Generalized Functions, Volume 2: Spaces of Fundamental and Generalized Functions

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go to site Letting , we have is a Cauchy sequence. On the other hand, replacing by in 4. Now letting we see by definition of that satisfies. Letting in 4. To prove the uniqueness of , we assume that is another function satisfying 4. Setting and in 4. Then it follows from 4. Letting , we have for all. This proves the uniqueness. Since is given by the uniform limit of the sequence , is also continuous on.

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Equations of Mathematical Physics. The first systematic theory of generalized functions also known as distributions was created in the early s, although some aspects were developed much earlier, most notably in the definition of the Green's function in mathematics and in the work of Paul Dirac on quantum electrodynamics in physics. Powered by. Thank you for posting a review! In Chapters 2 and 3 of the book, the authors transfer many results presented in Volume 1 to generalized functions corresponding to these more general spaces.

Letting we have the inequality. Now we consider the case. For this case, replacing by in 4. Following the similar method in case of , we see that. Now letting in 4. Suppose that in or satisfies the inequality. Hyers DH: On the stability of the linear functional equation. Rassias ThM: On the stability of the linear mapping in Banach spaces. Proceedings of the American Mathematical Society , 72 2 Transactions of the American Mathematical Society , 11 Aequationes Mathematicae , 44 Volume Rassias ThM: On the stability of functional equations and a problem of Ulam.

Acta Applicandae Mathematicae , 62 1 Rassias JM: Solution of the Ulam stability problem for quartic mappings. Bulletin of the Korean Mathematical Society , 40 4 Journal of Mathematical Analysis and Applications , 2 Park C-G: On the stability of the orthogonally quartic functional equation. Bulletin of the Iranian Mathematical Society , 31 1 Journal of Mathematical Analysis and Applications , 3 International Journal of Mathematics and Mathematical Sciences , 18 2 Czerwik S: On the stability of the quadratic mapping in normed spaces.

Abstract and Applied Analysis , Skof F: Local properties and approximation of operators. Rendiconti del Seminario Matematico e Fisico di Milano , Publications of the Research Institute for Mathematical Sciences , 30 2 Chung J, Lee S: Some functional equations in the spaces of generalized functions. Aequationes Mathematicae , 65 3 Chung J: Stability of functional equations in the spaces of distributions and hyperfunctions. Journal of Mathematical Analysis and Applications , 1 Lee Y-S: Stability of a quadratic functional equation in the spaces of generalized functions.

Journal of Inequalities and Applications , Matsuzawa T: A calculus approach to hyperfunctions. Nagoya Mathematical Journal , Publications of the Research Institute for Mathematical Sciences , 29 2 Download references. The second author was supported by the Special Grant of Sogang University in Correspondence to Young-Su Lee. Reprints and Permissions. Search all SpringerOpen articles Search.

Abstract We consider the general solution of quartic functional equations and prove the Hyers-Ulam-Rassias stability. Introduction One of the interesting questions concerning the stability problems of functional equations is as follows: when is it true that a mapping satisfying a functional equation approximately must be close to the solution of the given functional equation? General Solution of 1. Stability of 1. Theorem 3. References 1. Google Scholar 2. Google Scholar 5. Add to Basket. Book Description Academic Press, Condition: Very Good. New edition.

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Seller Inventory mon Sprache: Englisch Gewicht in Gramm: Seller Inventory Satisfaction Guaranteed! Book is in Used-Good condition. But since is positive and rt. Theorem V, Any conditionally positive generalized function F of order s on the space Z has the form P. At first glance the measure v,. As a matter of fact, Vf. Therefore the manifold consists of all hyperplanes. Therefore 15 is equivalent to In order to complete the proof of the lemma, it remains for us to show that the measure p. By Theorem 3' of Section 3, the measure Vg; which defines it is positive and tempered.

The convergence of follows from the fact that 9 a: is rapidly decreasing and that p. Let us prove the continuity of the functional Fj in the topology of Z. From the properties of the measure p. J for? First we prove that 24 defines a functional on Z. We showed on p. Therefore 24 defines a continuous linear functional on Z. Now let us show that F is conditionally positive of order s, i. For this we note the following. Using this equality, we can write 25 in the form F.

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The converse is also true. First we show that the vanishing of the moments and Cj, of 9. We proceed now to the description of the bilinear functional B p, ifi. This proves our auxiliary statement. Using the Fourier transformation, we obtain the following corollary of Theorem 6. It contains also the function dy,. He proved that every such function has the form. Povzner [reference 54 ]. A corresponding example will be given in Section 6.

We will obtain this result further on as a corollary of more general results connected with evenly positive-definite generalized functions. In order to establish the connection of this circle of topics with the theory of generalized functions, we remark the following. Volume 11, Chapter IV, Section 2. A brief definition is given on p. Let F be an evenly positive-definite generalized function on the space S';. Then F is the Fourier transform of a uniquely defined even positive measure fi, concentrated on the set 'JJ!

For functions which arc odd in at least one variable, in view of the evenness of F, F, y. The topology in S; induces a topology in O.

Since the set 93! Theorem 1". Since rji. Therefore the integral 12 converges. According to this lemma, for any function y! But, as we have seen above. Theorems 1, 1', and 1" are mutually equivalent. Therefore the proof of Theorem 1 is complete. The converse of Theorem 1 also holds. The proof of this assertion is trivial. In the course of proving Theorem 1 we omitted the proofs of Lemmas 2 and 3. Let us fill this gap. From the uniformity, in every bounded region, of the convergence of to and from 20 it follows also that the sequence converges to in the topology of Q.

This proves Lemma 2. Now we prove Lemma 3, i. Conversely, if p. Thus, the following theorem holds Theorem 3. The measure p. T he spice S, consists of inhnitely differentiable functions? We omit the details of the proof. Positive-Definite Generalized Functions and Groups of Linear Transformations The concept of an evenly positive-definite generalized function which we have studied is a special case of a more general concept, connected with groups of linear transformations.

For example, if the group G consists of all transformations in which some variables change sign, then the functions which are symmetric relative to G are the even functions. Amer Math. Since the order ""Cf i: c Titchniarsh. This problem will be solved in Section 6. A function is called positive, if p.

We will say that a function yi. The following theorem holds. Let L be a linear space whose elements are functions defined on some set iUi, and let F be a positive additive homogeneous functional on L. Then the functional F can be extended, without losing its positivity, to any linear space M consisting of functions which are subordinate to L.

Take any function ipi. In order to e. V, then. We start with the second converse part of the theorem. From the definition of the topology in Z cf. In addition, the values of these integrals will remain bounded as y; r; ranges over the neighbor- hood I.

We have therefore proven that F is bounded on U. Therefore we will denote it simply by p. Indeed, the func- tional F is continuous relative to the topology of Z. This neighborhood U is defined by an inequality of the form sup 1 1. We remark that this theorem is also valid for functions of several variables. Namely, the following assertion holds. The proof of this theorem repeats almost verbatim that of Theorem 4.

We have not stopped to prove Theorem 4' because we do not know, for functions of several variables, whether or not the concepts of positivity and multiplicative positivity are equivalent. For functions of one variable, as has been shown in Section 6. Thus, 3 describes not only positive, but also multi- plicatively positive generalized functions on i. We have already pointed out in Section 6 2 that the extension of a positive functional may be nonunique Therefore the positive measures fi, and fig defined according to Theorem 4 by a positive functional on Z are generally speaking not uniquely defined by this functional e present here an example of this nonuniqueness 6.

We define and in the same way. We now prove that I.


The first systematic theory of generalized functions (also known as distributions) was created in the early s, although some aspects were. Spaces of Fundamental and Generalized Functions, Volume 2, analyzes the general theory of linear topological spaces.

This will show that the distinct pairs fi], fXo and fil define the same positive functional on Z,. To prove 14 , we note that I exp[ —. Therefore its integral along the contour of this region equals zero. Formula 14 holds for these functions, and therefore it holds also for We have therefore constructed the sought for example. Topological Algebras with Involutions As we have alreadj' said, the basic concepts introduced and studied in Sections positivity, multiplicative positivity, positive definiteness pertain in essence to the theory of topological algebras with involutions.

Let us give the basic definitions concerning these algebras. A linear space L is called an algebra with an involution, if an operation of multiplication of elements and an operation of passing to the adjoint element. An algebra L is called comtiiutative if the multiplication of elements is commutative, i. As a rule, one considers algebras which are linear topological spaces. In this case the algebraic operations including the involution are required to be continuous in the topology of the space. Vn is some real number. However, certain algebras have other symmetric homomor- phisms.

We will denote the set of all symmetric homomorphisms of an algebra L by 'Hi. Obviously, to each element xeL there corresponds a function. To every positive finite measure a M defined on H! For any. Therefore the functional is multiplicatively positive. It can be shown that these functionals exhaust the set of multiplicative!

Precisely speaking, the following theorem holds. Every multiplicatively positive linear functional F on a commutative normed algebra L with involution can be represented, in a unique way, in the form F,. For the proof of this theorem the reader can consult, for example, M. This example is constructed in the following way. As is proved in algebraic geometry, an infinite set of third-order curves can be passed through these points, and all of these curves intersect in some ninth point Mg cf. Princeton Univ. Press, Princeton, New Jersey, Now the curves cpf.

But then, as was remarked above, they also pass through the point Mg, i. Let us now show that there e.

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This will prove the existence of a positive polynomial which cannot be represented as a sum of squares of polynomials. Let us show that these curves do not pass through Mg. But if curves of order m and 7t have more than mn common points, then they have a common component cf. V, y does not belong to the closure of this cone. This functional can be extended to the entire algebra of polynomials in such a way that it assumes positive values on all polynomials which are representable as sums of squares of polynomials. As a result one obtains a linear functional on the algebra of all polynomials in two variables which is multiplicatively positive but not positive.

The example which has been considered shows that for topological rings with involutions the concepts of positivity and multiplicative positivity of linear functionals do not, generally speaking, coincide. It would be very important to distinguish the class of topological rings in which these concepts coincide. As we have seen, the rings K, S, Z, S , and others belong to this class.

This problem consists in the following. Random Variables We assume that the reader is familiar with the basic concepts of probabilitj' theory. V2 , if. Defining the function P. X' that the value of the random variable i belong to the Borel set X. If we consider several random variables It is necessary to know also the probability P.

Of course, giving the probability distribution P Aq, We prove that both P. Vi, x and Pi. Certainly, then, the probability P. But it follows from the compatibility condition that P. V, CO , where P.

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Since P. V and P. Henceforth we will not distinguish between equal random variables. Yo , where P. We now proceed to the definition of the limit of a sequence of random variables. Suppose that we are given a sequence We will say that this sequence converges to the? If limt. In the case where K consists of functions of one variable, the corresponding random function rvill be called a generalized random process. In the case where K is a space of functions of several variables, 0 is called a generalized random field. Let us pause to consider the physical motivation for the concept of a generalized random function.

The usual concept of a random function, which we gave in Section 1. However, every actual measurement is accomplished by means of an apparatus w'hich has a certain inertia. These quantities are compatible and depend linearly upon 9. Moreover, small changes of the function 9 t cause small changes in the random variable 0 9 apparatuses which differ only slightly give close readings. One can construct examples of generalized random fields in the same way. Operations on Generalized Random Processes The operations which can be performed on generalized random processes are defined in a manner analogous to that by which they are defined for generalized functions.

The ordinary operations on generalized random processes are defined by means of the corresponding operations on the test functions 9 f. It is given by iii[0 9i. V,V, exp[- i Ax,. It consists of those points. The probability distribution in question is given, on by a formula similar to 1. We have obtained the probability distribution for proper Gaussian processes. We will not dwell further on this question. Along with real Gaussian processes, one can introduce complex Gaussian processes. Suppose now that 91, 9,, are linearly dependent. Since the functions 91, It can be shown that these probability distributions define a continuous linear random functional on K, i.

Taking into account that d. Let us now consider the general case. Derivatives of Generalized Gaussian Processes We now prove that the derivative of a generalized Gaussian random process is itself a Gaussian random process. By definition of the derivative of a random process, the random variable Let us find the form of the correlation functional of the Wiener process.

Let Blip, ip be the correlation functional of the Gaussian random process. According to Section 2. In other words, the process is stationary if the result of measurements on it by apparatuses characterized by the functions If 0 is stationary, then its mean is invariant under translation.

The Correlation Functional of a Stationary Process Let us now find the general form of the correlation functional of a complex stationary generalized random process. Thus, the correlation functional of a stationary generalized random process is a positive-definite bilinear Hermitean functional which is translation-invariant.

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Processes with Stationary Increments We proceed now to the study of generalized random processes with stationary wth-order increments. Namely, the following theorem holds. Suppose that 0 is a generalized random process with stationary nth-order increments, and Cq, If the Z A form a Gaussian family of random variables i.

Now we prove that the converse is true. We introduce a scalar product in K by means of the positive- definite functional B p, iji , the correlation functional of 0, i. We thus obtain an isometry between the spaces H and fi. But K and, consequently, S is dense in L-.

A having square mtcgrable moduli with respect to the measure a In particular contains the characteristic functions of all bounded Bore! Processes with Independent Values It is not possible, in the framework of the ordinary theory of random processes, to introduce processes with continuously varying time whose values at distinct times are independent random variables. In doing this we will establish a connec- tion between processes with independent values at every point and infinitely divisible random variables.

Physically, this means that the results of measuring the random quantity 0 in nonintersecting time intervals are mutually independent. An example of a process with independent values at every point is the velocity of a particle undergoing Brownian motion. It is always convenient to carry out the study of processes with in- dependent values at every point with the help of their characteristic functionals. However, there is little one can do with such processes. But on 3. The number n in formula 3 is called the order of the functional L p. For completeness, we present a proof of the theorem of Schur referred to.

If the Hermitean matrices a,. To prove this theorem, we need the following lemma. Vy, we obtain 6. Vj ]" will be positive-definite for n sufficiently large. Then the limit of this matrix, i. In Chapter II, Section 4. A Connection between Processes with Independent Values at Every Point and Infinitely Divisible Distribution Laws Theorem 2 implies a connection between processes with independent values at every point and infinitely divisible random variables.

Let x' a: be the characteristic function of an infinitely divisible random variable. The above mentioned connection between processes with independent valuesat every point and infinitely divisible random variables is established by the following theorem. The proof of this theorem proceeds in entirely the same way as the proof of the corresponding part of Theorem 2, and we omit it. Without carrying out its detailed proof, we state the following theorem. We have therefore proven that E,0.

Since F is concentrated on the diagonal. Hence the following theorem results. Theorem 9'. The nth-order moment of a generalized random process with independent values at every point is given by X J. Of course, not every such Gaussian process has the form since not every positive-definite functional of the form 24 can be represented in the form Basic Definitions Up to now, we have considered generalized random processes, i.

In this section we consider generalized random functions of several variable. In order to distinguish them from functions of one variable, we will call such func- tions generalized random fields. A substantial portion of the theory of generalized random fields is analogous to the corresponding portion of the theory of generalized random processes. In these cases we will restrict ourselves to the state- ments only of the corresponding results for example, in the theory of homogeneous fields, which is similar to the theory of stationary processes.

Then the joint probability distribution of the N — random variables is invariantt under the simultaneous trans- i. From this definition it is clear that if 0 is a generalized random field with homogeneous. This can be established in a way similar to that used in the case of random processes, by using the results of Chapter II, Section 4. Similarly, any bilinear functional, defiyjed for functions 9 d.

To do this we use the following lemma. Section 5. Let us now clarify the restrictions on the Hermitean form 9 which are imposed by relation But by the corollary in the appendix to Chapter II, Section 4, any function in K, whose moments up to and including order s — 1 equal zero, is the limit in JC of a sequence of such sums. Since the definition of the correlation functional included its conlmuiw in each variable separately, and therefore by Theorem 3 of Chapter I, Section 1.

But from this it follows that inequality fl2 can hold only if every coefficient is nonnegative. We omit the exact statement of the result thereby obtained, which is rather cumbersome. Multidimensional Generalized Random Fields In certain applications of the theory of random fields, it is not sufficient to consider only scalar fields. For example, the velocity of particles in a turbulent flow can be considered as a random quantity. Indeed, if gq, Therefore X T.

Weproceed now to the consideration of homogeneous multidimensional fields. A multidimensional generalized random field 0 is called hojuoge- ncous, if the probability distribution of the random matrix! We now introduce the concept of a vectorial field. Cylinder Sets In this chapter we study measures in linear topological spaces.

We will restrict ourselves to considering measures in spaces which are adjoint to some linear topological space 0. We will first study measures on the simplest sets in 0' — the cylinder sets. Following this, measures on sets of a more general form will be considered. Let us define the notion of a cylinder set in the space 0'. Thus the elements fi , As examples of cylinder sets we may consider the half-spaces in 0 defined by an inequality of the form F. One can give another definition of a cylinder set. Simplest Properties of Cylinder Sets Before studying the properties of cylinder sets, we stop to consider some simple assertions concerning linear topological spaces.

We will consider only locally convex linear topological spaces, i.

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The class of locally convex spaces is adequately broad; in particular, it contains all countably normed spaces. The following theorem on the extension of linear functionals holds for these spaces. Any linear functional F which is defined on a subspace W of a locally convex linear topological space 0 can be extended to a linear functional on all of 0.

Indeed, any element Fe0' is a linear functional on 0, and conse- quently on F. Thus, to every element ' The notion of a cylinder set can be introduced for any linear topological space 0. Namely, let IP be some closed linear subspace in 0, and A some set in the factor space 0l'f. However, all that we need are cylinder sets in 0 corresponding to annihilators of finite-dimensional subspaces. The following property is proven in entirely the same way. We see, thus, that the cylinder sets form an algebra of sets. This condition is also sufficient. In other words, the following assertion holds.

Suppose that is a system of Jiortnalized positive ineastires, regular in the sense of Caratheodory, in the factor spaces If Eq. Obviously fi Z is a cylinder set measure in 0', and all the measures iv are induced by fi. It can be shown that it is sufficient to verify 4 only for half-spaces in 0'. For the proof of this lemma cf.

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The Continuity Condition for Cylinder Set Measures We will henceforth consider only measures which satisfy a certain continuity condition. Obviously, if T carries the finite- dimensional subspace C into the finite-dimensional subspace! We will call fi2 the measure induced front by the mapping T. This extension can be carried out in the following way. We call the cylinder sets Borel sets of the 2 eroth class. Suppose that Borel sets of class jS have already been defined, where y6 is any transfinite number less than a.

Thus, Borel sets are defined for all transfinite numbers of the first and second classes. This result will be proven in Section 2. For the proof of this basic result we need certain results of measure theory. First of all we indicate the following simple criterion for the countable additivity of a measure. The necessity of the condition follows directly from the definition of countable additivity. As for the sufficiency, suppose that Z — Z. This theorem can be stated in another, equivalent, way. In order that a measure fj.

Here we introduce a condition for the count- able additivity of measures on the cylinder sets in spaces adjoint to countably Hilbert spaces, which is more convenient to use. Suppose that the cylinder set measure jx in the space 0' adjoint to a countably Plilbert space 0 is countably additive. Therefore p. Therefore 0' is a countable union of balls. Nordhoff, Groningen, Since the Tikhonov product of compact sets is compact, I is a compact set. Since a closed subset of a compact space is compact, and ct is a homeomorphism, then 5 2?

In other words, from any covering of 5 2? Let us now prove Theorem 2'. To see that p. Therefore we have S, 2? We introduce on the space Q of these unit vectors the measure t, defined naturally as normalized surface measure on the unit sphere. In the spherical coordinates. V, r, to dT to. The half-spaces which are involved in Lemma 2 are bounded by the planes. To prove Theorem 3, we need a lemma, similar to Lemma 2, in which the sphere of radius i? We precede this lemma by the following assertion concern- ing the average value of the square of the distance from the origin to the tangent planes of an ellipsoid.

Let fz be a positive normalized measure in w-dimensional. Let Q. Lemma 4'. Hence Theorem 2' implies that the measure p, is countably additive. This concludes the proof of Theorem 3. This assertion is proven by constructing, for any nonnuclear countably Hilbert space 0, a positive normalized cylinder set measure in 0' which is not countably additive, hloreover, this measure can be chosen to be a so-called Gaussian measure cf. Theorem 2 of Section 3. We have already mentioned that in certain questions one can consider, instead of nuclear spaces, two Hilbert spaces which are connected by a nuclear mapping.

Let us indicate here the analog of Theorem 3 which is obtained by this replacement. Section 1. That is, the following theorem holds. Theorem S. Since, according to Theorem 3, any positive normalized cylinder set measure, in the conjugate space of a nuclear space, which satisfies the continuity condition is countably additive, we obtain from Theorem 5 the following result. Then any positive normalized measure, on the cylinder sets of the conjugate space 0', which satisfies the continuity condition is countably additive.

Since the space K is the union space of its nuclear subspaces K n , any measure on the cylinder sets of K' which has the properties indicated above is countably additive. A Condition for the Countable Additivity of Measures on the Cylinder Sets in a Hilbert Space Theorem 3 gives a condition for the countable additivity of all cylinder set measures in the conjugate space of a countably Hilbert space.

Let Fj, Bg, We will call this topology the topology defined by the operators Thus we have constructed a sequence This means that fi is continuous relative to the topology in H defined by the B,.