Absolute convergence

Abstract: Let f (x) and K (x) be two functions integrable over the interval (-∞,+∞). It is very well known that their composition $$ \int\limits_{{ - \infty }}^{{ + \infty }} {f(t)K\left( {x - t} \right)dt} $$ exists, as an absolutely convergent integral, for almost every x. The integral can, however, exist almost everywhere even if K is not absolutely integrable. The mostinteresting special case is that of K (x) = 1/x. Let us set $$ \tilde{f}(x) = \frac{1}{\pi }\int\limits_{{ - \infty }}^{{ + \infty }} {\frac{{f(t)}}{{x - t}}dt} $$ .

Abstract: 1 Walsh Functions and Their Generalizations.- 1.1 The Walsh functions on the interval [0, 1).- 1.2 The Walsh system on the group.- 1.3 Other definitions of the Walsh system. Its connection with the Haar system.- 1.4 Walsh series. The Dirichlet kernel.- 1.5 Multiplicative systems and their continual analogues.- 2 Walsh-Fourier Series Basic Properties.- 2.1 Elementary properties of Walsh-Fourier series. Formulae for partial sums.- 2.2 The Lebesgue constants.- 2.3 Moduli of continuity of functions and uniform convergence of Walsh-Fourier series.- 2.4 Other tests for uniform convergence.- 2.5 The localization principle. Tests for convergence of a Walsh-Fourier series at a point.- 2.6 The Walsh system as a complete, closed system.- 2.7 Estimates of Walsh-Fourier coefficients. Absolute convergence of Walsh-Fourier series.- 2.8 Fourier series in multiplicative systems.- 3 General Walsh Series and Fourier-Stieltjes Series Questions on Uniqueness of Representation of Functions by Walsh Series.- 3.1 General Walsh series as a generalized Stieltjcs series.- 3.2 Uniqueness theorems for representation of functions by pointwise convergent Walsh series.- 3.3 A localization theorem for general Walsh series.- 3.4 Examples of null series in the Walsh system. The concept of U-sets and M-sets.- 4 Summation of Walsh Series by the Method of Arithmetic Mean.- 4.1 Linear methods of summation. Regularity of the arithmetic means.- 4.2 The kernel for the method of arithmetic means for Walsh- Fourier series.- 4.3 Uniform (C, 1) summability of Walsh-Fourier series of continuous functions.- 4.4 (C, 1) summability of Fourier-Stieltjes series.- 5 Operators in the Theory of Walsh-Fourier Series.- 5.1 Some information from the theory of operators on spaces of measurable functions.- 5.2 The Hardy-Littlewood maximal operator corresponding to sequences of dyadic nets.- 5.3 Partial sums of Walsh-Fourier series as operators.- 5.4 Convergence of Walsh-Fourier series in Lp[0, 1).- 6 Generalized Multiplicative Transforms.- 6.1 Existence and properties of generalized multiplicative transforms.- 6.2 Representation of functions in L1(0, ?) by their multiplicative transforms.- 6.3 Representation of functions in Lp(0, ?), 1 < p ? 2, by their multiplicative transforms.- 7 Walsh Series with Monotone Decreasing Coefficient.- 7.1 Convergence and integrability.- 7.2 Series with quasiconvex coefficients.- 7.3 Fourier series of functions in Lp.- 8 Lacunary Subsystems of the Walsh System.- 8.1 The Rademacher system.- 8.2 Other lacunary subsystems.- 8.3 The Central Limit Theorem for lacunary Walsh series.- 9 Divergent Walsh-Fourier Series Almost Everywhere Convergence of Walsh-Fourier Series of L2 Functions.- 9.1 Everywhere divergent Walsh-Fourier series.- 9.2 Almost everywhere convergence of Walsh-Fourier series of L2[0, 1) functions.- 10 Approximations by Walsh and Haar Polynomials.- 10.1 Approximation in uniform norm.- 10.2 Approximation in the Lp norm.- 10.3 Connections between best approximations and integrability conditions.- 10.4 Connections between best approximations and integrability conditions (continued).- 10.5 Best approximations by means of multiplicative and step functions.- 11 Applications of Multiplicative Series and Transforms to Digital Information Processing.- 11.1 Discrete multiplicative transforms.- 11.2 Computation of the discrete multiplicative transform.- 11.3 Applications of discrete multiplicative transforms to information compression.- 11.4 Peculiarities of processing two-dimensional numerical problems with discrete multiplicative transforms.- 11.5 A description of classes of discrete transforms which allow fast algorithms.- 12 Other Applications of Multiplicative Functions and Transforms.- 12.1 Construction of digital filters based on multiplicative transforms.- 12.2 Multiplicative holographic transformations for image processing.- 12.3 Solutions to certain optimization problems.- Appendices.- Appendix 1 Abelian groups.- Appendix 2 Metric spaces. Metric groups.- Appendix 3 Measure spaces.- Appendix 4 Measurable functions. The Lebesgue integral.- Appendix 5 Normed linear spaces. Hilbert spaces.- Commentary.- References.

Abstract: For a series of randomly discounted terms we give an integral criterion to distinguishbetween almost-sure absolute convergence and divergence in probability to $\infty$, these being the only possible forms of asymptotic behavior. This solves the existence problem for a one-dimensional perpetuity that remains from a 1979 study by Vervaat, and yields a complete characterization of the existence of distributional fixed points of a random affine map in dimension one.

Abstract: A novel power series representation of the generalized Mar- cum Q-function of positive order involving generalized Laguerre poly- nomials is presented. The absolute convergence of the proposed power series expansion is showed, together with a convergence speed analysis by means of truncation error. A brief review of related studies and some numerical results are also provided.