/2), c > 0. We have arrived at a contradiction, because / L1 (R+ , dσ). Thus, g0 ∈ L(r, R+ ) ≥ λ1 . The function g0∗ (7.2) is extremal, because λ(g0∗ ) = λ1 and J −1 g0 (t) ≥ 0, supp J −1 g0 ⊂ [0, r] (see [37]). Let us prove the uniqueness of the extremal function. Let g0 (λ) be the extremal function in the Logan problem on the half-line. Since g0 (λ) ≤ 0 for λ ≥ λ1 , we see that it converts inequality (7.3) into an equality. Hence, at the points λk (r/2), k ≥ 2, it has double zeros and, at the point λ1 , it has at least a zero of the first order. Let us consider the functions F (z) = ωd (z)g0 (z),

G(z) = ωd (z)g0∗ (z).

From the asymptotics of the Jacobi function (3.3), (3.4), the properties (3.2) of the entire function ωd (z), we have |G(iy)| |y|−2 e(r+2)y ,

y → +∞.

ψ(λ)g0∗ (λ),

where ψ is an even polynomial of degree not higher Using Lemma 3.6, we obtain g0 (λ) = than 2. Its degree cannot be 2 or else, by the asymptotics of the Jacobi function, g0 would not belong to L1 (R+ , dσ). Hence g0 (λ) = cg0∗ (λ), c > 0. Theorem 7.1 is proved. Suppose that y = (λ, ξ) ∈ Ωd , g(y) is a real and continuous function on Ωd ; put Λ(g) = sup{λ > 0 : g(λ, ξ) > 0, ξ ∈ Sd−1 }. The Logan problem on the hyperboloid. Calculate the quantity L(r, Hd ) = inf Λ(g) if g = Ff ∈ L1 (Ωd , dτ ),

g(y) ≡ 0,

f ∈ Cb (Hd ),

supp f ⊂ Br ,

f (x) ≥ 0.

(7.4)

As we have noted in the case of the Delsarte problem, if a function g satisfies conditions (7.4), then Qg satisfies conditions (7.1) and Λ(g) = λ(Qg); therefore, L(r, Hd ) = L(r, R+ ). Using Theorem 7.1 we obtain the following theorem. Theorem 7.2. Let d ≥ 2, r > 0, λ1 = λ1 (r/2). Then, in the Logan problem on the hyperboloid, L(r, Hd ) = λ1 , and the unique extremal function up to positive factor is g ∗ (y) =

ϕ2λ (r/2) , 1 − (λ/λ1 )2

y = (λ, ξ) ∈ Ωd ,

λ ∈ R+ ,

ξ ∈ Sd−1 .

ACKNOWLEDGMENTS This work was supported by the Russian Foundation for Basic Research under grant 16-01-00308. MATHEMATICAL NOTES

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REFERENCES ´ 1. V. V. Arestov and E. E. Berdysheva, “Turan’s problem for positive definite functions with supports in a hexagon,” Proc. Steklov Inst. Math., No. 1, Suppl, 20–29 (2001). ´ problem for a class of polytopes,” East J. Approx. 8 (3), 2. V. V. Arestov and E. E. Berdysheva, “The Turan 381–388 (2002). 3. E. E. Berdysheva, “Two related extremal problems for entire functions of several variables,” Math. Notes 66 (3–4), 271–282 (1999). 4. R. P. Boas and M. Kac, “Inequalities for Fourier transforms of positive functions,” Duke Math. J. 12, 189–206 (1945). 5. H. Bohman, “Approximate Fourier analysis of distribution functions,” Ark. Mat. 4, 99–157 (1960). 6. H. Cohn, “New upper bounds on sphere packings. II,” Geom. Topol. 6, 329–353 (2002). 7. W. Ehm, T. Gneiting, and D. Richards, “Convolution roots of radial positive definite functions with compact support,” Trans. Amer. Math. Soc. 356, 4655–4685 (2004). 8. D. V. Gorbachev, “Extremum problems for entire functions of exponential spherical type,” Math. Notes 68 (1–2), 159–166 (2000). 9. D. V. Gorbachev, “An extremal problem for an entire functions of exponential spherical type related to the Levenshtein estimate for the sphere packing density in Rn ,” Izv. Tul. Gos. Univ. Ser. Mat. Mekh. Inform. 6 (1), 71–78 (2000). 10. D. V. Gorbachev, “An extremal problem for periodic functions with supports in the ball,” Math. Notes 69 (3–4), 313–319 (2001). ´ esz, ´ ´ about positive definite functions,” Proc. 11. M. N. Kolountzakis and Sz. Gy. Rev “On a problem of Turan Amer. Math. Soc. 131, 3423–3430 (2003). ´ esz, ´ ´ 12. M. N. Kolountzakis and Sz. Gy. Rev “Turan’s extremal problem for positive definite functions on groups,” J. London Math. Soc. 74, 475–496 (2006). 13. B. F. Logan, “Extremal problems for positive-definite bandlimited functions. I. Eventually positive functions with zero integral,” SIAM J. Math. Anal. 14 (2), 249–252 (1983). 14. B. F. Logan, “Extremal problems for positive-definite bandlimited functions. II. Eventually negative functions,” SIAM J. Math. Anal. 14 (2), 253–257 (1983). ´ esz, ´ ´ 15. Sz. Gy. Rev “Turan’s extremal problem on locally compact abelian groups,” Anal. Math. 37 (1), 15–50 (2011). ¨ ¨ ¨ 16. C. L. Siegel, “Uber Gitterpunkte in konvexen Korpern und damit zusammenhangendes Extremal problem,” Acta Math. 65, 307–323 (1935). ¨ ´ “Uber 17. L. Fejer, trigonometrische Polynome,” J. Angew. Math. 146, 53–82 (1915). ´ Extremal Problem for Periodic Functions with Small Support 18. D. V. Gorbachev and A. S. Manoshina, “Turan and Its Applications,” Math. Notes 76 (5–6), 688–700 (2004). ´ problem for periodic functions with nonnegative Fourier 19. V. I. Ivanov and Yu. D. Rudomazina, “About Turan coefficients and small support,” Math. Notes 77 (5–6), 941–945 (2005). 20. V. I. Ivanov, D. V. Gorbachev, and Yu. D. Rudomazina, “Some extremal problems for periodic functions with conditions on their values and Fourier coefficients,” Proc. Steklov Inst. Math., No. 2 suppl, 139–159 (2005). ´ and Delsarte problems for periodic positive definite functions,” Math. Notes 80 21. V. I. Ivanov, “On the Turan (5–6), 875–880 (2006). ´ problems for periodic positive definite functions,” Annales Univ. Sci. 22. V. I. Ivanov and A. V. Ivanov, “Turan Budapest, Sec. Comp. 33, 219–237 (2010). 23. S. B. Stechkin, “An extremal problem for trigonometric series with nonnegative coefficients,” Acta Math. Acad. Sci. Hung. 23 (3-4), 289–291 (1972). 24. D. V. Gorbachev, Selected Problems in the Theory of Functions and Approximation Theory: Their Applications (Grif i K, Tula, 2005) [in Russian]. 25. D. V. Gorbachev, “Bohman extremal problem for Fourier–Hankel transform,” Izv. Tul. Gos. Univ. Ser. Estestv. Nauki, No. 4, 5–10 (2014). 26. A. V. Ivanov, “Some extremal problem for entire functions in weighted spaces,” Izv. Tul. Gos. Univ. Ser. Estestv. Nauki, No. 1, 26–44 (2010). 27. D. V. Gorbachev and V. I. Ivanov, “Bohman extremal problem for Dunkl transform,” Trudy Inst. Mat. Mekh. UrO RAN 21 (4), 115–123 (2015). 28. C. Frappier and P. Olivier, “A quadrature formula involving zeros of Bessel functions,” Math. Comp. 60, 303–316 (1993). 29. G. R. Grozev and Q. I. Rahman, “A quadrature formula with zeros of Bessel functions as nodes,” Math. Comp. 64, 715–725 (1995). 30. D. V. Gorbachev and V. I. Ivanov, “Gauss and Markov quadrature formulae with nodes at zeros of eigenfunctions of a Sturm–Liouville problem, which are exact for entire functions of exponential type,” Sbornik: Math. 206 (8), 1087–1122 (2015). MATHEMATICAL NOTES

Vol. 102 No. 4 2017

EXTREMAL PROBLEMS FOR THE FOURIER TRANSFORM ON THE HYPERBOLOID

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31. D. V. Gorbachev, V. I. Ivanov, and O. I. Smirnov, “The Delsarte Extremal Problem for the Jacobi Transform,” Math. Notes 100 (5–6), 677–686 (2016). 32. D. V. Gorbachev and V. I. Ivanov, “Bohman extremal problem for Jacobi transform,” Trudy Inst. Mat. Mekh. UrO RAN 22 (4), 115–123 (2016). ´ ´ 33. D. V. Gorbachev and V. I. Ivanov, “Turan’s and Fejer’s extremal problems for Jacobi transform,” Anal. Math. (in press) (2017). 34. N. J. Vilenkin, Special Functions and the Theory of Group Representations, Translations of mathematical monographs (Amer. Math. Soc., Providence, RI, 1978), Vol. 22. 35. D. V. Gorbachev, V. I. Ivanov, and R. A. Veprintsev, “Optimal argument in sharp Jackson’s inequality in the space L2 with the hyperbolic weight,” Math. Notes 96 (5–6), 338–348 (2014). 36. B. Ya. Levin, Distribution of Zeros of Entire Functions (Amer. Math. Soc., Providence, RI, 1980). 37. R. A. Veprintsev, “Approximation of the multidimensional Jacobi transform in L2 by partial integrals,” Math. Notes 97 (5–6), 831–845 (2015).

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