# A Closed Set of Normal Orthogonal Functions

@article{WalshACS, title={A Closed Set of Normal Orthogonal Functions}, author={Joseph L. Walsh}, journal={American Journal of Mathematics}, volume={45}, pages={5} }

A set of normal orthogonal functions {χ} for the interval 0 5 x 5 1 has been constructed by Haar†, each function taking merely one constant value in each of a finite number of sub-intervals into which the entire interval (0, 1) is divided. Haar’s set is, however, merely one of an infinity of sets which can be constructed of functions of this same character. It is the object of the present paper to study a certain new closed set of functions {φ} normal and orthogonal on the interval (0, 1); each… Expand

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#### References

On Cantor’s theorem concerning the coefficients of a convergent trigonometric series, with generalizations

- Mathematics
- 1909

* Presented to the Society April 24, 1909. tG. CANTOR, Uetier einen die trigonometrischen ReShen betre.ffieqlden Lehrsatz, Journal fur Mathematik, vol. 72 (1870), p. 130; Sur les series… Expand