Celebratio Mathematica

M. H. Stone: “On the order of an analytic function at a singular point,” Bull. Am. Math. Soc. 30 : 7 (1924), pp. 297. Abstract only. Initial research report for an article later published in Ann. Math. 26:1–2 (1924). JFM 50.0254.08 article

M. H. Stone: “An unusual type of expansion problem,” Bull. Am. Math. Soc. 30 : 7 (1924), pp. 297. Abstract only. Initial research report for an article later published in Trans. Am. Math. Soc. 26:3 (1924). JFM 50.0315.01 article

M. H. Stone: “On the order of an analytic function at a singular point,” Ann. Math. (2) 26 : 1–2 (September–December 1924), pp. 145–154. An initial research report was published in Bull. Am. Math. Soc. 30:7 (1924). MR 1502684 JFM 50.0226.01 article

If there is given a Taylor’s series converging, as we shall suppose, in the unit circle, it defines an analytic function \[ f(z) = a_0 + a_1z + a_2z^2 + \cdots,\quad z = re^{i\theta}, r < 1. \] The study of the singularities of \( f(z) \) on the circumference of the circle is clearly of the greatest importance. In his Thèse [1892], which is devoted to such a study, Hadamard associates with each point on the unit circle a quanitity which he calls the order of \( f(z) \) at that point. The researches of Hadamard and others [Dienes 1909, 1913; Bieberbach 1921; Fabry 1912] have shown that the behavior of \( f(z) \) in the neighborhood of a point on the circumference of the circle of convergence is very closely connected with its order at that point. It is the purpose of the present paper to give a proof of the existence of the order: writers on the subject appear to assume the existence as self-evident or content themselves with proofs which seem unsatisfactory to the writer.

M. H. Stone: “An unusual type of expansion problem,” Trans. Am. Math. Soc. 26 : 3 (1924), pp. 335–355. An initial research report was published in Bull. Am. Math. Soc. 30:7 (1924). MR 1501281 JFM 50.0313.01 article

In the majority of papers on the expansion problems associated with linear differential systems or with integral equations, the chief aim is a proof that a function, arbitrary within the limits of certain restrictions as light as possible, can be expanded in a uniformly convergent series of functions arising from the differential system or the integral equation. As exceptions to the rule we may cite a number of papers dealing with differential systems for which the boundary conditions are of irregular type [Jackson 1915–1916; Hopkins 1919]. In the following paragraphs we present a study of the linear differential system of the first order, without the customary limitation that a certain coefficient in the differential equation remain positive; the results obtained are in sharp contrast not only with those found for the usual expansion problems and those already mentioned in connection with irregular boundary conditions, but also with the facts for the precisely analogous second order differential system [Mason 1907; Lichtenstein 1914]. A second point of some interest is the appearance of a normal orthogonal set of trigonometric functions differing from the normalized Fourier set but as closely related to it as are the sine and cosine sets. It is interesting that this exceedingly simple differential system seems to have escaped attention entirely. So far as we know, the existence of this gap in the theory of boundary value and expansion problems was first pointed out by Professor Birkhoff in a recent course of lectures on the theory of linear differential equations.

M. H. Stone: “Three theorems on normal orthogonal sets,” Bull. Am. Math. Soc. 31 : 1–2 (1925), pp. 17–21. MR 1560984 JFM 51.0226.01 article

M. H. Stone: “A comparison of the series of Fourier and Birkhoff,” Bull. Am. Math. Soc. 31 : 5 (1925), pp. 225. Abstract only. Initial research report for an article later published in Trans. Am. Math. Soc. 28:4 (1926). JFM 51.0236.02 article

M. H. Stone: “Irregular differential systems of order two and the related expansion problems,” Bull. Am. Math. Soc. 31 : 5–6 (1925), pp. 225. Abstract only. Research announcement for an article published in Trans. Am. Math. Soc. 29:1 (1927). JFM 51.0343.03 article

M. H. Stone: “Integrals analogous to Fourier integrals,” Bull. Am. Math. Soc. 32 : 6 (1926), pp. 595. Abstract only. Research announcement for an article published in Math. Z. 28:1 (1928). JFM 52.0287.18 article

M. H. Stone: “The Borel summability of Fourier series,” Bull. Am. Math. Soc. 32 : 1 (1926), pp. 38. Abstract only. Research announcement for an article published in Am. J. Math. 48:2 (1926). JFM 52.0287.13 article

M. H. Stone: Ordinary linear homogeneous differential equations of order \( n \) and the related expansion problems. Ph.D. thesis, Harvard University, 1926. Advised by G. D. Birkhoff. MR 2936537 phdthesis

M. H. Stone: “Series of Legendre polynomials,” Bull. Am. Math. Soc. 32 : 2 (1926), pp. 130. Abstract only. JFM 52.0287.16 article

M. H. Stone: “The convergence of Bessel’s series,” Bull. Am. Math. Soc. 32 : 1 (1926), pp. 38. Abstract only. JFM 52.0464.08 article

M. H. Stone: “Developments in Legendre polynomials,” Ann. Math. (2) 27 : 4 (June 1926), pp. 315–329. MR 1502735 JFM 52.0279.03 article

Expansions in terms of Legendre polynomials have been discussed by many authors [Hobson 1908, 1909; Jackson 1912; Burkhardt 1909; Young 1919–1920]; yet, so far as the writer is informed, there exists no treatment which can fairly be described as “elementary”. The papers to which reference has been made are designed to be more or less exhaustive, and, in some instances, are based upon complicated theories. For one thoroughly familiar with the type of analysis employed in these researches, the problem of representing an “arbitrary” function by a series of Legendre poynomials has been fully handled, and, in a certain sense, solved. For those who have not specialised to this extent there is no available material on the convergence of Legendre series. Because of the importance of these developments in mathematical physics, a discussion of certain interesting cases with a minimum of analytical detail seems highly desirable. The present paper is intended as a step in this direction.

M. H. Stone: “The Borel summability of Fourier series,” Am. J. Math. 48 : 2 (1926), pp. 101–112. A research announcement for this article was published in Bull. Am. Math. Soc. 32:1 (1926). MR 1506577 JFM 52.0273.02 article

M. H. Stone: “A comparison of the series of Fourier and Birkhoff,” Trans. Am. Math. Soc. 28 : 4 (1926), pp. 695–761. An initial research report was published in Bull. Am. Math. Soc. 31:5 (1925). MR 1501372 JFM 53.0419.03 article

M. H. Stone: “Expansions in Bessel functions,” Ann. Math. (2) 28 : 1–4 (1926–1927), pp. 271–290. MR 1502780 JFM 53.0337.02 article

M. H. Stone: “The normal frequency function and general frequency functions,” Bull. Am. Math. Soc. 33 : 3 (1927), pp. 261. Abstract only. JFM 53.0523.12 article

M. H. Stone: “Expansions associated with a regular differential systems of the second order,” Bull. Am. Math. Soc. 33 : 1 (1927), pp. 11–12. Abstract only. JFM 53.0429.04 article

M. H. Stone: “A characteristic property of certain sets of trigonometric functions,” Bull. Am. Math. Soc. 33 : 3 (1927), pp. 261. Abstract only. Research announcement for an article published in Am. J. Math. 49:4 (1927). JFM 53.0277.05 article

M. H. Stone: “A characteristic property of certain sets of trigonometric functions,” Am. J. Math. 49 : 4 (1927), pp. 535–542. A research announcement for this article was published in Bull. Am. Math. Soc. 33:3 (1926). MR 1506643 JFM 53.0423.01 article

M. H. Stone: “Irregular differential systems of order two and the related expansion problems,” Trans. Am. Math. Soc. 29 : 1 (1927), pp. 23–53. A research announcement for this article was published in Bull. Am. Math. Soc. 31:5–6 (1925). MR 1501375 JFM 53.0429.01 article

M. H. Stone: “The expansion problems associated with regular differential systems of the second order,” Trans. Am. Math. Soc. 29 : 4 (1927), pp. 826–844. MR 1501416 JFM 53.0420.01 article

M. H. Stone: “The summability of Fourier series,” Bull. Am. Math. Soc. 33 : 6 (1927), pp. 721–732. MR 1561455 JFM 53.0250.01 article

M. H. Stone: “Developments in Hermite polynomials,” Ann. Math. (2) 29 : 1–4 (1927–1928), pp. 1–13. MR 1502813 JFM 53.0271.01 article

M. H. Stone: “Certain integrals analogous to Fourier integrals,” Math. Z. 28 : 1 (1928), pp. 654–676. A research announcement for this article was published in Bull. Am. Math. Soc. 32:6 (1926). MR 1544983 JFM 54.0476.01 article

M. H. Stone: “Linear transformation in Hilbert space, II: Analytical aspects,” Proc. Natl. Acad. Sci. U.S.A. 15 : 5 (May 1929), pp. 423–425. article

M. H. Stone: “Unitary transformations in abstract Hilbert space,” Bull. Am. Math. Soc. 35 : 2 (1929), pp. 167. Abstract only. JFM 55.0245.01 article

M. H. Stone: “Expansions associated with self-adjoint elliptic partial differential equations,” Bull. Am. Math. Soc. 35 : 2 (1929), pp. 166–167. Abstract only. JFM 55.0290.07 article

M. H. Stone: “Hermitian symmetric operators in abstract Hilbert space,” Bull. Am. Math. Soc. 35 : 1 (1929), pp. 6. Abstract only. JFM 55.0244.19 article

M. H. Stone: “Theory of Hermitian transforms,” Bull. Am. Math. Soc. 35 : 2 (1929), pp. 167. Abstract only. JFM 55.0244.20 article

M. H. Stone: “Linear transformation in Hilbert space, I: Geometrical aspects,” Proc. Natl. Acad. Sci. U.S.A. 15 : 3 (March 1929), pp. 198–200. JFM 55.0824.01 article

M. H. Stone: “Gauss’ law of error in several variables,” Bull. Am. Math. Soc. 35 : 2 (1929), pp. 167. Abstract only. JFM 55.0303.17 article

M. H. Stone: “Linear transformations in Hilbert space, III: Operational methods and group theory,” Proc. Natl. Acad. Sci. U.S.A. 16 : 2 (February 1930), pp. 172–175. JFM 56.0357.01 article

M. H. Stone: “Hausdorff’s theorem concerning Hermitian forms,” Bull. Am. Math. Soc. 36 : 4 (1930), pp. 259–261. MR 1561929 JFM 56.0127.04 article

J. D. Tamarkin and M. H. Stone: “Remarks on a paper of Dr. Tautz,” Acta Math. 57 : 1 (1931), pp. 459–463. MR 1555342 JFM 57.0529.02 article

Jacob David Tamarkin	Related
Georg Lukas Tautz	Related

M. H. Stone: “A note on the theory of infinite series,” Ann. Math. (2) 32 : 2 (April 1931), pp. 233–238. MR 1502993 JFM 57.0334.01 Zbl 0002.18802 article

M. H. Stone: “On one-parameter unitary groups in Hilbert space,” Ann. Math. (2) 33 : 3 (July 1932), pp. 643–648. MR 1503079 JFM 58.0424.01 Zbl 0005.16403 article

M. H. Stone: Linear transformations in Hilbert space and their applications to analysis. AMS Colloquium Publications 15. American Mathematical Society (New York), 1932. Republished (with abbreviated title) in 1990. JFM 58.0420.02 Zbl 0005.40003 book

M. H. Stone: “On the structure of Boolean algebras,” Bull. Am. Math. Soc. 39 : 3 (1933), pp. 200. Abstract only. JFM 59.0067.03 article

M. H. Stone: “Boolean algebras and their application to topology,” Proc. Natl. Acad. Sci. U.S.A. 20 : 3 (March 1934), pp. 197–202. JFM 60.0108.02 Zbl 0010.08104 article

M. H. Stone: “Subsumption of the theory of Boolean algebras under the theory of rings,” Proc. Natl. Acad. Sci. U.S.A. 21 : 2 (February 1935), pp. 103–105. JFM 61.0054.03 Zbl 0011.05104 article

In a previous communication [1934], we have sketched an analogy between Boolean algebras and abstract rings, with particular reference to the theory of ideals and ideal arithmetic. In the present note we shall show that Boolean algebras are actually special instances, rather than analogs, of the general algebraic systems known as rings. In establishing this result we must choose the fundamental operations in a Boolean algebra in an appropriate manner, indicated below. The algebraic theory developed in the previous communication is thus a particular instance of the theory of rings and can be deduced in part from the known theorems concerning rings. These facts were discovered just as the detailed exposition of our independent theory of Boolean algebras was on the point of completion; and they now compel a radical revision which will necessarily delay publication of the complete theory.

J. von Neumann and M. H. Stone: “The determination of representative elements in the residual classes of a Boolean algebra,” Fundam. Math. 25 (1935), pp. 353–378. JFM 61.0976.02 Zbl 0012.24403 article

If \( A \) is any abstract ring and \( \mathfrak{a} \) is a left- and right-ideal in \( A \), we may consider the problem of selecting a single representative element from each of the residual classes \( (\operatorname{mod} \mathfrak{a}) \) in such a manner that sums and products of representative elements are themselves representative elements. If such a selection is possible, the representative elements evidently constitute a subsystem of \( A \) which is isomorphic to the quotient-ring \( A/\mathfrak{a} \) under the correspondence carrying each residual class \( (\operatorname{mod} \mathfrak{a}) \) into its representative element. In this paper we shall confine our attention to rings in which every element is idempotent — that is, in which the law \( aa = a \) obtains. These rings will be seen to have the formal properties of certain algebras of classes, and will therefore be termed Boolean rings. A particular case of the representation problem for Boolean rings has previously been discussed by one of us [von Neumann 1931]. Here we shall examine the problem on an abstract basis, giving sufficient conditions for the existence of a solution, special cases in which no solution exists, and special cases in which a solution exists if and only if \[ \aleph_{n+1} = 2^{\aleph_n} .\] The sufficient conditions given here can be applied to the particular case mentioned above.

M. H. Stone: “Postulates for Boolean algebras and generalized Boolean algebras,” Am. J. Math. 57 (October 1935), pp. 703–732. MR 1507106 JFM 61.0975.05 Zbl 0012.29002 article

M. H. Stone: “The theory of representations for Boolean algebras,” Trans. Am. Math. Soc. 40 : 1 (July 1936), pp. 37–111. MR 1501865 JFM 62.0033.04 Zbl 0014.34002 article

M. H. Stone: “Applications of Boolean algebras to topology,” Rec. Math. Moscou, n. Ser. 1(43) : 5 (1936), pp. 765–771. JFM 62.0687.02 Zbl 0016.18201 article

M. H. Stone: “Applications of Boolean algebras to general topology,” Bull. Am. Math. Soc. 42 : 7 (1936), pp. 489. Abstract only. JFM 62.0048.17 article

B. A. Lengyel and M. H. Stone: “Elementary proof of the spectral theorem,” Ann. Math. (2) 37 : 4 (October 1936), pp. 853–864. MR 1503314 JFM 62.0450.02 Zbl 0016.03001 article

M. H. Stone: “Topological representations of distributive lattices and Brouwerian logics,” Čas. Mat. Fys. 67 (1937), pp. 1–25. JFM 63.0830.01 Zbl 0018.00303 article

M. H. Stone and J. D. Tamarkin: “Elementary proofs of some known theorems of the theory of complex Euclidean spaces,” Duke Math. J. 3 : 2 (1937), pp. 294–302. MR 1545987 JFM 63.0366.03 Zbl 0016.40501 article

M. H. Stone: “Some remarks on non-separable Banach spaces,” pp. 120 in Comptes rendus du Congrès International des Mathématiciens, Oslo, 1936 [Proceedings of the International Congress of Mathematicians, Oslo, 1936] (Oslo, 13–18 July 1936), vol. II: Conferénces de Sections. A.W. Brøggers (Oslo), 1937. Abstract only. JFM 63.0367.02 incollection

M. H. Stone: “Applications of the theory of Boolean rings to general topology,” Trans. Am. Math. Soc. 41 : 3 (1937), pp. 375–481. MR 1501905 JFM 63.1173.01 Zbl 0017.13502 article

M. H. Stone: “Note on formal logic,” Am. J. Math. 59 : 3 (July 1937), pp. 506–514. MR 1507259 JFM 63.0025.03 Zbl 0017.14504 article

It has been observed that the theory of Boolean algebras assumes a particular satisfactory algebraic form when developed in terms of the symmetric difference \( a + b \) and the product \( a\cdot b \) as fundamental operations: for Boolean algebras are then characterized as rings with unit in which every element is idempotent [1936]. The close connection between Boolean algebras and the formal (Aristotelian) logic of propositions therefore suggests that a logistic system built up from corresponding operations would be of some interest, and would have the special advantage of reducing the proofs of most logical theorems to simple and essentially familiar algebraic calculations. In the present note we shall develop such a system, based on results of Leśniewski [1929] and Bernstein [1936].

M. H. Stone: “Algebraic characterizations of special Boolean rings,” Fundam. Math. 29 (1937), pp. 223–303. JFM 63.0872.02 Zbl 0017.33902 article

In a paper entitled The Theory of Representations for Boolean Algebras [1936], we have introduced and discussed a certain classification of the ideals in a Boolean ring (or generalized Boolean algebra). Here we propose to carry out a detailed study of that classification, with the particular purpose of discovering what types of Boolean ring can be characterized by properties of the ideal-structure. In order to make our examination complete, we have to consider many details of a somewhat tedious and uninteresting nature. For the convenience of the reader who prefers to pass over such details, we adopt a synthetic, rather than analytic, form of presentation; and formulate our results in a series of theorems and tables which, we hope, can be easily and rapidly surveyed.

M. H. Stone: “The representation of Boolean algebras,” Bull. Am. Math. Soc. 44 : 12 (1938), pp. 807–816. MR 1563881 JFM 64.0030.04 Zbl 0020.34204 article

M. H. Stone: The theory of real functions. Edwards Brothers (Ann Arbor, MI), 1940. Lithographic copy of author’s manuscript, based on Harvard University lecture notes. book

M. H. Stone: “Characteristic functions of families of sets,” Duke Math J. 7 : 1 (1940), pp. 453–457. Reprinted in Fund. Math. 33 (1945). MR 0003690 JFM 66.0966.02 Zbl 0024.10302 article

M. H. Stone: “A general theory of spectra, I,” Proc. Natl. Acad. Sci. U. S. A. 26 : 4 (April 1940), pp. 280–283. MR 0002023 JFM 66.1284.03 Zbl 0063.07208 article

The mathematical theory of spectra deals with the characteristic value problem (Eigenwertproblem) for linear operators, and provides a general unifying treatment for typical instances of the problem occurring in applied mathematics. Of recent years the tendency to emphasize the algebraic aspects of the spectral theory has become more and more pronounced. This tendency is quite as evident in the applications as in the purely mathematical developments, being a characteristic feature of the quantum theory and of the Heaviside calculus. In the present note we sketch further steps in the direction of a thorough “algebraization” of the spectral theory: we shall show that without the mediation of any theory of integration it is possible to define general functions of operators and to elaborate their calculus.

M. H. Stone: “A general theory of spectra, II,” Proc. Natl. Acad. Sci. U. S. A. 27 : 1 (January 1941), pp. 83–87. MR 0004092 JFM 67.0413.03 Zbl 0063.07209 article

N. Dunford and M. H. Stone: “On the representation theorem for Boolean algebras,” Revista Ci., Lima 43 : 437 (1941), pp. 447–453. MR 0007002 Zbl 0060.06502 article

M. H. Stone: “On characteristic functions of families of sets,” Fund. Math. 33 (1945), pp. 27–33. Reprinted from Duke Math. J. 7:1 (1940). MR 0015446 article

M. H. Stone: “Remarks on metrizability,” Bull. Am. Math. Soc. 53 : 4 (1947), pp. 289–291. MR 0021311 Zbl 0032.31402 article

M. H. Stone: “Pseudo-norms and partial orderings in Abelian groups,” Ann. Math. (2) 48 : 4 (October 1947), pp. 851–856. MR 0022221 Zbl 0029.10501 article

M. Stone: “Science and statecraft,” Science 105 (1947), pp. 507–510. Address of the retiring vice-president, Section A, AAAS, 1942, delivered at Boston, December 27, 1946. article

M. H. Stone: “Notes on integration, I,” Proc. Natl. Acad. Sci. U. S. A. 34 : 7 (July 1948), pp. 336–342. MR 0025552 Zbl 0031.01402 article

The theory of integration, because of its central rôle in mathematical analysis and geometry, continues to afford opportunities for serious investigation. The need for extending and rounding out the classical studies of Riemann, Stieltjes and Lebesgue has stimulated considerable interest not only in new aspects of the theory but also in the simplification and perfection of the old. The present series of communications is intended to outline a treatment which, while exploiting fully the possibilities for simplification, will attain a high degree of generality. Among the many contributions to the mathematical literature which have provided material for our handling of the subject we wish to cite above all an important paper of Daniell [1918]. In spite of the fact that the basic ideas in the present discussion are the common property of mathematicians, some of our results appear to be novel. If they are, it is because we have chosen to introduce and exploit an adaptation of the concept of an “upper integral,” making this the technical foundation of the whole theory.

M. H. Stone: “Note on integration, II,” Proc. Natl. Acad. Sci. U. S. A. 34 : 9 (September 1948), pp. 447–455. MR 0026102 Zbl 0034.02903 article

M. H. Stone: “The generalized Weierstrass approximation theorem (continued),” Math. Mag. 21 : 5 (May–June 1948), pp. 237–254. Second of two parts, continued from previous issue. article

M. H. Stone: “On a theorem of Pólya,” J. Indian Math. Soc. (N.S.) 12 (1948), pp. 1–7. MR 0027447 Zbl 0031.05702 article

M. H. Stone: “Notes on integration, III,” Proc. Natl. Acad. Sci. U. S. A. 34 : 10 (October 1948), pp. 483–490. MR 0027040 Zbl 0034.03001 article

M. H. Stone: “The generalized Weierstrass approximation theorem,” Math. Mag. 21 : 4 (March–April 1948), pp. 167–184. First of two parts, continued in following issue. MR 0027121 article

M. H. Stone: “On the compactification of topological spaces,” Ann. Soc. Polon. Math. 21 (1948), pp. 153–160. MR 0026316 Zbl 0031.41701 article

M. H. Stone: “Boundedness properties in function-lattices,” Canadian J. Math. 1 (1949), pp. 176–186. MR 0029091 Zbl 0032.16901 article

The continuous real functions on a topological space \( X \) are partially ordered in a natural way by putting \( f\leq g \) if and only if \( f(x) \leq g(x) \) for all \( x \) in \( X \). With respect to this partial ordering these functions constitute a lattice, the lattice operations \( \cup \) and \( \cap \) being defined by the relations \begin{align*} (f \cup g)(x) &= \max(f(x),g(x)),\\ (f \cap g)(x) &= \min(f(x),g(x)) . \end{align*} The lattice character of any partially ordered system merely expresses the existence of least upper and greatest lower bounds for any finite set of elements in the system. Many partially ordered systems enjoy much stronger boundedness properties than these: for example, every subset with an upper bound may have a least upper bound, as in the case of the real number system. It is our purpose in this paper to determine the conditions under which the function-lattice described above exhibits behaviour of this kind. As an illustration of the results established we may take the important case where \( X \) is a compact Hausdorff space: here we find that the boundedness property required of the function-lattice is equivalent to the condition that \( X \) be the Boolean space associated with a Boolean algebra which has a certain strong additivity property; and many curious and useful relationships among the functions of Baire on the space \( X \) appear as a consequence of this condition, which implies that each bounded function of Baire on \( X \) differs from a uniquely determined continuous function only on a set of the first category.

M. H. Stone: “Postulates for the barycentric calculus,” Ann. Mat. Pura Appl. (4) 29 : 1 (1949), pp. 25–30. MR 0036014 Zbl 0037.25002 article

M. H. Stone: “The algebraization of harmonic analysis,” Math. Student 17 (1949), pp. 81–92. MR 0037928 Zbl 0037.35301 article

M. H. Stone: “Notes on integration, IV,” Proc. Natl. Acad. Sci. U.S.A. 35 : 1 (January 1949), pp. 50–58. MR 0027830 Zbl 0034.03101 article

The theory of integration presented in the earlier notes [1948a, 1948b, 1948c] postulates an elementary integral in terms of which all other concepts are defined. We must therefore undertake to compare the general integrals and other mathematical objects which are associated with different elementary integrals. In this note we offer such a comparison, giving the appropriate general forms of such classical results as the Lebesgue decomposition theorem and the Radon–Nikodym theorem. This is also a suitable context in which to consider an unpublished definition of a general integral recently introduced by N. Bourbaki. As we shall see below, the general integral of Bourbaki differs in few of its essential properties from the one treated in these notes. By comparing the two integrals we obtain solutions to a number of problems raised by Theorem 18 of our second note and mentioned in the corresponding footnote.

M. H. Stone: “Algebraic formulation of the problem of measure,” pp. 69–74 in Leopoldo Fejér et Frederico Riesz: LXX annos natis dedicatus, pars B [Leopoldo Fejér and Frederico Riesz: 70th birthday dedication, part B], published as Acta Sci. Math. Szeged 12 : B. Bolyai Institute (Szeged, Hungary), 1950. MR 0034817 Zbl 0037.07401 incollection

Frigyes Riesz	Related
Lipót Fejér	Related

M. H. Stone: “On unbounded operators in Hilbert space,” J. Indian Math. Soc. (N.S.) 15 (1951), pp. 155–192. MR 0052042 Zbl 0047.11102 article

M. H. Stone: “On the foundations of harmonic analysis,” Medd. Lunds Univ. Mat. Sem. 1952 : Tome Supplementaire (1952), pp. 207–227. Also published in K. Fysiogr. Sällsk. Lund. Förh. 21:17 (1952). MR 0051342 article

M. H. Stone: “On the theorem of Gelfand–Mazur,” Ann. Soc. Polon. Math. 25 (1952), pp. 238–240. MR 0056822 Zbl 0049.08204 article

M. H. Stone: “On the foundations of harmonic analysis,” K. Fysiogr. Sällsk. Lund. Förh. 21 : 17 (1952), pp. 152–172. Also published in Medd. Lunds Univ. Mat. Sem. 1952 (1952). MR 0048460 Zbl 0045.38403 article

M. H. Stone: “The introduction to the theory of analytic functions,” Bol. Soc. Mat. Mexicana 10 : 3–4 (1953), pp. 29–30. MR 0064862 Zbl 0053.23703 article

H. Rubin and M. H. Stone: “Postulates for generalizations of Hilbert space,” Proc. Am. Math. Soc. 4 : 4 (1953), pp. 611–616. MR 0056199 Zbl 0051.33802 article

M. H. Stone: “Free Boolean rings and algebras,” An. Acad. Brasil. Ci. 26 (1954), pp. 9–17. MR 0066342 Zbl 0057.02401 article

M. H. Stone: “The future of mathematics,” J. Math. Soc. Japan 9 : 4 (1957), pp. 493–507. An address delivered before the Mathematical Society of Japan, May 21, 1956. Excerpts were translated into Polish and published in Pokroky Mat. Fys. Astronom. 16:2 (1971). MR 0094294 Zbl 0079.00101 article

M. H. Stone: “Mathematics and the future of science,” Bull. Am. Math. Soc. 63 : 2 (1957), pp. 61–76. 1956 Gibbs Lecture. A Polish translation was published in Wiadom. Mat. 10 (1961). An Italian translation was published in Archimede 10 (1958). A Russian translation was published in Mat. Prosveshchenie 4 (1959). MR 0086013 Zbl 0077.00202 article

M. H. Stone: “An essay on computation,” J. Madras Univ., Sect. B 27 (1957), pp. 143–156. Zbl 0077.11501 article

M. H. Stone: “La matematica e il futuro della scienza” [Mathematics and the future of science], Archimede 10 (1958), pp. 1–16. Italian translation of an article published in Bull. Am. Math. Soc. 63:2 (1957). MR 0096578 Zbl 0080.00101 article

M. Stone: “Mathematics and the future of science” [Matematika i buduŝee nauki], Mat. Prosveshchenie 4 (1959), pp. 111–127. Russian translation of an article published in Bull. Am. Math. Soc. 63:2 (1957). Zbl 0088.00201 article

M. H. Stone: “Matematyka i przyszłość nauki” [Mathematics and the future of science], Wiadom. Mat. (2) 4 (1961), pp. 161–175. Polish translation of an article published in Bull. Am. Math. Soc. 63:2 (1957). MR 0122673 Zbl 0105.24101 article

M. H. Stone: “Hilbert space methods in conformal mapping,” pp. 409–425 in Proceedings of the international symposium on linear spaces (Jerusalem, 5–12 July 1960). Pergamon Press (Oxford), 1961. MR 0136716 Zbl 0118.29903 incollection

M. Stone: “The revolution in mathematics,” Am. Math. Mon. 68 : 8 (October 1961), pp. 715–734. MR 0130792 Zbl 0100.00105 article

M. H. Stone: “Mathematics in continental China, 1949–1960,” Notices Am. Math. Soc. 8 (1961), pp. 209–215. MR 0117146 article

M. Stone: “A generalized Weierstraß approximation theorem,” pp. 30–87 in Studies in modern analysis, 2nd edition. Edited by R. C. Buck. MAA Studies in Mathematics 1. Prentice-Hall (Englewood Cliffs, NJ), 1962. Zbl 0147.11702 incollection

M. H. Stone: “Topological aspects of conformal mapping theory,” pp. 343–346 in General topology and its relations to modern analysis and algebra (Prague, 1–8 September 1961). Edited by J. Novák. Academic Press (New York), 1962. MR 0176048 Zbl 0109.30304 incollection

M. H. Stone: Lectures on preliminaries to functional analysis. Matscience Report 19. Institute of Mathematical Sciences (Madras), 1963. Notes by B. Ramachandran. book

M. H. Stone: “Reforma em educação matemática” [Reform in mathematics education], Bol. Soc. Parana. Mat. 6 : 1 (1963), pp. 19–24. Zbl 0107.24103 article

M. Stone: “The real number system reviewed,” Enseign. Math. (2) 15 (1969), pp. 261–267. To the memory of J. Karamata. A Catalan translation was published in Stochastica 2:3 (1977). Zbl 0177.01301 article

Functional analysis and related fields: Proceedings of a conference in honor of Professor Marshall Stone (Chicago, 20–24 May 1968). Edited by F. E. Browder. Springer (New York), 1970. MR 0268635 Zbl 0211.00102 book

M. H. Stone: “Some comments on the spectral theorem,” pp. 67–75 in Dedicado al Prof. Alberto González Domínguez [Dedicated to Prof. Alberto González Domínguez], published as Rev. Un. Mat. Argentina 25 : 1–2. Union Matematica Argentina (Bahia Blanca), 1970–1971. MR 0318945 Zbl 0238.47015 incollection

The spectral theorem for self-adjoint operators in a Hilbert space (with real, complex, or quaternionic scalars) generalizes the classical theorems on the canonical reduction of quadratic or hermitian forms and their matrices. Usually two steps are needed, the first passing from finite-dimensional spaces to bounded operators in general spaces, the second from bounded operators to unbounded. There has always been a certain interest (see [Riesz 1930; Lengyel and Stone 1936; Tseng 1935; Bernau 1968; Wouk 1966]) in carrying out this generalization by “pure” Hilbert space methods–that is to say, by using only intrinsic algebraic and geometric properties of abstract Hilbert space without recourse to special theorems drawn from classical analysis. For bounded operators the spectral theorem was treated in this spirit by F. Riesz [1930] and by Lengyel and Stone [1936], for unbounded operators by Y. Y. Tseng [1935]. The present paper, while closely related to Tseng’s, expounds a variant of his approach that may appear somewhat simpler and may shed some additional light on the techniques required.

M. H. Stone: “Budoucnost matematiky” [The future of mathematics], Pokroky Mat. Fys. Astronom. 16 : 2 (1971), pp. 57–63. Translated excerpts from a transcribed address published in J. Math. Soc. Japan 9:4 (1957). MR 0274238 article

M. Stone: “A reminiscence on the extension of the Weierstrass approximation theorem,” Historia Math. 3 : 3 (August 1976), pp. 328. Excerpt from a 1976 letter to Israel Halperin. MR 0505003 article

M. Stone: “International relations in mathematics,” pp. 31–39 in Men and institutions in American mathematics. Edited by J. D. Tarwater, J. T. White, and J. D. Miller. Graduate Studies 13. Texas Tech University, 1976. incollection

J. Dalton Tarwater	Related
John Thomas White	Related
John David Miller	Related

M. H. Stone: “Reminiscences of mathematics at Chicago,” Univ. Chicago Mag. (Autumn 1976), pp. 30. article

M. H. Stone: “El sistema dels nombres reals revisat” [The system of real numbers revisited], Stochastica 2 : 3 (1977), pp. 1–14. To Professor P. P. Calleja. Catalan translation of an article published in Enseign. Math. 15 (1969). MR 562427 Zbl 0418.04004 article

M. H. Stone: “Book Review: ‘Courant in Göttingen and New York: The story of an improbable mathematician’,” Bull. Amer. Math. Soc. 84 : 2 (1978), pp. 234–241. MR 1567039 article

M. H. Stone: “Stone, Marshall Harvey,” pp. 164–166 in McGraw-Hill modern scientists and engineers, revised and expanded edition, vol. 3: R–Z. McGraw-Hill (New York), 1980. incollection

M. Stoun: “Memories of Academician P. S. Aleksandrov,” Uspekhi Mat. Nauk 39 : 5(239) (1984), pp. 7–9. MR 764006 article

F. E. Browder: “Stone Age of mathematics on the Midway,” Math. Intell. 11 : 3 (June 1989), pp. 22–25. Also published in A century of mathematics in America, part 2 (1989). article

F. E. Browder: “The Stone Age of mathematics on the Midway,” pp. 191–193 in A century of mathematics in America, part 2. History of Mathematics 2. American Mathematical Society (Providence, RI), 1989. Also published in Math. Intell. 11:3 (1989). MR 1003128 Zbl 0682.01012 incollection

G. W. Mackey: “Marshall Harvey Stone. 1903–1989,” Notices Am. Math. Soc. 36 : 3 (1989), pp. 221–223. MR 0987152 Zbl 1194.01113 article

G. Kolata: “M. H. Stone, acclaimed mathematician, dies at 85,” New York Times (11 January 1989), pp. B10. article

M. H. Stone: “Reminiscences of mathematics at Chicago,” Math. Intell. 11 : 3 (1989). With an appendix by Felix E. Browder. Reprinted from Univ. Chicago Mag. (Autumn, 1976). MR 1007035 Zbl 0681.01014 article

M. H. Stone: “Reminiscences of mathematics at Chicago,” pp. 183–190 in A century of mathematics in America, part 2. Edited by P. Duren, R. A. Askey, and U. C. Merzbach. History of Mathematics 2. American Mathematical Society (Providence, RI), 1989. Reprinted from Univ. Chicago Mag. (Autumn, 1976). MR 1003127 Zbl 0682.01014 incollection

Peter L. Duren	Related
Uta Caecilia Merzbach	Related
Richard Allen Askey	Related	Volume

M. H. Stone: Linear transformations in Hilbert space. AMS Colloquium Publications 15. American Mathematical Society (Providence, RI), 1990. Republication (with abbreviated title) of the 1932 original. MR 1451877 Zbl 0933.47001 book

J. D. Zund: “Marshall Harvey Stone,” pp. 864–865 in American national biography, vol. 20. Edited by J. A. Garraty and M. C. Carnes. Oxford University Press (New York), 1999. incollection

Mark Christopher Carnes	Related
Joseph D. Zund	Related
John Arthur Garraty	Related

G. W. Mackey: “Marshall H. Stone: Mathematician, statesman, advisor, and friend,” pp. 15–25 in Operator algebras, quantization, and noncommutative geometry: A centennial celebration honoring John von Neumann and Marshall H. Stone (Baltimore, MD, 15–16 January 2003). Edited by R. S. Doran and R. V. Kadison. Contemporary Mathematics 365. American Mathematical Society (Providence, RI), 2004. MR 2106814 Zbl 1059.01539 incollection

Robert Stuart Doran	Related
Richard V. Kadison	Related
George Whitelaw Mackey	Related
John von Neumann	Related

Operator algebras, quantization, and noncommutative geometry: A centennial celebration honoring John von Neumann and Marshall H. Stone (Baltimore, MD, 15–16 January 2003). Edited by R. S. Doran and R. V. Kadison. Contemporary Mathematics 365. American Mathematical Society (Providence, RI), 2004. MR 2106812 Zbl 1056.58001 book

John von Neumann	Related
Robert Stuart Doran	Related
Richard V. Kadison	Related

K. H. Parshall: “A mathematical ‘good neighbor’: Marshall Stone in Latin America (1943),” pp. 19–31 in Festschrift Ubiratan D’Ambrosio: Em comemoração ao 75. aniversário, published as Rev. Bras. Hist. Mat. 2007 : special issue no. 1. Issue edited by S. Nobre. Sociedade Brasileira de História da Matemática (Rio Claro, Brazil), December 2007. MR 2417729 Zbl 1206.01058 incollection

Ubiratan D'Ambrosio	Related
Karen V. Hunger Parshall	Related
Sergio Roberto Nobre	Related

K. H. Parshall: “Marshall Stone and the internationalization of the American mathematical research community,” Bull. Am. Math. Soc. 46 : 3 (2009), pp. 459–482. MR 2507278 Zbl 1170.01007 article

The American mathematical research community celebrated, symbolically at least, its fiftieth anniversary in 1938. Many of those fifty years had marked a period of consolidation and growth at home of programs in mathematics at institutions of higher education supportive of high-level research as well as of a corps of talented researchers capable of making seminal contributions in a variety of mathematical areas. By the middle decades of the twentieth century — the 1930s, 1940s, and 1950s — members of that community, like members of the broader American public, began increasingly to look outward beyond the national boundaries of the United States and toward a larger international arena. This paper explores the contexts within which the American mathematical research community, in general, and the American mathematician Marshall Stone, in particular, deliberately worked in the decades around mid-century to effect the transformation from a national community to one actively participating in an internationalizing mathematical world.

Marshall Harvey Stone

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