We consider the analogue of the $\Lambda(p)-$problem for subsets of the Legendre polynomials or more general ultraspherical polynomials. We obtain the ``best possible'' result that if $2

MSC Classifications:

42C10, 33C45, 46B07 show english descriptions Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions]
Local theory of Banach spaces 42C10 - Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
33C45 - Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions]
46B07 - Local theory of Banach spaces

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