(b. Columbia, Missouri, 26 November 1894; d. Stockholm, Sweden, 18 March 1964
Wiener was the son of Leo Wiener, who was born in Byelostok, Russia, and Bertha Kahn. Although a child prodigy, he matured into a renowned mathematician rather slowly. At first he was taught by his father. He entered high school at the age of nine and graduated two years later. After four years in college, he enrolled at the Harvard Graduate School at the age of fifteen in order to study zoology. That soon turned out to be a wrong choice. He next tried philosophy at Cornell. “A philosopher in spite of himself,” Wiener took a Ph.D. at Harvard in 1913 with a dissertation on the boundary between philosophy and mathematics. A Harvard traveling fellowship paid his way to Europe. Bertrand Russell was his chief mentor at Cambridge and advised him to learn more mathematics. Neither the examples of Hardy and Littlewood at Cambridge,however, nor those of Hilbert and Landau at Göttingen, converted him to mathematics. Back in the United States in 1915, Wiener tried various jobs teaching philosophy, mathematics, and engineering. In the spring of 1919 he got a position in the mathematics department of the Massachusetts Institute of Technology, not then particularly distinguished in that discipline. An assistant professor in 1924, associate in 1929, and full professor in 1932, he remained at MIT until his retirement. Although his genius contributed to establishing the institute’s present reputation, he could never comfort himself over the failure of other American universities, and particularly of Harvard to show much interest in him. He traveled a great deal, to Europe and to Asia, and his visits to Germany in the interwar years left their traces in many anecdotes told in Continental circles. His Cybernetics made him a public figure. President Lyndon Johnson awarded him the National Medal of Science two months before his death. He died during a trip to Sweden and left two daughters. His wife was the former Margaret Engemann.
In appearance and behavior, Norbert Wiener was a baroque figure, short, rotund, and myopic, combining these and many qualities in extreme degree. His conversation was a curious mixture of pomposity and wantonness. He was a poor listener. His self-praise was playful, convincing, and never offensive. He spoke many languages but was not easy to understand in any of them. He was a famously bad lecturer.
Wiener was a great mathematician who opened new perspectives onto fields in which the activity became intense, as it still is. Although most of his ideas have become standard knowledge, his original papers, and especially his books, remain difficult to read. His style was often chaotic. After proving at length a fact that would be too easy if set as an exercise for an intelligent sophomore, he would assume without proof a profound theorem that was seemingly unrelated to the preceding text, then continue with a proof containing puzzling but irrelevant terms, next interrupt it with a totally unrelated historical exposition, meanwhile quote something from the “last chapter” of the book that had actually been in the first, and so on. He would often treat unrelated questions consecutively, and although the discussion of any one of them might be lucid, rigorous, and beautiful, the reader is left puzzled by the lack of continuity. All too often Wiener could not resist the temptation to tell everything that cropped up in his comprehensive mind, and he often had difficulty in separating the relevant mathematics neatly from its scientific and social implications and even from his personal experiences. The reader to whom he appears to be addressing himself seems to alternate in a random order between the layman, the undergraduate student of mathematics, the average mathematician, and Wiener himself.
Wiener wrote a most unusual autobiography. Although it conveys an extremely egocentric view of the world, I find it an agreeable story and not offensive, because it is naturally frank and there is no pose, least of all that of false modesty. All in all it is abundantly clear that he never had the slightest idea of how he appeared in the eyes of others. His account of the ill-starred trip to Europe in 1926–1927 is a particularly good example. Although he says almost nothing about the work of the mathematicians whom he met, he recalled after twenty-five years meeting J. B. S. Haldane and setting him straight over an error in his book The Gold-Makers: Haldane had used a Danish name for a character supposed to be an Icelander (l Am a Mathematician, 160). In his autobiography Wiener comes through as a fundamentally good-na-tured person, realistic about his human responsibilities and serious enough to be a good friend, a good citizen, and a good cosmopolite. Despite his broad erudition, the philosophical interludes are no more than common sense, if not downright flat. Unlike many autobiographers, he never usurps the role of a prophet who long ago predicted the course that things have taken. A good biography ought to be written of him, one that would counterbalance his autobiography and do him more justice than anyone can do in a book about himself.
According to his own account, Wiener’s understanding of modern mathematics began in 1918, when he came across works on integration, functionals, and differential equations among the books of a young Harvard student who had died. At that time he met I. A. Barnett, who by suggesting that he work on integration in function spaces, put Wiener on the track that would lead him to his greatest achievements, the first of which was differential space. It was already characteristic of Wiener’s openness of mind that, rather than being satisfied with a general integration theory, he looked for physical embodiments to test the theory. The first he tried, turbulence, was a failure; but the next, Brownian motion (1921), studied earlier by Einstein, was a success. Wiener conceived a measure in the space of one-dimensional paths that leads to the application of probability concepts in that space (see Selected Papers, no. 2). The construction is surprisingly simple. Take the set of continuous functions x(t) of t ≧ 0 with X(0) = 0 and require that the probability of x passing for ti between αi and β(i= 1,. . .,K is provided by the Einstein-Smoluchowski formula that gives for the probability density of a point at x staying at y after a lapse of time tthe expression
In later work Wiener made this measure more explicit by a measure-preserving mapping of the real number line on function space. He also proved that almost all paths are nondifferentiable and that almost all of them satisfy a Lipschitz condition of any degree <1/2, although almost none does so with such a condition of degree >1/2. “Differential space” is a strange term for this function space with a measure, promising a measure defined not by finite but by differential methods. Although vaguely operative on the background, this idea was never made explicit by Wiener when he resumed use of the term “differential space” in later work.
In 1923–1925 Wiener published papers that greatly influenced potential theory: Dirichlet’s problem, in its full generality (see Selected Papers, no. 3). The exterior problem of a compact set K in 3-space led him to the capacitory potential of a measure with support K as a basic tool.
From Brownian motion Wiener turned to the study of more general stochastic processes, and the mathematical needs of MIT’s engineering department set him on the new track of harmonic analysis. His work during the next five years culminated in a long paper (1930) on generalized harmonic analysis (see Selected Papers, no. 4), which as a result of J. D. Tamarkin’s collaboration is very well written. Rather than on the class L2, Wiener focused on that of measurable functions f with
existing for all x, which is even broader than that of almost periodic functions. He borrowed the function π from physics as a key to harmonic analysis and connected it later to communication theory. Writing π as a Fourier transform,
he obtained what is now called the spectral distribution S. The most difficult step was to connect S to the integrated Fourier transform g of f by an analogue of the classical formula dλ. A brilliant example is: If f(x) = ±1 for xn ≤ x < xn+1’ where the signs are fixed by spinning a coin, then the spectral distribution of f is almost certainly continuous.
A key formula in this field was placed by Wiener on the cover of the second part of his autobiography:
When Wiener attempted to prove this, A. E. Ingham led him to what Hardy and Littlewood had called Tauberian theorems; but Wiener did more than adapt their results to his own needs. He gave a marvelous example of the unifying force of mathematical abstraction by recasting the Tauberian question as follows (see Selected Papers, no. 5): To prove the validity of
by which kind of more tractable kernel K1, K may be replaced. The answer is (for K and K1훆 L1): If the Fourier transform of K1 vanishes nowhere, the validity with K1 implies that with K. Tauberian theorems have lost much of their interest today, but the argument by which Wiener proved his theorem is still vigorous. Wiener showed that in L1 the linear span of the translates of a function is dense if its Fourier transform vanishes nowhere. This, again, rests on the remark that the Fourier transform class L1 is closed with respect to division (as far as possible). Wiener’s work in this area became the historical source of the theory of Banach algebras. The “Wiener problem,” that is, the problem of deciding whether it is true that in L1 a function f,1 belongs to the closure of the span of the translates of f2 if and only if the Fourier transform of f1 always vanishes together with that of f2, greatly influenced modern harmonic analysis; it was proved to be wrong by Paul Malliavin in 1959.
Fourier transforms and Tauberian theorems were also the subject of Wiener and R. E. A. C. Paley’s collaboration, which led to Fourier Transforms (1934). Another cooperative achievement was the study of the Wiener-Hopf equation (see Selected Papers, no. 6),
generalizing Eberhard Hopf’s investigation on radiation equilibrium. In I Am a Mathematician (p. 177), Wiener remarked that although originally accounting for the discontinuity of two physical media at x=0, it can even better serve to embody the discontinuity of knowledge at the boundary of future and past. The previous work on the Wiener Hopf equation became influential in Wiener’s prediction theory.
Until the late 1930’s stochastic processes, as exemplified by Brownian motion, and harmonic analysis were loose ends in the fabric of Wiener’s thought. To be sure, they were not isolated from each other: the spectrally analyzed function f was thought of as a single stochastic happening, and the earlier cited example shows that such a happening could even be conceived as embedded in a stochastic process. Work of others in the 1930’s shows the dawning of the idea of spectral treatment of stationary stochastic processes; at the end of the decade it became clear that the “Hilbert space trick” of ergodic theory could serve this aim also. Initially Wiener had neglected ergodic theory; in 1938–1939 he fully caught up (see Selected Papers, nos. 7–8), although in later work he did not avail himself of these methods as much as he might have done.
Communication theory, which for a long time had been Wiener’s background thought, became more prominent in his achievements after 1940. From antiaircraft fire control and noise filtration in radar to control and communication in biological settings, it was technical problems that stimulated his research Although linear prediction was investigated independently by A. N. Kolmogorov, Wiener’s approach had the merit of dealing with prediction and filtering under one heading. If on the strength of ergodicity of the stationary stochastic process f(ft ε L2), the covariances φ(t) = (ft, f0) are supposed to be provided by the data of the past, linear predicting means estimating the future of f by its projection on the linear span of the past ft On the other hand, linear filtering means separating the summands “message” and “noise” in f = where again the autocovariances and cross covariances Φ(t) = (ft, f0) and are supposed to be known and the message is estimated by its projection on the linear span of the past signals ft Both tasks lead to Wiener-Hopf equations for a weighting distribution w,
The implications of these fundamental concepts were elaborated in a wartime report that was belatedly published in 1949; it is still difficult to read, although its contents have become basic knowledge in communication theory. Nonlinear filtering was the subject of Wiener’s unpublished memorandum (1949) that led to combined research at MIT, as reported by his close collaborator Y. W. Lee (see Selected Papers, pp. 17–33). A series of lectures on this subject was published in 1958. One of its main subjects is the use of an orthogonal development of nonlinear (polynomial) Volterra functionals by R. H. Cameron and W. T. Martin (1947) in a spectral theory and in the analysis and synthesis of nonlinear filters, which, rather than with trigonometric inputs, are probed with white Gaussian inputs.
After this brief exposition of Wiener’s mathematics of communication, it remains to inspect the broad field that Wiener himself vaguely indicated as cybernetics; he tells how he coined this term, although it had not been unusual in the nineteenth century to indicate government theory. Whilestudying antiaircraft fire control, Wiener may have conceived the idea of considering the operator as part of the steering mechanism and of applying to him such notions as feedback and stability, which had been devised for mechanical systems and electrical circuits. No doubt this kind of analogy had been operative in Wiener’s mathematical work from the beginning and sometimes had even been productive. As time passed, such flashes of insight were more consciously put to use in a sort of biological research for which Wiener consulted all kinds of people, except mathematicians, whether or not they had anything to do with it. Cybernetics, or the Control and Communication in the Animal and the Machine (1948) is a rather eloquent report of these abortive attempts, in the sense that it shows there is not much to be reported. The value and influence of Cybernetics, and other publications of this kind, should not, however, be belittled. It has contributed to popularizing a way of thinking in communication theory terms, such as feedback, information, control, input, output, stability, homeostasis, prediction, and filtering . On the other hand, it also has contributed to spreading mistaken ideas of what mathematics really means Cybernetics suggests that it means embellishing a nonmathematical text with terms and formulas from highbrow mathematics. This is a style that is too often imitated by those who have no idea of the meaning of the mathematical words they use. Almost all so-called applications of information theory are of this kind.
Even measured by Wiener’s standards, Cybernetics is a badly organized work–a collection of misprints, wrong mathematical statements, mistaken formulas, splendid but unrelated ideas, and logical absurdities. It is sad that this work earned Wiener the greater part of his publicrenown, but this is an afterthought. At that time mathematical readers were more fascinated by the richness of its ideas than by its shortcomings. Few, if any, reviewers voiced serious criticism.
Wiener published more writings of this kind. The last was a booklet entitled God and Golem, Inc. It would have been more appropriate as the swan song of a lesser mathematician than Wiener.
I. Original Works. Many of Wiener’s writings were brought together in his Selected Papers (Cambridge, Mass., 1964), which includes contributions by Y. W. Lee, N. Levinson, and W. T. Martin. Among his works are Fourier Transforms in the Complex Domain (New York, 1934), written with Raymond E. A. C. Paley; Cybernetics, or the Control and Communication in the Animal and the Machine (Paris – Cambridge, Mass., 1948); Extrapolation, interpolation and Smoothing of Stationery Time Series, With Engineering Applications (Cambridge, Mass.–New York – London, 1949); Ex-Prodigy–My Childhood and Youth (New York, 1953; Cambridge, Mass., 1955); I Am a Mathematician-the Later Life of a Prodigy (Garden City, N .Y .,1956; repr. Cambridge, Mass., 1964); Nonlinear Problems in Random Theory (Cambridge, Mass.–New York–London, 1958) God and Golem, Inc;.(Cambridge, Mass., 1964); and Differential Space, Quantum Systems, and Prediction (Cambridge, Mass., 1966), written with Armand Siegel, Bayard Rankin, and William Ted Martin.
II. Secondary Literature. See “Norbert Wiener,” Bulletin of the American Mathematical Society, spec. iss., 72 no,. 1, pt. 2 (1966), with contributions by N. Levinson, W. Rosenblith and J. Wiesner, M. Brelot, J.P. Kahane, S. Mandelbrojt, M. Kac, J. L. Doob, P. Masani, and W. L. Root, with bibliography of 214 items (not including posthumous works.) See also Constance Reid Hilbert (Berlin, 1970), esp. 169–170.
Norbert Wiener was born in Columbia, Missouri, in 1894 and died while visiting Stockholm, Sweden, in 1964. A child prodigy, he became a widely respected mathematician and teacher. During the last twenty years of his life, he became known throughout the world as a founder of and spokesman for the new science that he had named “cybernetics.”
Cybernetics, in a narrow sense, is the study of the relationship between information processing and purposeful behavior, both in machines and in animals. In a wider sense the concepts of cybernetics apply to social systems as well and suggest new ways to analyze complex social organizations in terms of the flow and processing of information. However, the more fundamental promise of cybernetics lies not in its ability to help explain the behavior of complex systems, but rather in the fact that the explanations are framed in the new language of information and control. Because of this, the real revolution stimulated by Wiener’s notions on cybernetics is a conceptual one that reaches deep into the foundation and structure of the sciences.
Wiener’s education and intellectual outlook were enormously influenced by his father, Leo Wiener, who himself had been an intellectually precocious child. Leo Wiener, who was born in Russia and educated in Europe, arrived in the United States at the age of 21 and later became a teacher. Eventually he became professor of Slavic languages and literature at Harvard, where he taught for thirty years before his retirement. Leo Wierner, like James Mill, the father of an earlier child prodigy, had his own theories about educating children. Under his father’s rigorous tutelage and discipline, Norbert Wiener at the age of seven was reading books on biology and physics that were beyond even his father’s scope. He entered high school at the age of nine and graduated three years later. He then entered Tufts College and graduated—cum laude in mathematics—at 15. After a false start toward advanced work in biology, Wiener studied philosophy; he received his PH.D. from Harvard in 1913, at the age of 18.
Upon leaving Harvard, Wiener secured a postdoctoral fellowship that allowed him to travel to Cambridge University, where he studied epistemology and logic with Bertrand Russell and mathematics with G. H. Hardy. After his stay at Cambridge, he went to Gottingen, where he studied mathematics with Landau and David Hilbert and philosophy with Husserl. During a brief period following his return to America he was a writer for the Encyclopedia Americana, a mathematician computing ballistic tables for the U.S. Army at the Aberdeen proving grounds, and also a journalist for the Boston Herald. Then, in the spring of 1919, Wiener accepted a position in the mathematics department of the Massachusetts Institute of Technology, where he remained, eventually to become a full professor and later Institute professor.
His first work at M.I.T. was on the theory of Brownian motion. This work, which was influenced by the notions of J. Willard Gibbs on statistical mechanics and Lebesgue on probability, shaped his subsequent statistical treatment of the problems of information and communication. Wiener’s early mathematical work on harmonic analysis had a later impact on his notions about the filtering and predicting of time series. Thus, much of the mathematical work that he developed during the first part of his career later influenced his ideas on cybernetics.
About 1940, when the United States was gearing itself for a possible war, Wiener became involved in the problem of designing fire control equipment. It was around this complex set of problems that his ideas on information processing and control coalesced to form the basis of cybernetics. The problem of fire control is to design a machine that, when fed radar tracking data, will compute how to aim a gun so that its projectile will intersect the path of the moving target at the appropriate time. This involves not only a theory of prediction and a mechanism to embody the theory but also a theory of stability and control.
In the course of this work, Wiener and his colleague Julian Bigelow (who was later to direct the construction of the first von Neumann-type electronic computer at Princeton University) recognized the critical role of feedback in the organization of a control system. This recognition led to the conjecture that the kinds of information processing and feedback loops necessary to control a mechanical system might resemble those in the cerebellum that control purposeful human behavior. If this conjecture was true, then similar kinds of breakdowns in the internal information-processing mechanisms of a man and of a mechanical control system would produce similar pathological behavior. These ideas were recorded in a paper jointly authored by Wiener, Bigelow, and A. Rosenblueth called “Behavior, Purpose and Teleology,” which was published in 1943. It makes explicit the thesis that the brain can be viewed, in a mechanical way, as a kind of computing machine, and that the concepts of information and control are adequate to explain purposeful motor behavior. Left implicit, however, is the further conjecture that the concepts of information and control will be adequate to explain the mechanisms and processes underlying the behavioral correlates of so-called “higher mental functions” involved in thinking. Wiener was not able to publish a fuller treatment of these ideas until the end of World War ii .
In 1948 Wiener published his book Cybernetics: Or Control and Communication in the Animal and the Machine, which became a best seller and was reprinted many times and translated into many languages. In that now-famous book, Wiener attempted to bring together the concepts underlying information processing, communication, and control. He described the relationship of these cybernetic concepts to other disciplines ranging from neurophysiology, mathematical logic, and computer science to psychology and sociology. His book had an impact on many scientists in these fields, stimulating them to take a fresh look at their own work from a cybernetic point of view. It suggested to psychologists that the behavioral correlates of thinking, remembering, learning, and so forth could be analyzed in terms of the underlying information processes. And much work on the computer simulation of behavior has emerged from that suggestion. The notion of viewing the brain as a kind of computing machine stimulated neurophysiologists not merely to make comparisons between components and coding in both systems but also to try to interpret the logical organization of the brain in terms of information processing and control mechanisms.
The diversity of cybernetic applications in different fields sheds light on the unifying aspect of its basic concepts. Traditionally, a chasm has separated work on the psychology of complex behavior from work on those physiological mechanisms that produce behavior. The gap between these two fields is, in fact, a communication gap caused by the semantic mismatch of concepts from the languages of physiology and psychology. The concepts of behavioristic psychology are too gross and elaborate to fit with the more atomistic concepts from the language of physiology. This same kind of gap would make it impossible to explain the behavior of a digital control computer in terms of the basic physics of its switches, wires, and so forth. Because cybernetics deals with a set of concepts intermediate between psychology and physiology it can provide a conceptual bridge to span both disciplines. The deeper meaning of cybernetics, which lies in the structure of its language and its role in analyzing complex systems in terms of information processing, communication, and control, has yet to be fully unfolded by philosophers of science.
The extent to which Wiener himself saw this philosophical dimension of his work is not clear. However, he did see clearly some of the social-scientific implications of cybernetics. He believed, and others have subsequently developed the notion, that the economy can be viewed as a control system aimed at maintaining certain conditions of economic growth and that economic instability in the form of period booms and slumps is similar to oscillations in a poorly designed mechanical control system. In a similar vein Wiener argued that society can be examined and understood in terms of the flow and processing of information between individuals and social groups.
Wiener was particularly fearful of the expanding role of the computing machine. He recognized very early that machines could and would eventually displace an increasing number of workers both in the factory and in the office, and he thought that if economic incentives pushed automation ahead of our understanding of its consequences, technological unemployment could shatter social and economic stability. He was also concerned about the potential misuse of computers in decision making and feared that as machines became increasingly complex their users would be less aware of the consequences of their instructions to the machines. As a result, a decision maker might cause a machine to initiate some action the consequences of which might, in fact, be contrary to his actual desires. During the last ten years of his life Wiener traveled widely, lecturing and writing about cybernetics and the potential dangers to a society vastly influenced by computers and automation.
Earlier in his life Wiener had received recognition for his contributions to mathematics, and shortly before his death he was awarded the National Medal of Science by the president of the United States.
M. E. Maron
[See alsoconcept formation; cybernetics; information THEORY; SIMULATION; SYSTEMS ANALYSIS;and the biographies ofbabbage; von neumann.]
WORKS BY WIENER
1943 ROSENBLUETH, A.; WIENER, Norbert; and BlGElow, J. Behavior, Purpose and Teleology. Philosophy of Science 10, no. 1:18-24.
(1948) 1961 Cybernetics: Or, Control and Communication in the Animal and the Machine. 2d ed. Cambridge, Mass.: M.I.T. Press.
(1950) 1954 The Human Use of Human Beings: Cybernetics and Society. 2d ed. Boston: Houghton Mifflin.
1953 Ex-prodigy: My Childhood and Youth. New York: Simon & Schuster.
1956 I Am a Mathematician; The Later Life of a Prodigy: An Autobiographical Account of the Mature Years and Career of Norbert Wiener and a Continuation of the Account of His Childhood in Ex-prodigy. Garden City, N.Y.: Doubleday. → A paperback edition was published in 1964 by M.I.T. Press.
Norbert Wiener, 1894-1964. 1966 American Mathematical Society, Bulletin 72, no. 1, part 2. → A bibliography appears on pages 135-145.
The American mathematician Norbert Wiener (1894-1964) studied computing and control devices. Out of these studies he created the science of cybernetics.
Norbert Wiener was born on Nov. 26, 1894, at Cambridge, Mass. His father, Leo Wiener, professor of Slavonic languages and literature at Harvard University, determined to train the boy actively and single-mindedly as a scholar. Norbert was driven hard on the way to becoming a prodigy; fortunately he had the intellect and energy to emerge without undue suffering. He graduated with a bachelor's degree from Tufts College at the age of 14 and obtained his doctorate at Harvard four years later.
Wiener was awarded a traveling fellowship which he spent at the two centers where learning, especially in the mathematical and physical sciences, was perhaps the most significant and the most exciting in Europe: the University of Cambridge, England, and the University of Göttingen, Germany. After a varied career during World War I, he joined the Massachusetts Institute of Technology in 1919 as an instructor in the department of mathematics, and he remained on its staff for the whole of his career. There he was introduced to the work of the chemist Josiah Willard Gibbs, whose research on statistical mechanics, published in 1902, was a decisive influence in the development of Wiener's intellectual career.
Wiener had been instructed in the Lebesgue integral by G. H. Hardy at Cambridge, and with this grounding and his recognition of the importance of Gibbs's writings, he attacked the problem of the Brownian motion and produced one of his first major contributions to research. About the same time he began work on harmonic analysis. He brought to bear on this problem the method of Tauberian theorems and by this means refined his theory of harmonic analysis and also produced simple proofs of the prime-number theorem. He also worked on Fourier transforms and wrote Fourier Transforms in a Complex Domain.
At the same time that he pursued these studies into the field of quasi-analytic functions, Wiener was developing his interest in electrical circuits. The knowledge he gained on the problems of feedback control was of use when he became engaged in World War II on fire-control apparatus for antiaircraft guns. His interest in the parallels between feedback control in circuits and mental processes led to the creation of a new discipline which he called cybernetics, the study of control, communication, and organization. In Cybernetics (1948), his most influential work outside the field of pure mathematics, he propounded a new approach to the study of man in his technological environment, a science of man as component of an age of automation. On March 18, 1964, Wiener died in Stockholm.
The best sources of biographical material are Wiener's two volumes of memoirs, Ex Prodigy: My Childhood and Youth (1953) and I Am a Mathematician: The Later Life of a Prodigy(1956). See also Mitchell Wilson, American Science and Invention (1954). □
American mathematician best known for establishing the science of cybernetics, which is concerned with the mathematical analysis of, analogies between, and information flow within mechanical and biological systems. This work was influenced by his previous application of statistical methods to antiaircraft fire control and communications engineering. Wiener also derived a physical definition of information related to entropy. He developed a mathematical theory of Brownian motion and advanced the study of integrals, quantum mechanics, and harmonic analysis.