12 Result(s)
-
Chapter
Walsh Series with Monotone Decreasing Coefficients
Let {c n }, n = 0, 1, …, be a monotone decreasing sequence of real numbers, i. e., Δc n ≡ c n - cn+1 ≥ 0 for n = 0, 1, …...
-
Chapter
Divergent Walsh-Fourier Series. Almost Everywhere Convergence of Walsh-Fourier Series of L2 Functions
In Chapter 2 we saw that even for a continuous function it is necessary to impose additional conditions to insure that its Walsh- Fourier series converges at every point. Without such conditions, as we remarke...
-
Chapter
Summability of Walsh Series by the Method of Arithmetic Means
As we saw in §2.3, there exist continuous functions whose Walsh- Fourier series diverge at a given point. We shall show in Chapter 9 that there exist integrable functions whose Walsh-Fourier series diverge eve...
-
Chapter
Generalized Multiplicative Transforms
Let 1 ≤ p < ∞. A complex valued function f(x) is said to belong to L p (0, ∞) if ∫0∞|f(x)| p dx > ∞. The norm of f(x) in the space L p ...
-
Chapter
Lacunary Subsystems of the Walsh System
The Rademacher system, {r k (x)} = {w2 k , k = 0,1,…, which was used to define the Walsh system (see §1.1), is a typical example of what is called a lac...
-
Chapter
Other Applications of Multiplicative Functions and Transformations
Filtering is one of the fundamental techniques of digital signal processing. By a digital filter we shall mean a transformation which takes a sequence of numbers {x(n)}, called the input, in another sequence {y(n
-
Chapter
Operators in the Theory of Walsh-Fourier Series
In this chapter, and the next, we shall obtain several results about Walsh-Fourier series by using properties of operators which take one space of measurable functions to another. We begin with definitions and...
-
Chapter
Applications of Multiplicative Series and Transforms to Digital Information Processing
In the last decade interest has increased significantly in applications of the Walsh system and its generalizations, especially applications to digital information processing. This interest stems from a peculi...
-
Chapter
Walsh-Fourier Series. Basic Properties
Within the collection of all Walsh series, Walsh-Fourier series play a crucial role. These are the series of the form (1.4.1) whose coefficients are given by the formula
-
Chapter
Approximations by Walsh and HAAR Polynomials
Let f(t) be a function continuous on the interval [0, 1]. Consider the
10.1.1 $$\begin{array}{*{20}{c}} {{{E}_{n}}(f) = \mat... -
Chapter
Walsh Functions and Their Generalizations
Consider the function defined on the half open unit interval [0, 1) by $${{r}_{0}}(x) = \left\{ {\begin{array}{*{20}{c}} 1 & {for x \in [0, ...
-
Chapter
General Walsh Series and Fourier-Stieltjes Series. Questions on Uniqueness of Representations of Functions by Walsh Series
In this chapter we shall consider general Walsh series, i. e., series whose coefficients are not necessarily Walsh-Fourier coefficients of some function. For the study of these series, the function which is th...
