Generalized Walsh transforms

Selfridge, R. G.

doi:10.2140/pjm.1955.5.451

Cited by 37 publications

(10 citation statements)

References 4 publications

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“…I will briefly discuss the history of the Walsh functions in Section 2. In Section 3, I will present a brief discussion and comparison of Fourier-based and c.o functions were developed by Chrestenson (1955), andSelfridge (1955) combined these generalizations to develop a theory of Walsh transforms.…”

Section: Minutes Of Sleepmentioning

confidence: 99%

Walsh-Fourier Analysis and its Statistical Applications

Stoffer

1991

Journal of the American Statistical Association

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T~e aim of .th~s article is to acqu.ain~statisticians and practitioners, whose research activities require statistical methodology, With th~statistical. theory and apphc~tlons~f Walsh-Fourier analysis. It has been suggested that Walsh spectral analysis is suited t~(albe.lt~~t restricted .to) the analysis of discrete-valued and categorical-valued time series, and of time series that contain sharp dlscon~n.U1tIes. I explain~he need for Walsh-Fourier analysis, review the history and properties of Walsh functions, and outline the existmg .Walsh-Founer theory. for real-time stationary time series. I discuss various statistical applications based on the Walsh-Founer transform and provide an annotated bibliography.

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Section: Minutes Of Sleepmentioning

confidence: 99%

Walsh-Fourier Analysis and its Statistical Applications

Stoffer

1991

Journal of the American Statistical Association

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“…In 1955 Selfridge [11] developed the Walsh transform on L(R+) (beyond that initially introduced by Fine [4]) and discussed some of its properties. Among other things, he discussed the modified Fejér mean (1) lim f (l-[^)f(u)ipx{u)du, where / is the Walsh transform of /, [u] is the greatest integer less than or equal to u , and y/x is the p-adic Walsh function with index x (x G R+), and showed that for some integrable functions (1) does not converge a.e.…”

Section: Introductionmentioning

confidence: 99%

“…follows immediately. The notation in this paper is standard and follows that (for example) of [2,11]. where tpk(x) are the p-adic Rademacher functions.…”

Section: Introductionmentioning

confidence: 99%

The type of the maximal operators of a class of Walsh convolution operators

He¹,

Mustard²

1992

Proc. Amer. Math. Soc.

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Abstract.This paper discusses the properties of a class of p-adic Walsh convolution operators. The class consists of those 1-parameter sets of operators with kernels that can be represented as the p-adic Walsh-Fourier integral of a uniformly quasi-convex function. The paper proves that the maximal operators associated with each 1-parameter set are all of strong type (00,00) and of weak type (1,1).

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“…The purposes of this note are twofold. These generalized functions are used.in Fine integral transforms [5], [8]. The second purpose is to give a straightforward method for constructing the generalized Walsh functions which include the Walsh functions as defined by the majority of authors.…”

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confidence: 99%

“…Finally it is clear how to construct the generalized functions over the integers module for >_ 2 (see [3], [8]). …”

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confidence: 99%