# Generalized functions

## Generalized functions Topics

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### Staircase function

A function composed of a set of equally spaced jumps of equal length, such as the ceiling function , floor function , or nearest integer function .

### Sokhotsky's formula

Let be a linear functional acting according to the formula(1)(2)where and is the space of test functions. Then Sokhotsky's formula states that(3)where is the delta function (Vladimirov 1971, pp. 75-76).

### Heaviside step function

The Heaviside step function is a mathematical function denoted , or sometimes or (Abramowitz and Stegun 1972, p. 1020), and also known as the "unit step function." The term "Heaviside step function" and its symbol can represent either a piecewise constant function or a generalized function.When defined as a piecewise constant function, the Heaviside step function is given by(1)(Abramowitz and Stegun 1972, p. 1020; Bracewell 2000, p. 61). The plot above shows this function (left figure), and how it would appear if displayed on an oscilloscope (right figure).When defined as a generalized function, it can be defined as a function such that(2)for the derivative of a sufficiently smooth function that decays sufficiently quickly (Kanwal 1998).The Wolfram Language represents the Heaviside generalized function as HeavisideTheta, while using UnitStep to represent the piecewise function Piecewise[1,..

### Shah function

The shah function is defined by(1)(2)where is the delta function, so for (i.e., is not an integer). The shah function is also called the sampling symbol or replicating symbol (Bracewell 1999, p. 77), and is implemented in the Wolfram Language as DiracComb[x].It obeys the identities(3)(4)(5)(6)The shah function is normalized so that(7)The "sampling property" is(8)and the "replicating property" is(9)where denotes convolution.The two-dimensional sampling function, sometimes called the bed-of-nails function, is given by(10)which can be adjusted using a series of weights as(11)where is a reliability weight, is a density weight (weighting function), and is a taper. The two-dimensional shah function satisfies(12)(Bracewell 1999, p. 85).

### Generalized function

The class of all regular sequences of particularly well-behaved functions equivalent to a given regular sequence. A distribution is sometimes also called a "generalized function" or "ideal function." As its name implies, a generalized function is a generalization of the concept of a function. For example, in physics, a baseball being hit by a bat encounters a force from the bat, as a function of time. Since the transfer of momentum from the bat is modeled as taking place at an instant, the force is not actually a function. Instead, it is a multiple of the delta function. The set of distributions contains functions (locally integrable) and Radon measures. Note that the term "distribution" is closely related to statistical distributions.Generalized functions are defined as continuous linear functionals over a space of infinitely differentiable functions such that all continuous functions have derivatives..

### Sawtooth wave

The sawtooth wave, called the "castle rim function" by Trott (2004, p. 228), is the periodic function given by(1)where is the fractional part , is the amplitude, is the period of the wave, and is its phase. (Note that Trott 2004, p. 228 uses the term "sawtooth function" to describe a triangle wave.) It therefore consists of an infinite sequence of truncated ramp functions concatenated together.The sawtooth wave is implemented in the WolframLanguage as SawtoothWave[x].If , , and , then the Fourier series is given by(2)and the function can be written(3)(4)(5)(6)(7)where is the floor function.

### Triangle wave

Analytic representations the symmetric triangle wave with period 2 and varying between and 1 include(1)(2)(3)where is the fractional part of .The triangle wave is implemented in the WolframLanguage as TriangleWave[x].The Fourier series for the triangle waveis given by(4)which can be summed to yield the analytic expression(5)where is a Lerch transcendent.A form of triangle wave ranging between 0 and 1 with period 2 is given by(6)(Trott 2004, p. 228), where is the nearest integer function.

### Rectangle function

The rectangle function is a function that is 0 outside the interval and unity inside it. It is also called the gate function, pulse function, or window function, and is defined by(1)The left figure above plots the function as defined, while the right figure shows how it would appear if traced on an oscilloscope. The generalized function has height , center , and full-width .As noted by Bracewell (1965, p. 53), "It is almost never important to specify the values at , that is at the points of discontinuity. Likewise, it is not necessary or desirable to emphasize the values in graphs; it is preferable to show graphs which are reminiscent of high-quality oscillograms (which, of course, would never show extra brightening halfway up the discontinuity)."The piecewise version of the rectangle function is implemented in the Wolfram Language as UnitBox[x] (which takes the value 1 at ), while the generalized function version is implemented..

### Delta sequence

A delta sequence is a sequence of strongly peaked functionsfor which(1)so that in the limit as , the sequences become delta functions.Examples include(2)(3)(4)(5)(6)(7)(8)(Arfken 1985, pp. 482 and 488-489).

### Delta function

The delta function is a generalized function that can be defined as the limit of a class of delta sequences. The delta function is sometimes called "Dirac's delta function" or the "impulse symbol" (Bracewell 1999). It is implemented in the Wolfram Language as DiracDelta[x].Formally, is a linear functional from a space (commonly taken as a Schwartz space or the space of all smooth functions of compact support ) of test functions . The action of on , commonly denoted or , then gives the value at 0 of for any function . In engineering contexts, the functional nature of the delta function is often suppressed.The delta function can be viewed as the derivativeof the Heaviside step function,(1)(Bracewell 1999, p. 94).The delta function has the fundamental property that(2)and, in fact,(3)for .Additional identities include(4)for , as well as(5)(6)More generally, the delta function of a function of is given by(7)where..

### Step function

A function on the reals is a step function if it can be written as a finite linear combination of semi-open intervals . Therefore, a step function can be written aswhere , if and 0 otherwise, for , ..., .

### Null function

A null function satisfies(1)for all , so(2)Like a delta function, they satisfy(3)

### Square wave

The square wave, also called a pulse train, or pulse wave, is a periodic waveform consisting of instantaneous transitions between two levels. The square wave is sometimes also called the Rademacher function. The square wave illustrated above has period 2 and levels and 1/2. Other common levels for square waves include and (digital signals).Analytic formulas for the square wave with half-amplitude , period , and offset include(1)(2)(3)where is the floor function, is the sign function, and is the inverse hyperbolic tangent.The square wave is implemented in the WolframLanguage as SquareWave[x].Let the square wave have period . The square wave function is odd, so the Fourier series has and(4)(5)(6)(7)The Fourier series for the square wave with period , phase offset 0, and half-amplitude 1 is therefore(8)..

### Impulse pair

The even impulse pair is the Fourier transform of ,(1)It satisfies(2)where denotes convolution, and(3)The odd impulse pair is the Fourier transform of ,(4)