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  • Appendix Fundamental Properties of Generalized Functions

    A.1 Definition of generalized functions

    First of all, let us give some material from mathematics, which is necessary for defining the concept of a generalized function. Let us consider the following linear functional :

    T [ϕ(x)] = ∫ f (x)ϕ(x)dx. (A.1)

    Here ϕ(x) is the argument of the functional, and f (x) is its kernel, which deter- mines the properties of the functional. Almost everywhere below, we consider these functions being real, mapping points x of the number line R onto R. Thereby, the linear functional T [ϕ(x)] maps the “points” ϕ(x) of a certain functional space D (ϕ(x) ∈ D) onto points of the real line R. Let us call the above mentioned “func- tional points” ϕ(x) test functions, and D — a space of test functions. Let us also note that indefinite integrals, in our notation, symbolize definite integrals over the entire x-axis, i.e. having the integration limits ±∞.

    As soon as the space D of test functions is chosen, the conjugate to it set D ′ of linear continuous functionals T [ϕ ], underlying the following definition of general- ized functions, becomes automatically defined.

    The space D of test functions is subject to several natural requirements: it needs to be sufficiently large, in order to identify any continuous kernel f (x) through val- ues of the integral (A.1). In other words, as soon as values of the functional T [ϕ ] are known for a certain set of test functions ϕ ∈ D , values of the kernel f (x) may be defined for all x. On the other hand, the space D of test functions is subjected to quite strict limitations, in order to obtain as rich a set D ′ of linear continuous functions T [ϕ ] as possible.

    It so happens that the family of infinitely differentiable compact functions is a suitable candidate for the space D . Below, we reserve the notation D precisely for this functional space. Recall that a function ϕ(x) is called compact, if it has a bounded support. In its turn, the support supp ϕ of a function ϕ(x) is the closure of the set of the points on the x-axis, where ϕ(x) = 0. Now we are fully prepared to formulate the definition of a generalized function.

  • 442 Fundamental Properties of Generalized Functions

    By definition: any linear functional T [ϕ ], which is continuous on the set D of infinitely differentiable compact functions is called a generalized function.

    Let us give a few comments on the above-formulated definition of a generalized function:

    1. Recall that a functional T [ϕ ] is called linear, if, for all test functions ϕ(x) ∈D and ψ(x) ∈D , the following equality holds:

    T [αϕ +βψ ] = αT [ϕ ]+βT [ϕ ] ,

    where α and β are arbitrary constants. 2. A functional T on D is called continuous, if for any functional sequence

    {ϕk(x)}, whose elements belong to the space D and converge at k→ ∞ to a certain test function ϕ(x) ∈ D , the corresponding numerical sequence {T [ϕ k]} converges to the number T [ϕ ].

    Thereby convergence of the sequence {ϕ k(x)} is understood in the following sense:

    a) The supports of all elements of the sequence {ϕk} belong to some bounded set of the x-axis.

    b) At k→ ∞, the functions ϕk(x) and all their derivatives ϕ (n)k (x) uniformly con- verge to the function ϕ(x) and its derivatives, respectively.

    If it is possible to express a linear continuous functionalT [ϕ(x)] in the form of the integral (A.1), whose kernel f (x) is an everywhere continuous, bounded function, then such a functional is called a regular generalized function and it is identified with the kernel f (x). In this sense, all ordinary continuous functions are regular generalized functions.

    There are, however, linear continuous functionals T [ϕ ], which cannot be iden- tified with a certain continuous kernel f (x). In this case, such a linear continuous functional T [ϕ(x)] is called a singular generalized function.

    Perhaps the most important example of a singular generalized function is the functional

    δ [ϕ(x)] = ϕ(0), (A.2)

    mapping the test function ϕ(x) ∈ D onto its value at x = 0. This, obviously linear and continuous, functional is called a delta function. Although this functional cannot be expressed by means of the usual integral (A.1), it is often written in the following symbolic integral form:

    δ [ϕ(x)] = ∫ δ (x)ϕ(x)dx= ϕ(0), (A.3)

    where δ (x) is the symbol for the delta function, which is defined by its sifting prop- erty, according to which the value ϕ(0) corresponds to the “integral” in (A.3).

    The advantage of such a symbolic notation consists in its clarity during actual computations using delta functions. Let us illustrate the “transparency” of the sym- bolic integral notation of generalized functions, related to the delta function, by

  • A.2 Fundamental sequences 443

    expressing the functional δa[ϕ(x)] = ϕ(a) by means of the equality∫ δ (x−a)ϕ(x)dx= ϕ(a) (A.4)

    via a delta function δ (x−a) with a shifted argument. In the following, we will pre- dominantly use the mathematically not quite correct, but convenient while solving applied problems, symbolic integral notation of the form (A.3), (A.4).

    While discussing properties of generalized functions, an important role is played by the concept of the support of a generalized function. Namely, a linear continuous functional T is considered to be equal to zero in an open domain B of the x-axis, if T [ϕ(x)] = 0 for all test functions ϕ(x), whose supports belong to the set B. The closure of the largest open domain, where a generalized function T is not equal to zero is called the support of the generalized function T , and it is denoted by supp T . From the definition, it immediately follows that the support of the delta function consists of the only point x= 0, i.e.

    supp δ (x) = {0}. (A.5)

    Note that if a generalized function T is compact, i.e. it has a bounded support, then it is possible to relax the conditions, imposed on the set of test functions ϕ(x), by requiring only their infinite differentiability (ϕ ∈C∞). For instance, it is possible to assign in (A.6) ϕ(x) ≡ 1, which leads to the normalization condition of the delta function ∫

    δ (x)dx= 1. (A.6)

    In conclusion, let us discuss an important for physical applications, but ignored by mathematicians, property of the delta function. It belongs to a comparatively narrow class of scale-invariant functions, whose argument may be dimensional; e.g., it may be a spatial coordinate x or time t; and which has a non-zero dimension depending on the dimension of the argument. Namely, the delta function, which has the spatial coordinate x as its argument, has the dimension of the reciprocal of x:

    [δ (x)] = L−1,

    as it is seen from the normalization condition (A.6). Thus δ (x) has the same dimen- sion as the power function 1/x.

    A.2 Fundamental sequences

    Although the delta function is, in principle, non-representable in the integral form (A.3), it can, nevertheless, be obtained as the limit of a sequence of ordinary inte- grals

    Tk[ϕ(x)] = ∫ fk(x)ϕ(x)dx, (A.7)

  • 444 Fundamental Properties of Generalized Functions

    corresponding to the sequence of locally integrable functions { f k(x)}. This limit is understood in the sense that, for some sequence { f k(x)}, k = 1,2, . . . , and for any test function ϕ(x) ∈D , the following limiting equality holds:

    Tk[ϕ(x)]→ ϕ(0) (k→ ∞).

    It this is indeed so, it is said that the sequence { fk(x)} weakly converges to the delta function.

    The choice of a weakly converging to the delta function δ (x) sequence of regular linear functionals {Tk}, defined by their kernels { fk(x)}, is not unique. Therefore it is useful to discuss several such sequences, which give an opportunity better to understand the “internal structure” of delta functions.

    Example 1. Let us consider a family of Gaussian functions (see Fig. A.1)

    γε(x) = 1√ 2πε


    ( − x


    ) , (A.8)

    depending on the parameter ε > 0, and take the sequence of functions f k(x) = γ1/k(x), k = 1,2, . . . as a weakly convergent sequence. Note that the factor in front of the exponential function on the right-hand side of Eq. (A.8) is chosen so that the normalization condition holds: ∫

    γε (x)dx= 1.

    At k→ ∞, the value of the parameter ε tends to zero, and the Gaussian func- tion, “approximating the delta function,” becomes progressively higher and nar- rower, “concentrating” at the origin, while conserving the total area under the Gaus- sian curve so that, for all elements of the Gaussian-function sequence, the above- mentioned normalization condition is satisfied.

    Example 2. Another, important for theory and applications, weakly convergent to the delta function sequence is generated by the Cauchy distribution

    Fig. A.1 Three elements of the sequence of Gaus- sian functions (A.8), weakly convergent to the delta function, for k = 1;2;4 (ε = 1;1/2;1/4).

  • A.2 Fundamental sequences 445

    cε (x) = 1 π

    ε x2 + ε2

    . (A.9)

    Although, at first sight, the profiles of the Cauchy distribution seem similar to the Gaussian curves discussed above, there are significant distinctions between them. The main one is the fact that the Cauchy distribution tends to zero at x→±∞ rather slowly, according to the power law

    cε(x)∼ επx2 .

    Nevertheless, it can be rigorously proved that for any continuous function ϕ(x) with a bounded support, the following limitin