Franklin system

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One of the classical orthonormal systems of continuous functions. The Franklin system (see  or ) is obtained by applying the Schmidt orthogonalization process (cf. Orthogonalization method) on the interval to the Faber–Schauder system, which is constructed using the set of all dyadic rational points in ; in this case the Faber–Schauder system is, up to constant multiples, the same as the system , where is the Haar system. The Franklin system was historically the first example of a basis in the space of continuous functions that had the property of orthogonality. This system is also a basis in all the spaces , (see ). If a continuous function on has modulus of continuity , and is the partial sum of order of the Fourier series of with respect to the Franklin system, then Here the Fourier–Franklin coefficients of satisfy the inequalities  and the conditions

a) , ;

b) , ;

c) , ; are equivalent for .

If the continuous function is such that then the series converges uniformly on , and if then All these properties of the Franklin system are proved by using the inequalities  The Franklin system is an unconditional basis in all the spaces  and, moreover, in all reflexive Orlicz spaces (see ). If belongs to , , then one has the inequality where denotes the norm in , and the constants depend only on .

The Franklin system has had important applications in various questions in analysis. In particular, bases in the spaces (see ) and (see ) have been constructed using this system. Here is the space of all continuously-differentiable functions on the square with the norm and , the disc space, is the space of all functions that are analytic in the open disc in the complex plane and continuous in the closed disc with the norm The questions of whether there are bases in and were posed by S. Banach .