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}
}
  • J. Walsh
  • Mathematics
  • American Journal of Mathematics
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
ON THE UNIFORM CONVERGENCE OF DOUBLE FOURIER–WALSH SERIES
Introduction. The problems of the existence of so called “universal functions” and the “universal series” are classical, and there is an extensive literature on the theory of functions, which areExpand
REPRESENTATION OF MEASURABLE FUNCTIONS ALMOST EVERYWHERE BY CONVERGENT SERIES
In this paper it is proved that for a certain class of systems (systems of type (X)) one may construct a series (1)having the following properties:1) uniformly on the interval .2) For any measurableExpand
Summability of Fourier series on the quaternions of norm one
1. Introduction. The Peter-Weyl theorem asserts that if G is a compact group, then the matrix elements of a complete system of inequivalent irreducible unitary representations form a completeExpand
On the existence of universal functions with respect to the double Walsh system for classes of integrable functions
Abstract. It is shown that there exists a function U ∈ L([0, 1)) such that for each ε > 0 one can find a measurable set Eε ⊂ [0, 1) with |Eε| > 1−ε such that U is universal for the space L(Eε) withExpand
On the Coefficients of Multiple Walsh-Fourier Series with Small Gaps
For a Lebesgue integrable complex-valued function f defined over the m-dimensional torus I := [0, 1), let f̂(n) denote the multiple Walsh-Fourier coefficient of f , where n = ( n, . . . , n ) ∈ (Z),Expand
multiple-real-valued Walsh functions
A set of functions are introduced that take on any specified prime number m of real values on the interval from 0 to 1. They are similar in structure to the conventional Walsh functions and in factExpand
Localized bases in L2(0, 1) and their use in the analysis of Brownian motion
TLDR
A recursive scheme for a basis construction in the Hilbert space L^2(0,1) which is analogous to that of Haar and Walsh is introduced and a new decomposition theory for the Hilbertspace of square-integrable functions on the unit-interval is found. Expand
A complete system of orthogonal step functions
We educe an orthonormal system of step functions for the interval [0, 1]. This system contains the Rademacher functions, and it is distinct from the Paley-Walsh system: its step functions use theExpand
Universal functions with respect to the double Walsh system for classes of integrable functions
The paper addresses questions on existence and structure of universal functions for spaces L 1 ( E ), E ⊂ [0, 1) 2 , with respect to the doubleWalsh system in the sense of signs of FourierExpand
Fourier series on certain solenoids
interval ft : t 2 R;a t bg, and let ]a;b[ be the open interval ft : t 2 R;a < t < bg. The intervals [a;b[ and ]a;b] are dened similarly. Given a real number t, the symbol [t] denotes the largestExpand
...
1
2
3
4
5
...

References

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