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

Some extremal problems for Fourier transform on 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 1 / 36

Extremal problems for Fourier transform on R d R d f ( x ) e − i ( x , y ) dx be the Fourier transform. � • Let F ( f )( y ) = Turan problem. For central symmetric convex body V ⊂ R d • it is necessary to calculate the quantity � T ( V , R d ) = sup R d f ( x ) dx , f ∈ C b ( R d ) , if 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: � 1 � � � T ( V , R d ) = 2 V � = dx , f V = χ 1 2 V ∗ χ 1 2 V . � � 1 2 V 2 / 36

er problem. For central symmetric convex body V ⊂ R d it • Fej´ is necessary to calculate the quantity F ( V , R d ) = sup g (0) , if g ∈ L 1 ( R d ) ∩ C b ( R d ) , g ( y ) � 0 , 1 � supp F − 1 ( g ) ⊂ V . R d g ( y ) dy = 1 , (2 π ) d • 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 , R d ) = F ( V , R d ) . • L. Fej´ er (1915, [9]), R.P. Boas and M. Kac (1945, d = 1, [2]) 3 / 36

• Delsarte problem. Calculate the quantity � D ( B s , R d ) = sup R d f ( x ) dx , f ∈ L 1 ( R d ) ∩ C b ( R d ) , f (0) = 1 , f ( x ) � 0 , | x | � s , F ( f )( y ) � 0 . if • 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 , B s , R d ) = sup R d g ( y ) dy , g ∈ L 1 ( R d ) ∩ C b ( R d ) , if g (0) = 1 , g ( y ) � 0 , | y | ≥ s , supp F − 1 ( g ) ⊂ B r , F − 1 ( g )( y ) � 0 . 2 q d / 2 • Unique case: r = , J d / 2 ( q d / 2 ) = 0. s • V.I. Levenshtein (1979, [12]), V.A. Yudin (1989, [13]), D.V. Gorbachev (2000, [14]), H. Cohn (2002, [15]) 4 / 36

• Bohman problem. Calculate the quantity � B ( B r , R d ) = inf R d | y | 2 g ( y ) dy , if � g ∈ L 1 ( R d ) ∩ C b ( R d ) , g ( y ) � 0 , R d g ( y ) dy = 1 , supp F − 1 ( g ) ⊂ B r . • 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 ( B r , R d ) = inf Λ( g ) , if g ∈ L 1 ( R d ) ∩ C b ( R d ) , supp F − 1 ( g ) ⊂ B r , F − 1 ( g )( y ) � 0 , . g �≡ 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]) 5 / 36

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 J α ( t ) is the Bessel function of the order α , j α ( t ) = 2 α Γ( α +1) J α ( t ) � � S d − 1 e i ( x ,ξ ) d ω ( ξ ) , | x | = t � j d / 2 − 1 ( t ) = t α is the normalized Bessel function, q α is minimal positive zero of J α , d ν α ( t ) = (2 α Γ( α + 1)) − 1 t 2 α +1 dt is the power measure on the half-line R + , and � ∞ H α ( λ ) = f ( t ) j α ( λ t ) d ν α ( t ) 0 is the Hankel transform. Note that H − 1 = H α . The restriction of α the Fourier transform on radial functions leads to the Hankel transform with α = d 2 − 1. 6 / 36

• Let χ r ( t ) be characteristic function of the segment [0 , r ]. • Turan problem. Calculate the quantity � ∞ T α ( r , R + ) = sup f ( t ) d ν α ( t ) , 0 if f ∈ C b ( R + ) , f (0) = 1 , supp f ⊂ [0 , r ] , H α ( f )( λ ) � 0 . • er problem. Calculate the quantity Fej´ F α ( r , R + ) = sup g (0) , g ∈ L 1 ( R + , d ν α ) ∩ C b ( R + ) , if g ( y ) � 0 , � ∞ g ( λ ) d ν α ( λ ) = 1 , supp H α ( g ) ⊂ [0 , r ] . 0 • 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 . � r / 2 • Theorem 1. T α ( r , R + ) = F α ( r , R + ) = d ν α ( t ) and 0 g r ( λ ) = c H α ( f r )( λ ) = j 2 f r ( t ) = ( χ r / 2 ∗ χ r / 2 )( t ) , α +1 ( λ r / 2) . 7 / 36

• Delsarte problem. Calculate the quantity � ∞ D α ( s , R + ) = sup f ( t ) d ν α ( t ) , 0 if f ∈ L 1 ( R + , d ν α ) ∩ C b ( R + ) , f (0) = 1 , f ( t ) ≤ 0 , t � s , H α ( f )( λ ) � 0 . • This problem is solved only for α = − 1 / 2 , 3 , 11. • Modified Delsarte problem. Calculate the quantity � ∞ D α ( r , s , R + ) = sup g ( λ ) d ν α ( λ ) , 0 if g ∈ L 1 ( R + , d ν α ) ∩ C b ( R + ) , g (0) = 1 , g ( λ ) � 0 , λ � s , supp H α ( g ) ⊂ [0 , r ] , H α ( g )( λ ) � 0 . � − 1 �� r / 2 Theorem 2. D α ( r , 2 q α +1 • , R + ) = d ν α ( λ ) and r 0 j 2 α +1 ( λ r / 2) g r ( λ ) = � 2 . � 1 − λ r / 2 q α +1 8 / 36

• Bohman problem. Calculate the quantity � ∞ λ 2 g ( λ ) d ν α ( λ ) , B α ( r , R + ) = inf 0 if g ∈ L 1 ( R + , d ν α ) ∩ C b ( R + ) , g ( λ ) � 0 , � ∞ g ( λ ) d ν α ( λ ) = 1 , supp H α ( g ) ⊂ [0 , r ] . 0 � 2 � 2 q α • Theorem 3. B α ( r , R + ) = and r j 2 α ( λ r / 2) g r ( λ ) = � 2 � 2 . � � 1 − λ r / 2 q α 9 / 36

• Let g be real continuous function, and let Λ( g ) = sup { λ : g ( λ ) > 0 } . • Logan problem. Calculate the quantity L α ( r , R + ) = inf Λ( g ) , if g ∈ L 1 ( R + , d ν α ) ∩ C b ( R + ) , g ( λ ) �≡ 0 , supp H α ( g ) ⊂ [0 , r ] , H α ( g )( λ ) � 0 . Theorem 4. L α ( r , R + ) = 2 q α • and r j 2 α ( λ r / 2) g r ( λ ) = � 2 . � 1 − λ r / 2 q α • Theorems 1-4 were proved by D.V. Gorbachev ([14, 3, 23, 25, 26]). He proved the uniqueness of extremal functions. 10 / 36

• 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 L 1 ( R + , d ν α ), and let 0 < q α, 1 < . . . < q α, n < . . . be positive zeros of J α ( t ). Theorem 5. For any function g ∈ E r • 1 the Gauss quadrature formula with positive weights holds � ∞ ∞ � g ( λ ) d ν α ( λ ) = γ α, k ( r ) g (2 q α, k / r ) . (1) 0 k =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 ) d ν α ( λ ) = γ ′ γ ′ α, 0 ( r ) g (0) + α, k ( r ) g (2 q α +1 , k / r ) . (2) 0 k =1 The series in (2) converges absolutely. 11 / 36

• Let us give an example of the application of the Gauss quadrature formula in the solution of the Bohman problem. Since 1 , λ 2 g ∈ E r an admissible function g ∈ E r 1 , g ( λ ) � 0, and � ∞ 0 g ( λ ) d ν α ( λ ) = 1, then applying the Gauss quadrature formula two times, we obtain � ∞ ∞ λ 2 g ( λ ) d ν α ( λ ) = � γ α, k ( r )(2 q α, k / r ) 2 g (2 q α, k /τ ) 0 k =1 ∞ � (2 q α, 1 / r ) 2 � γ α, k ( r ) g (2 q α, k / r ) k =1 � ∞ = (2 q α, 1 / r ) 2 g ( λ ) d ν α ( λ ) = (2 q α, 1 / r ) 2 . 0 • The extremal function g r ( λ ) has at the points 2 q α, k / r , k � 2, doubling zeros, therefore the following function is extremizer j 2 α ( λ r / 2) g τ ( λ ) = � 2 � 2 . � � 1 − λ r / 2 q α 12 / 36

• 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 ) = t 2 α +1 , α � − 1 / 2, and hyperbolic weight w ( t ) = (sinh t ) 2 α +1 (cosh t ) 2 β +1 , α � β � − 1 / 2 . • Let λ 0 � 0, and suppose that the Shturm–Lioville problem ∂ w ( t ) ∂ � � λ 2 + λ 2 � � ∂ t u λ ( t ) + w ( t ) u λ ( t ) = 0 , 0 ∂ t ∂ u λ u λ (0) = 1 , ∂ t (0) = 0 , λ, t ∈ R + , has spectral measure d σ ( λ ) = s ( λ ) d λ , s ( λ ) ≍ λ 2 α +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 < λ 1 ( t ) < . . . < λ k ( t ) < . . . be positive zeros of ϕ ( t , λ ) with respect to λ . 13 / 36