## Gap Series and Measures on Spheres |

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2n+l a e G Abel summable absolutely convergent algebra of continuous Banach algebra Banach space Banach-Steinhaus theorem Bohr set Borel measure bounded E-function bounded multiplicative linear compact Abelian group compact group continuous functions countable defined denote dual group E-polynomial easily seen F is admissible F(or finite constant finite number Fourier series function f g e G G is Abelian GAP SERIES given Haar measure Hadamard set Hence i e M(S implies integers irreducible characters irreducible unitary L2 G Laplace series LEMMA Let F Let G maximal ideal space multiplicative linear functional n-oo normal subgroup normal subhypergroup orthogonal polynomial Proof satisfies Schwarz inequality sequence Sidon set spherical harmonics subgroup of G subhypergroup of G subset summable to zero theorem thesis WpHI x e S2 yields z0 g