[PPT] - Some extremal problems for Fourier transform on hyperboloid 6th PowerPoint Presentation

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Some extremal problems for Fourier transform

n hyperboloid

6th Workshop on Fourier Analysis and Related Fields P´ ecs, Hungary 24–31 August 2017

D. V. Gorbachev, V. I. Ivanov, O. I. Smirnov

Tula State University, Tula, Russia

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Extremal problems for Fourier transform on Rd

Let F(f )(y) =
Rd f (x)e−i(x,y) dx be the Fourier transform.
Turan problem. For central symmetric convex body V ⊂ Rd

it is necessary to calculate the quantity T(V , Rd) = sup

Rd f (x) dx,

if f ∈ Cb(Rd), f (0) = 1, supp f ⊂ V , F(f )(y) 0.

Euclidean ball: C.L. Siegel (1935, d 1, [1]),

R.P. Boas and M. Kac (1945, d = 1, [2]), D.V. Gorbachev (2001, d > 1, [3]), M.N. Kolountzakis and Sz.Gy. R´ ev´ esz (2003, d > 1, [6])

Another bodies:

V.V. Arestov and E.E. Berdysheva (2001, 2002, tiles polytopes, [4, 5]), M.N. Kolountzakis and Sz.Gy. R´ ev´ esz (2003, spectral domains, [6, 7, 8])

In all known cases:

T(V , Rd) =

1

2V

=
1

2 V

dx, fV = χ 1

2 V ∗ χ 1 2 V . 2 / 36

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Fej´

er problem. For central symmetric convex body V ⊂ Rd it is necessary to calculate the quantity F(V , Rd) = sup g(0), if g ∈ L1(Rd) ∩ Cb(Rd), g(y) 0, 1 (2π)d

Rd g(y) dy = 1,

supp F−1(g) ⊂ V .

Remark. By Paley-Wiener theorem the set of admissible

functions coincides with the set of nonnegative entire functions of exponential type, defined by the dual body.

T(V , Rd) = F(V , Rd).
L. Fej´

er (1915, [9]), R.P. Boas and M. Kac (1945, d = 1, [2])

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Delsarte problem. Calculate the quantity

D(Bs, Rd) = sup

Rd f (x) dx,

if f ∈ L1(Rd)∩Cb(Rd), f (0) = 1, f (x) 0, |x| s, F(f )(y) 0.

M. Viazovska (2016, d=8, [10]),
H. Cohn, A. Kumar, S.D. Miller, D. Radchenko, M. Viazovska

(2016, d=24, [11])

Modified Delsarte problem. Calculate the quantity

D(E r

1, Bs, Rd) = sup

Rd g(y) dy,

if g ∈ L1(Rd) ∩ Cb(Rd), g(0) = 1, g(y) 0, |y| ≥ s, supp F−1(g) ⊂ Br, F−1(g)(y) 0.

Unique case: r =

2qd/2 s

, Jd/2(qd/2) = 0.

V.I. Levenshtein (1979, [12]), V.A. Yudin (1989, [13]),

D.V. Gorbachev (2000, [14]), H. Cohn (2002, [15])

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Bohman problem. Calculate the quantity

B(Br, Rd) = inf

Rd |y|2g(y) dy,

if g ∈ L1(Rd)∩Cb(Rd), g(y) 0,

Rd g(y) dy = 1, supp F−1(g) ⊂ Br.
H. Bohman (1960, d = 1, [16]),
V. A. Yudin (1976, d > 1,

[17]),

W. Ehm, T. Gneiting, D. Richards (2004, d > 1, [18])
Let g be real continuous function, and let

Λ(g) = sup{|y| : g(y) > 0}.

Logan problem. Calculate the quantity

L(Br, Rd) = inf Λ(g), if g ∈ L1(Rd)∩Cb(Rd), g ≡ 0, supp F−1(g) ⊂ Br, F−1(g)(y) 0, .

B.F. Logan (1983, d = 1, [19, 20]),

N.I. Chernykh (1967, d = 1, [21]), V.A. Yudin (1981, d > 1, [22]), D.V. Gorbachev (2000, d > 1, [23]), E.E. Berdysheva (1999, cube, [24])

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Extremal problems for Hankel transform on R+

Extremal functions in these extremal problems for the ball are
radial. By averaging functions over the Euclidean sphere the

problems are reduced to analogous problems for the Hankel transform.

Let α −1/2, and suppose that Jα(t) is the Bessel function of

the order α, jα(t) = 2αΓ(α+1)Jα(t) tα

jd/2−1(t) =
Sd−1 ei(x,ξ) dω(ξ), |x| = t
is the normalized Bessel function, qα is minimal positive zero of Jα,

dνα(t) = (2αΓ(α + 1))−1t2α+1 dt is the power measure on the half-line R+, and Hα(λ) = ∞ f (t)jα(λt)dνα(t) is the Hankel transform. Note that H−1

α

= Hα. The restriction of the Fourier transform on radial functions leads to the Hankel transform with α = d

2 − 1.

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Let χr(t) be characteristic function of the segment [0, r].
Turan problem. Calculate the quantity

Tα(r, R+) = sup ∞ f (t) dνα(t), if f ∈ Cb(R+), f (0) = 1, supp f ⊂ [0, r], Hα(f )(λ) 0.

Fej´

er problem. Calculate the quantity Fα(r, R+) = sup g(0), if g ∈ L1(R+, dνα) ∩ Cb(R+), g(y) 0, ∞ g(λ) dνα(λ) = 1, supp Hα(g) ⊂ [0, r].

Remark. By Paley-Wiener theorem for the Hankel transform

the set of admissible functions coincides with the set of even nonnegative entire functions of exponential type at most r.

Theorem 1. Tα(r, R+) = Fα(r, R+) =

r/2 dνα(t) and fr(t) = (χr/2 ∗ χr/2)(t), gr(λ) = cHα(fr)(λ) = j2

α+1(λr/2).

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Delsarte problem. Calculate the quantity

Dα(s, R+) = sup ∞ f (t) dνα(t), if f ∈ L1(R+, dνα)∩Cb(R+), f (0) = 1, f (t) ≤ 0, t s, Hα(f )(λ) 0.

This problem is solved only for α = −1/2, 3, 11.
Modified Delsarte problem. Calculate the quantity

Dα(r, s, R+) = sup ∞ g(λ) dνα(λ), if g ∈ L1(R+, dνα) ∩ Cb(R+), g(0) = 1, g(λ) 0, λ s, supp Hα(g) ⊂ [0, r], Hα(g)(λ) 0.

Theorem 2. Dα(r, 2qα+1

r

, R+) = r/2 dνα(λ) −1 and gr(λ) = j2

α+1(λr/2)

1 −

λr/2qα+1

2 .

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Bohman problem. Calculate the quantity

Bα(r, R+) = inf ∞ λ2g(λ) dνα(λ), if g ∈ L1(R+, dνα) ∩ Cb(R+), g(λ) 0, ∞ g(λ) dνα(λ) = 1, supp Hα(g) ⊂ [0, r].

Theorem 3. Bα(r, R+) =
2qα

r

2 and gr(λ) = j2

α(λr/2)

1 −
λr/2qα

22 .

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Let g be real continuous function, and let

Λ(g) = sup{λ : g(λ) > 0}.

Logan problem. Calculate the quantity

Lα(r, R+) = inf Λ(g), if g ∈ L1(R+, dνα) ∩ Cb(R+), g(λ) ≡ 0, supp Hα(g) ⊂ [0, r], Hα(g)(λ) 0.

Theorem 4. Lα(r, R+) = 2qα

r

and gr(λ) = j2

α(λr/2)

1 −

λr/2qα

2 .

Theorems 1-4 were proved by D.V. Gorbachev

([14, 3, 23, 25, 26]). He proved the uniqueness of extremal functions.

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A unified method for solving of these problems is to use the

Gauss and Markov quadrature formulae on the half-line with nodes at zeros of the Bessel function (C. Frappier and P. Oliver (1993, [27]), G.R. Grozev and Q.I. Rahman (1995, [28]), R.B. Ghanem and C. Frappier (1998, [29])).

Let E r

1 be the set of even entire functions of exponential type

at most r, whose restrictions on R+ belong to L1(R+, dνα), and let 0 < qα,1 < . . . < qα,n < . . . be positive zeros of Jα(t).

Theorem 5. For any function g ∈ E r

1 the Gauss quadrature

formula with positive weights holds ∞ g(λ) dνα(λ) =

∞

k=1

γα,k(r)g(2qα,k/r). (1) The series in (1) converges absolutely.

Theorem 6. For any function g ∈ E r

1 the Markov quadrature

formula with positive weights holds ∞ g(λ) dνα(λ) = γ′

α,0(r)g(0) + ∞

k=1

γ′

α,k(r)g(2qα+1,k/r).

(2) The series in (2) converges absolutely.

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Let us give an example of the application of the Gauss

quadrature formula in the solution of the Bohman problem. Since an admissible function g ∈ E r

1, λ2g ∈ E r 1, g(λ) 0, and

∞

0 g(λ) dνα(λ) = 1, then applying the Gauss quadrature formula

two times, we obtain ∞ λ2g(λ) dνα(λ) =

∞

k=1

γα,k(r)(2qα,k/r)2g(2qα,k/τ) (2qα,1/r)2

∞

k=1

γα,k(r)g(2qα,k/r) = (2qα,1/r)2 ∞ g(λ) dνα(λ) = (2qα,1/r)2.

The extremal function gr(λ) has at the points 2qα,k/r, k 2,

doubling zeros, therefore the following function is extremizer gτ(λ) = j2

α(λr/2)

1 −
λr/2qα

22 .

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Recently (2015, [30]) we proved the Gauss and Markov

quadrature formulae on the half-line with nodes at zeros of eigenfunctions of the Shturm–Lioville problem under some natural conditions on weight function w, which, in particular, are fulfilled for the power weight w(t) = t2α+1, α −1/2, and hyperbolic weight w(t) = (sinh t)2α+1(cosh t)2β+1, α β −1/2.

Let λ0 0, and suppose that the Shturm–Lioville problem

∂ ∂t

w(t) ∂

∂t uλ(t)

+
λ2 + λ2
w(t)uλ(t) = 0,

uλ(0) = 1, ∂uλ ∂t (0) = 0, λ, t ∈ R+, has spectral measure dσ(λ) = s(λ) dλ, s(λ) ≍ λ2α+1, λ → +∞, and an eigenfunction ϕ(t, λ), which is an even and analytic function of t on R and even entire function of exponential type |t| with respect to λ. Let 0 < λ1(t) < . . . < λk(t) < . . . be positive zeros of ϕ(t, λ) with respect to λ.

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Let ϕ0(t) = ϕ(t, 0), let u(t, λ) = ϕ(t, λ)/ϕ0(t), let

0 < λ′

1(t) < . . . < λ′ k(t) < . . . be positive zeros of ∂ ∂t u(t, λ) with

respect to λ, and let E r

1 be the set of even entire functions of

exponential type at most r, whose restrictions on R+ belong to L1(R+, dσ).

Theorem 7. For any function g ∈ E r

1 the Gauss quadrature

formula with positive weights holds ∞ g(λ) dσ(λ) =

∞

k=1

γk(r)g(λk(r/2)). (3) The series in (3) converges absolutely.

Theorem 8. For any function g ∈ E r

1 the Markov quadrature

formula with positive weights holds ∞ g(λ) dσ(λ) = γ′

0(r)g(0) + ∞

k=1

γ′

k(r)g(λ′ k(r/2)).

(4) The series in (4) converges absolutely.

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Extremal problems for Jacobi transform on R+

In the case of hyperbolic weight

w(t) = 22ρ(sinh t)2α+1(cosh t)2β+1, t ∈ R+, α ≥ β ≥ −1/2, where ρ = α + β + 1 = λ0, eigenfunction ϕλ(t) is the Jacobi function ϕλ(t) = F ρ + iλ 2 , ρ − iλ 2 ; α + 1; −(sinh t)2 .

Let dµ(t) = w(t) dt, and let dσ(λ) = s(λ) dλ,

s(λ) = (2π)−1

2ρ−iλΓ(α + 1)Γ(iλ)

Γ((ρ + iλ)/2)Γ((ρ + iλ)/2 − β)

−2

, be the spectral measure. The direct and inverse Jacobi transforms are defined by equalities J f (λ) = ∞ f (t)ϕλ(t) dµ(t), J −1g(t) = ∞ g(λ)ϕλ(t) dσ(λ).

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Turan problem. Calculate the quantity

Tα,β(r, R+) = sup J (f )(0) = sup ∞ f (t)ϕ0(t) dµ(t), if f ∈ Cb(R+), f (0) = 1, supp f ⊂ [0, r], J (f )(λ) 0.

Fej´

er problem. Calculate the quantity Fα,β(r, R+) = sup g(0), if g ∈ L1(R+, dσ) ∩ Cb(R+), g(λ) 0, ∞ g(λ) dσ(λ) = 1, supp J −1(g) ⊂ [0, r].

Remark. By Paley-Wiener theorem for the Jacobi transform

the set of admissible functions coincides with the set of even nonnegative entire functions of exponential type at most r.

Let uλ(t) = ϕλ(t)/ϕ0(t), and let ∆(t) = ϕ2

0(t)w(t).

Theorem 9. [33]

Tα,β(r, R+) = Fα,β(r, R+) = r/2 ∆(t) dt and fr(t) = (ϕ0χr/2∗ϕ0χr/2)(t), gr(λ) = cJ (fr)(λ) = ∂

∂t uλ(r/2)

λ2 2 .

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Delsarte problem. Calculate the quantity

Dα,β(s, R+) = sup J (f )(0) = sup ∞ f (t)ϕ0(t) dµ(t), if f ∈ L1(R+, dµ)∩Cb(R+), f (0) = 1, f (t) 0, t s, J (f )(λ) 0.

Modified Delsarte problem for entire functions. Calculate

the quantity Dα,β(r, s, R+) = sup J −1(g)(0) = sup ∞ g(λ) dσ(λ), if g ∈ L1(R+, dσ) ∩ Cb(R+), g(0) = 1, g(λ) 0, λ s, supp J −1(g) ⊂ [0, r], J −1(g)(λ) 0.

Theorem 10. [31]

Dα,β(r, λ′

1(r/2), R+) =

r/2 ∆(t) dt −1 and gτ(λ) =

λ−2 ∂

∂t uλ(r/2)

2 1 −

λ/λ′

1(r/2)

2 .

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Bohman problem. Calculate the quantity

Bα,β(r, R+) = inf ∞ (λ2 + ρ2)g(λ) dσ(λ), if g ∈ L1(R+, dσ) ∩ Cb(R+), g(λ) 0, ∞ g(λ) dσ(λ) = 1, supp J −1(g) ⊂ [0, r].

Theorem 11. [32]

Bα,β(r, R+) = λ2

1(τ/2) + ρ2

and gr(λ) = ϕ2

λ(r/2)

1 −
λ/λ1(r/2)

22 .

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Recall that Λ(g) = sup{λ : g(λ) > 0}.
Logan problem. Calculate the quantity

Lα,β(r, R+) = inf Λ(g), if g ∈ L1(R+, dσ) ∩ Cb(R+), g(λ) ≡ 0, supp J −1(g) ⊂ [0, r], J −1(g)(λ) 0.

Theorem 12. [34]

Lα,β(r, R+) = λ1(r/2) and gr(λ) = ϕ2

λ(r/2)

1 −

λ/λ1(r/2)

2 .

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Extremal problems for Fourier transform on Hd

Let d ∈ N, d ≥ 2, and suppose that Rd is d-dimensional real

Euclidean space with inner product (x, y) = x1y1 + . . . + xdyd, 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] = −x1y1 − . . . − xdyd + xd+1yd+1, Hd = {x ∈ Rd,1 : [x, x] = 1, xd+1 > 0} is the upper sheet of two sheets hyperboloid,

d(x, y) = arc cosh[x, y] = ln([x, y] +
[x, y]2 − 1) is the

distance between x, y ∈ Hd.

The pair
Hd, d(·, ·)
is known as the Lobachevskii space. Let

x0 = (0, . . . , 0, 1) ∈ Hd, d(x, x0) = d(x), r > 0, and let Bτ = {x ∈ Hd−1 : d(x) ≤ r} be the ball in the Lobachevskii space.

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Let t > 0, η ∈ Sd−1, x = (sinh t η, cosh t) ∈ Hd, and let

dµ(t) = w(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 rotation group SO(d) and the measure dν is invariant under hyperbolic rotation group SO0(d, 1).

Let λ ∈ R+ = [0, ∞), ξ ∈ Sd−1,

y = (λ, ξ) ∈ R+ × Sd−1 = Ωd, and let dσ(λ) = s(λ) dλ = 23−2dΓ−2d 2

Γ

d−1

2

+ iλ

Γ(iλ)
2

dλ, dτ(y) = dσ(λ)dω(ξ) be the Lebesgue measures on R+ and Ωd.

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The direct and inverse Fourier transforms are defined by

equalities Ff (y) =

Hd f (x)[x, ξ′]− d−1

2 −iλ dν(x),

F−1g(x) =

Ωd g(y)[x, ξ′]− d−1

2 +iλ dτ(y),

where ξ′ = (ξ, 1), ξ ∈ Sd−1.

Let

ϕλ(t) = F (d − 1)/2 + iλ 2 , (d − 1)/2 − iλ 2 ; d 2 ; −(sinh t)2 be the Jacobi function (α = (d − 2)/2, β = −1/2). We have ϕλ(t) =

Sd−1[x, ξ′]− d−1

2 ±iλ dω(ξ),

where x = (sinh t η, cosh t), η, ∈ Sd−1, ξ′ = (ξ, 1).

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Two averaging operators over sphere

Pf (t) =

Sd−1 f (x) dω(η), x = (sinh t η, cosh t) ∈ Hd,

Qg(λ) =

Sd−1 g(y) dω(ξ), y = (λ, ξ) ∈ Ωd

give us spherical functions on Hd and Ωd. They are used both for the setting and for the solving of extremal problems.

If f (x) = f0(d(x)) = f0(t) and g(y) = g0(λ) are spherical

functions, then Ff (y) = J f0(λ), F−1g(x) = J −1g0(t).

Let ∆(t) = ϕ2

0(t)w(t), uλ(t) = ϕλ(t)/ϕ0(t).

Some facts from the harmonic analysis on the hyperboloid can

be found in [35].

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SLIDE 24

Tur´

an problem. Calculate the quantity T(r, Hd) = sup Q(Ff )(0), if f ∈ Cb(Hd), f (x0) = 1, supp f ⊂ Br, Ff (y) 0.

Fej´

er problem. Calculate the quantity F(r, Hd) = sup Qg(0), if g ∈ L1(Ωd, dτ) ∩ Cb(Ωd), g(y) 0,

Ωd g(y) dτ(y) = 1,

supp F−1(g) ⊂ Br.

Remark. Admissible functions in the Fej´

er problem are even entire functions of exponential type at most r with respect to λ.

Theorem 13. [34]

T(r, Hd) = F(r, Hd) = r/2 ∆(t) dt and fr(x) = (ϕ0χr/2∗ϕ0χr/2)(t), gr(y) = cF(fr)(y) = ∂

∂t uλ(r/2)

λ2 2 , x = (sinh t η, cosh t) ∈ Hd, y = (λ, ξ) ∈ Ωd.

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Delsarte problem. Calculate the quantity

D(s, Hd) = sup Q(Ff )(0), if f ∈ Hd, f (x0) = 1, f (x) 0, d(x) s, F(f )(y) 0.

Modified Delsarte problem. Calculate the quantity

D(r, s, Hd) = sup

Ωd g(y) dτ(y),

if g ∈ L1(Ωd, dτ) ∩ Cb(Ωd), Qg(0) = 1, g(λ, ξ) 0, λ s, supp F−1(g) ⊂ Br, F−1(g)(x) 0.

Theorem 14. [34]

D(r, λ′

1(r/2), Hd) =

r/2 ∆(t) dt −1 and gr(y) =

λ−2 ∂

∂t uλ(r/2)

2 1 −

λ/λ′

1(r/2)

2 , y = (λ, ξ) ∈ Ωd.

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Let ρ = α + β + 1 = (d − 2)/2 − 1/2 + 1 = (d − 1)/2.
Bohman problem. Calculate the quantity

B(r, Hd) = inf

Ωd
λ2 + ρ2

g(y) dτ(y), y = (λ, ξ), if g ∈ L1(Ωd, dτ) ∩ Cb(Ωd), g(y) 0,

Ωd g(y) dτ(y) = 1,

supp F−1(g) ⊂ Br.

Theorem 15. [34]

B(r, Hd) = λ2

1(r/2) + ρ2

and gr(y) = ϕ2

λ(r/2)

1 −
λ/λ1(r/2)

22 , y = (λ, ξ) ∈ Ωd.

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Let y = (λ, ξ) ∈ Ωd, let g(y) be a real, continuous function on

Ωd, and let Λ(g) = sup{λ > 0 : g(λ, ξ) > 0, ξ ∈ Sd−1}.

Logan problem. Calculate the quantity

L(r, Hd) = inf Λ(g), if g ∈ L1(Ωd, dτ) ∩ Cb(Ωd), g(y) ≡ 0, supp F−1(g) ⊂ Br F−1(g)(x) 0.

Theorem 16. [34]

L(r, Hd) = λ1(r/2) and gr(y) = ϕ2

λ(r/2)

1 −

λ/λ1(r/2)

2 , y = (λ, ξ) ∈ Ωd.

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Thank you for attention to the talk!

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SLIDE 29

References I

C. L. Siegel, ¨

Uber Gitterpunkte in konvexen K¨

rpern und damit

zusammenh¨ angendes Extremal problem, Acta Math. 65 (1935), no. 1, 307-323.

R. P. Boas and M. Kac, Inequalities for Fourier Transforms of

positive functions, Duke Math. J. 12 (1945), 189–206.

D. V. Gorbachev, An extremal problem for periodic functions

with supports in the ball, Math. Notes 69 (2001), no 3-4, 313–319.

V. V. Arestov and E. E. Berdysheva, Tur´

an’s problem for positive definite functions with supports in a hexagon, Proc. Steklov Inst. Math. (2001), no 1 suppl, 20–29.

V. V. Arestov and E. E. Berdysheva, The Tur´

an problem for a class of polytopes, East J. Approx. 8 (1977), no 3, 381–388.

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References II

M. N. Kolountzakis and Sz. Gy. R´

ev´ esz, On a problem of Tur´ an about positive definite functions, Proc. Amer. Math. Soc. 131 (2003), 3423–3430.

M. N. Kolountzakis and Sz. Gy. R´

ev´ esz, Tur´ an’s extremal problem for positive definite functions on groups, J. London

Math. Soc. 74 (2006), 475–496.
Sz. Gy. R´

ev´ esz, Tur´ an’s extremal problem on locally compact abelian groups, Anal. Math. 37 (2011), no 1, 15–50.

L. Fej´

er, ¨ Uber trigonometrische Polynome, J. Angew. Math. 146 (1915), 53–82.

M. Viazovska, The sphere packing problem in dimension 8,

arXiv:1603.04246 [math.NT] (2016).

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References III

H. Cohn, A. Kumar, S. D. Miller, D. Radchenko,
M. Viazovska, The sphere packing problem in dimension 24,

arXiv:1603.04246 [math.NT] (2016).

V. I. Levenshtein, Bounds for packings in n-dimensional

Euclidean space, SovietMath. Dokl. 20 (1979), 417-421.

V. A. Yudin, Packings of balls in Euclidean space, and extremal

problems for trigonometric polynomials, DiscreteMath. Appl. 1 (1991), no. 1, 6972.

D. V. Gorbachev, An extremal problem for an entire functions
f exponential spherical type related to the Levenshtein

estimate for the sphere packing density in Rn, Izv. TulGU Ser.

Mat. 6 (2000), no 1, 71-78.
H. Cohn, New upper bounds on sphere packings II, Geom.
Topol. 6 (2002), 329–353.

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References IV

H. Bohman, Approximate Fourier analysis of distribution

functions, Ark. Mat. 4 (1960), 99–157.

V. A. Yudin, The multidimensional Jackson theorem, Math.

Notes 20 (1976), no. 3, 801-804.

W. Ehm, T. Gneiting, D. Richards, Convolution roots of radial

positive definite functions with compact support, Trans. Amer.

Math. Soc. 356 (2004), 4655–4685.
B. F. Logan, Extremal problems for positive-definite

bandlimited functions. I. Eventually positive functions with zero integral, SIAM J. Math. Anal. 14 (1983), no 2, 249–252.

B. F. Logan, Riesz transform and Riesz potentials for Dunkl

transform, SIAM J. Math. Anal. 14 (1983), no 2, 253–257.

32 / 36

SLIDE 33

References V

N. I. Chernykh, On Jacksons inequality in L2, Proc. Steklov
Inst. Math. 88 (1967), 75–78.
V. A. Yudin, Multidimensional Jackson theorem in L2, Math.

Notes 29 (1981), no. 2, 158-162.

D. V. Gorbachev, Extremum problems for entire functions of

exponential spherical type, Math. Notes. 68 (2000), no. 2, 159-166.

E. E. Berdysheva, Two related extremal problems for entire

functions of several variables, Math. Notes. 66 (1999), no. 3, 271–282.

D. V. Gorbachev, Selected Problems in the Theory of

Functions and Approximation Theory: Their Applications, Tula: Grif i K, 2005.

33 / 36

SLIDE 34

References VI

D. V. Gorbachev, Boman extremal problem for Fourier–Hankel

transform, Izv. Tul. Gos. Univ., Ser. Estestv. Nauki. (2014),

no. 4, 5–10.
C. Frappier, P. Olivier, A quadrature formula involving zeros of

Bessel functions, Math. Comp. 60 (1993), 303–316.

G. R. Grozev, Q. I. Rahman, A quadrature formula with zeros
f Bessel functions as nodes, Math. Comp. 64 (1995),

715–725.

R. B. Ghanem, C. Frappier, Explicit quadrature formulae for

entire functions of ex- ponential type, J. Approx. Theory. 92 (1998), no. 2, 267–279.

34 / 36

SLIDE 35

References VII

D. V. Gorbachev, 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

f exponential type, Sbornik: Math. 206 (2015), no. 8,

1087–1122.

D. V. Gorbachev, V. I. Ivanov, O. I. Smirnov, The Delsarte

Extremal Problem for the Jacobi Transform, Math. Notes. 100 (2016), no. 5, 677–686.

D. V. Gorbachev, V. I. Ivanov, Boman extremal problem for

Jacobi transform, Trudy Inst. Mat. Mekh. UrO RAN 22 (2016), no. 4, 115–123.

D. V. Gorbachev, V. I. Ivanov, Tur´

an’s and Fej´ er’s extremal problems for Jacobi transform, Anal. Math. (2017), (In press).

35 / 36

SLIDE 36

References VIII

D. V. Gorbachev, V. I. Ivanov, O. I. Smirnov, Some extremal

problems for Fourier transform on hyperboloid, Math. Notes. 102 (2017), (In press).

N. J. Vilenkin, Special Functions and the Theory of Group

Representations, Translations of mathematical monographs 22 Providence, RI: Amer. Math. Soc., 1978.

36 / 36