广义Walsh变式与一极值问题

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Spectral Decomposition of Cyclic Operators in Discrete Harmonic Analysis

For the operators of the discrete Fourier transform, the discrete Vilenkin–Christenson transform, and all linear transpositions of the discrete Walsh transform, we obtain their spectral decompositions and calculate the dimensions of eigenspaces. For complex operators, namely, the discrete Fourier transform and the Vilenkin–Christenson transform, we obtain real projectors on eigenspaces. For the discrete Walsh transform, we consider in detail the Paley and Walsh orderings and a new ordering in which the matrices of operators are symmetric. For operators of linear transpositions of the discrete Walsh transforms with nonsymmetric matrices, we obtain a spectral decomposition with complex projectors on eigenspaces. We also present the Parseval frame for eigenspaces of the discrete Walsh transform.     

Inferring Boolean functions via higher-order correlations

Both the Walsh transform and a modified Pearson correlation coefficient can be used to infer the structure of a Boolean network from time series data. Unlike the correlation coefficient, the Walsh transform is also able to represent higher-order correlations. These correlations of several combined input variables with one output variable give additional information about the dependency between variables, but are also more sensitive to noise. Furthermore computational complexity increases exponentially with the order. We first show that the Walsh transform of order 1 and the modified Pearson correlation coefficient are equivalent for the reconstruction of Boolean functions. Secondly, we also investigate under which conditions (noise, number of samples, function classes) higher-order correlations can contribute to an improvement of the reconstruction process. We present the merits, as well as the limitations, of higher-order correlations for the inference of Boolean networks.     

An example of a zero series expansion in the Walsh system

We construct an example of a zero series expansion in the Walsh system which converges to zero outside some closed M set of zero measure and converges to + at each point of this set. This shows, in particular, that in the theorem which says that a Walsh series which converges everywhere to a finite symmetric function is a Fourier series it is impossible to omit the requirement of finiteness and allow convergence of the series on a set of zero measure to an infinity of specified sign.Translated from Matematicheskie Zametki, Vol. 19, No. 2, pp. 179–186, February, 1976.     

Uniqueness for multiple trigonometric and Walsh series with convergent rearranged square partial sums

If at each point of a set of positive Lebesgue measure every rearrangement of a multiple trigonometric series square converges to a finite value, then that series is the Fourier series of a function to which it converges uniformly. If there is at least one point at which every rearrangement of a multiple Walsh series square converges to a finite value, then that series is the Walsh-Fourier series of a function to which it converges uniformly.

     

Uniqueness for Cesàro summable double Walsh series

We prove a Tauberian theorem for Walsh series of two variables, and use it to obtain several results about uniqueness of Cesàro summable double Walsh series. Namely, we show that up to sets of measure zero, Cesàro summability of double Walsh series is the same as convergence of the square dyadic partial sums and, under a suitable growth condition, that uniqueness holds for Cesàro summable double Walsh series.     

Uniqueness of multiple Walsh series for the convergence on binary cubes

We consider uniqueness problems for multiple Walsh series convergent on binary cubes on a multidimensional binary group. We find conditions under which a given finite or countable set is a set of uniqueness.     

Digital nets and sequences constructed over finite rings and their application to quasi-Monte Carlo integration

A detailed study of digital (t, m, s)-nets and digital (T,s)-sequences constructed over finite rings is carried out. We present general existence theorems for digital nets and sequences and also explicit constructions. Special attention is devoted to the case where the finite ring is a residue class ring of the integers. This study is motivated by the problem of numerical integration of multivariate Walsh series by quasi-Monte Carlo methods, for which we also provide a general error bound.The third author was supported by the CEI Project PACT, WP5.1.2.1.3.     

迭代方程与离散叙率Walsh函数

在J.L.Shanks的基础上,给出了产生离散叙率Walsh函数的迭代方程,由迭代 方程推出了离散Walsh 函数的表达式和Walsh变换的速算 法(FWWT).     

On the Existence and Structure of Universal Functions

Doklady Mathematics - The existence of functions the Fourier–Walsh series of which are universal in the sense of signs for the class of almost everywhere finite measurable functions, and the...     

λ-Convergence of multiple Walsh–Paley series and sets of uniqueness

Mathematical Notes - λ-convergent multiple Walsh–Paley series on a multidimensional dyadic group are studied. It is proved that, for all λ &gt; 1, any arbitrary finite union of...     

On the <Emphasis Type="Italic">p</Emphasis>-ary (cubic) bent and plateaued (vectorial) functions

Plateaued functions play a significant role in cryptography, sequences for communications, and the related combinatorics and designs. Comparing to their importance, those functions have not been studied in detail in a general framework. Our motivation is to bring further results on the characterizations of bent and plateaued functions, and to introduce new tools which allow us firstly a better understanding of their structure and secondly to get methods for handling and designing such functions. We first characterize bent functions in terms of all even moments of the Walsh transform, and then plateaued (vectorial) functions in terms of the value distribution of the second-order derivatives. Moreover, we devote to cubic functions the characterization of plateaued functions in terms of the value distribution of the second-order derivatives, and hence this reveals non-existence of homogeneous cubic bent (and also (homogeneous) cubic plateaued for some cases) functions in odd characteristic. We use a rank notion which generalizes the rank notion of quadratic functions. This rank notion reveals new results about (homogeneous) cubic plateaued functions. Furthermore, we observe non-existence of a function whose absolute Walsh transform takes exactly 3 distinct values (one being zero). We finally provide a new class of functions whose absolute Walsh transform takes exactly 4 distinct values (one being zero).     

On the arithmetic Walsh coefficients of Boolean functions

We generalize to the arithmetic Walsh transform (AWT) some results which were previously known for the Walsh–Hadamard transform of Boolean functions. We first generalize the classical Poisson summation formula to the AWT. We then define a generalized notion of resilience with respect to an arbitrary statistical measure of Boolean functions. We apply the Poisson summation formula to obtain a condition equivalent to resilience for one such statistical measure. Last, we show that the AWT of a large class of Boolean functions can be expressed in terms of the AWT of a Boolean function of algebraic degree at most three in a larger number of variables.     

斜变换ST的演化生成与快速算法

１．引言 含有“斜”基向量的正交变换（斜变换 ＳＴ）概念是由 Ｅｎｏｍｏｔｏ ＆ Ｓｈｉｂａｔａ（１９７１）提出的［１］．斜向量是一个在其范围内呈均匀阶梯下降的离散锯齿波形．对于亮度逐渐改变的图象,用斜向量来表示是适合的． Ｅｎｏｍｏｔｏ 　＆ 　Ｓｈｉｂａｔａ仅考虑了斜向量长度为 ４和 ８的情况．Ｐｒａｔｔ等人利用递推性将 ＳＴ推广到 Ｎ＝ ２ｍ阶的情形,给出了 ＳＴ的一般定义［２］,并与其它变换进行了比较［３］．ＳＴ已成功地用在图象编码上,而且在非正弦类交换编码的应用中,斜变换的效果最好［２,３］． Ａｈ…     

Divergence of the Fourier series by generalized Haar systems at points of continuity of a function

We obtain a connection between the Dirichlet kernels and partial Fourier sums by generalized Haar and Walsh (Price) systems. Based on this, we establish an interrelation between convergence of the Fourier series by generalized Haar and Walsh (Price) systems. For any unbounded sequence we construct a model of continuous function on a group (and even on a segment [0, 1]), whose Fourier series by generalized Haar system generated by this sequence, diverges at some point.     

Walsh函数的一个迭代方程体系

Shanks曾用迭代方程产生离散佩利编号Walsh函数。作者在[4]中,给出了产生离散沃尔什编号Wa1sh函数的迭代方程. 本文在上述基础上,提出了一个产生离散哈德玛编号Walsh函数的迭代方程,推出了离散哈德玛编号Walsh函数的表示式及其变换(FWHT)的快速计算公式. 上述三个极为类似的迭代方程,已构成了离散Walsh函数的迭代方程体系.连续的Walsh函数,也能用迭代方程这种形式来描述.     

A partial ordering that generates filtered Walsh series

The second author recently introduced modified Walsh–Dirichlet kernels which generated filtered Walsh–Fourier series. The purpose of this article is to show that those filtered Walsh series can be generated by a partial ordering on the natural numbers which may be of interest in its own right.     

Smooth PONS

PONS^tm is a basis which satisfies all of the fundamental properties of the Walsh functions (each element is piecewise constant, takes on only the values ±1, and can be efficiently computed via a fast transform) plus three additional properties that are false for the Walsh functions: PONS is optimal with respect to a global uncertainty principle; all PONS elements have uniformly bounded crest factors; and all PONS elements are QMF’s. In 1991, Ingrid Daubechies asked whether there exists a smooth basis satisfying the global uncertainty principle property. In this article we show how to transform any basis into another basis by applying the PONS construction, thereby providing an affirmative answer to this question.     

Walsh函数的一种新定义及快速算法设计

传统的Walsh函数是以Rademacher函数为基函数生成 .本文运用对称复制的观点 ,定义了一种新函数 G函数 ,并以G函数为基础 ,定义了四种序的Walsh函数 ,同时 ,运用序码分析方法 ,实现了两种序Walsh变换的快速算法设计 .