Gap Series and Measures on Spheres |
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Page 32
... let f be a bounded E - function . Let ( f ; x ) be the second Cesaro N means of S ( f ) = Σ Ε Then n = 0 ( 2 ) 02 • ( f ; x ) = N f ( x ) ( x ) a ( N ; n ) = n √ f ( y ) K ( < x , y > ) dy n = 0 S2 where a ( N ; n ) → 1 as N ...
... let f be a bounded E - function . Let ( f ; x ) be the second Cesaro N means of S ( f ) = Σ Ε Then n = 0 ( 2 ) 02 • ( f ; x ) = N f ( x ) ( x ) a ( N ; n ) = n √ f ( y ) K ( < x , y > ) dy n = 0 S2 where a ( N ; n ) → 1 as N ...
Page 38
... Let F be a - polynomial on Tw Then F ( x ) = Σ Ê ( y ) y ( x ) ∞ γερ Now if f ( t ) = Σ c ( n ) n -it n = 1 = 2 Σ c ( n ) y ( x ) n = 1 then | f || = || F || by ( 6 ) . If E is a ll ∞ Bohr set , ( 7 ) Σπ | Ê ( x ) | = γε Ε Σ | c ( n ) | ...
... Let F be a - polynomial on Tw Then F ( x ) = Σ Ê ( y ) y ( x ) ∞ γερ Now if f ( t ) = Σ c ( n ) n -it n = 1 = 2 Σ c ( n ) y ( x ) n = 1 then | f || = || F || by ( 6 ) . If E is a ll ∞ Bohr set , ( 7 ) Σπ | Ê ( x ) | = γε Ε Σ | c ( n ) | ...
Page 68
... F on S2 such that ( 7 ) 1 if x € W F ( x ) = 0 if if x V and or Fge whenever ge Proof . Suppose W = { x : < x , Yo > > cos h1 } and V = { x : < x , yo > ∞ > cos h2 } where 0 < h1 < h2 < π . Let f C [ -1 , 1 ] be such that ( 8 ) 1 if t cos ...
... F on S2 such that ( 7 ) 1 if x € W F ( x ) = 0 if if x V and or Fge whenever ge Proof . Suppose W = { x : < x , Yo > > cos h1 } and V = { x : < x , yo > ∞ > cos h2 } where 0 < h1 < h2 < π . Let f C [ -1 , 1 ] be such that ( 8 ) 1 if t cos ...
Common terms and phrases
A₁ Abel summable Abelian group Banach algebra Bohr set Borel measure bounded E-function bounded multiplicative linear closed normal subgroup compact Abelian group compact group continuous functions convergent d(a_ d²(a defined denote dual group dµ(x E-function E-polynomial F is admissible finite constant finite number follows Fourier function f GAP SERIES Haar measure Hadamard set Hence implies irreducible characters L² G Laplace series LEMMA lên Let F LP norm maximal ideal maximal ideal space multiplicative linear functional n₁ n₂ norm normal subgroup normal subhypergroup orthogonal P₁ P₂ polynomial Proof rotates rotation-invariant Rudin Schwarz inequality sequence Sidon set spherical harmonics subset summable to zero Suppose theorem thesis xe S₂ yields Z_(x α α α α β αε αι απ γε μ α