Ivar Fredholm

How to use this course

Ivar Fredholm published little, and one paper made him permanent. In Sur une classe d’équations fonctionnelles (1903) he solved the linear integral equation with fixed limits completely. He did it by an analogy that every student of linear algebra can follow: an integral is a limit of sums, so an integral equation is a limit of linear systems, and the solution should be a limit of Cramer’s rule. He made that limit rigorous, found an entire function that plays the role of the determinant, and proved that the integral equation behaves exactly like a square matrix: it is uniquely solvable unless the determinant vanishes, and when it vanishes, solvability is governed by the transposed problem.

The paper reached Göttingen in 1901, before its final form appeared, and set David Hilbert to work. Hilbert space, the spectral theory of compact operators, Frigyes Riesz’s abstract theory and the modern Fredholm index all descend from it. So does a line of work on determinants that now reaches random matrices and integrable systems.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proofs or proof sketches, and exercises with folded solutions. Lessons 04 to 08 follow Fredholm’s own work from 1890 to 1908. Lessons 09 and 10 follow his ideas into the twentieth and twenty-first centuries. Throughout, the equation of the second kind is written

with a continuous kernel \(K\) on \([a,b]^2\). Fredholm himself wrote the equation with a plus sign; the sign used here is the one in most current textbooks.

Timeline

Stockholm, 1866–1886

Erik Ivar Fredholm was born in Stockholm on 7 April 1866. His father, Ludvig Oscar Fredholm, was a merchant who had made a fortune; his mother, Catharina Paulina Stenberg, was also from a merchant family. The household was prosperous and cultivated, and Ivar received the best schooling the city offered. He had a younger brother, John Oscar, born in 1875.

He attended the Beskowska School, a private school in Stockholm, and took his baccalaureate on 16 May 1885. Music mattered to him as much as mathematics. He played the flute as a boy and later the violin, and Johann Sebastian Bach was his lifelong favourite. Much later he built a violin from half a coconut shell.

After school he spent a year at the Royal Technological Institute in Stockholm. The year was short, but it left a durable interest in mechanics and in mathematics that does something. His later papers almost all begin with a physical problem: elastic bodies, heat, potentials, vibrations.

Uppsala and Mittag-Leffler, 1886–1898

In 1886 Fredholm enrolled at Uppsala, then the only Swedish university that could award doctorates, and took his first degree there on 28 May 1888. The most exciting mathematics in Sweden, however, was in Stockholm. Gösta Mittag-Leffler, a student of Karl Weierstrass, held the chair at the young Stockholms Högskola, had founded Acta Mathematica in 1882 and had brought Sofia Kovalevskaya to Stockholm as professor. Fredholm stayed registered at Uppsala but went to Stockholm to study with Mittag-Leffler, who became his teacher in everything but the formal sense.

A first paper

His first paper (1890) answered a question about the heat equation and analytic continuation. He gave a power series, of the form \(\sum_k a^k z^{k^2}\) with \(0<a<1\), that converges on the closed unit disc together with all its derivatives, yet has the unit circle as a natural boundary: it cannot be continued analytically past any point of it. Mittag-Leffler sent a copy to Henri Poincaré.

The thesis

In 1895 and 1896 Fredholm studied the equation \(\Delta u+\lambda u=h\) and could not complete the argument he wanted; the missing piece became central to his later theory of integral equations. His doctoral thesis, defended at Uppsala in 1898 and published in 1900 in Acta Mathematica as Sur les équations de l’équilibre d’un corps solide élastique, treated the system of partial differential equations for the equilibrium of an anisotropic elastic body, such as a crystal. He wrote these equations in a compact operator notation of his own and found their fundamental solutions (Lesson 08). The same year he became docent, a lecturer, in mathematical physics at Stockholms Högskola.

Paris and the discovery, 1899–1903

In 1899 Fredholm spent several months in Paris. The central problem in potential theory there was the Dirichlet problem: find a harmonic function in a region with prescribed boundary values. Carl Neumann had solved it for convex regions by a series method; Poincaré had pushed that method further in 1896 and shown that the solution of the associated integral equation depends meromorphically on a parameter. Fredholm talked with Poincaré, Émile Picard and Jacques Hadamard. Poincaré presented a note of his on partial differential equations to the Academy, and on 22 December 1899 Fredholm wrote to thank him, describing his plans for the Dirichlet problem.

Two other ingredients were ready to hand. His compatriot Helge von Koch had developed the theory of infinite determinants, begun by the astronomer George William Hill for lunar theory. Hadamard had proved in 1893 his inequality bounding a determinant by the product of the lengths of its rows. Fredholm combined these with Vito Volterra’s view, from 1896, that an integral equation is the limit of a linear system.

The result came in two steps. A short note of 1900 in the proceedings of the Royal Swedish Academy of Sciences, Sur une nouvelle méthode pour la résolution du problème de Dirichlet, announced the method. The full theory appeared in Acta Mathematica 27 in 1903. In between, in 1901, the Uppsala mathematician Erik Holmgren lectured on Fredholm’s work in Hilbert’s seminar at Göttingen. Hilbert saw at once what it meant. He spent much of the next decade on integral equations, and his student Erhard Schmidt rebuilt the theory again. Lessons 04 to 07 tell the mathematical story.

Integral equations before 1900

For the general reader. In an integral equation the unknown is a whole function, and it appears inside an integral. Such equations arose throughout the nineteenth century, from the motion of a bead on a curve to the electric potential of a conductor. Mathematicians could solve them by successive approximation, but only when a parameter was small. Fredholm’s achievement was to remove that restriction.

A short history

Niels Henrik Abel solved an equation of the first kind in 1823, the tautochrone problem. Joseph Liouville turned boundary-value problems for second-order differential equations into integral equations in the 1830s and solved them by iteration. Carl Neumann, in the 1870s, solved the Dirichlet problem for convex regions by an iteration he called the method of the arithmetic mean. Paul du Bois-Reymond coined the term “integral equation” in 1888. Volterra treated equations with a variable upper limit in 1896, and Poincaré showed in 1896 that Neumann’s series, viewed as a function of a parameter \(\lambda\), continues to a meromorphic function.

The Neumann series

Write \((K\varphi)(x)=\int_a^bK(x,y)\varphi(y)\,dy\) and define the iterated kernels \(K_1=K\), \(K_{n+1}(x,y)=\int_a^bK(x,z)K_n(z,y)\,dz\).

Proof. By induction \(|K_n(x,y)|\le M^n(b-a)^{n-1}\), so the resolvent series is dominated by a geometric series with ratio \(|\lambda|M(b-a)<1\). Substituting shows that \(\varphi\) solves the equation. If \(\psi=\lambda K\psi\), then \(\|\psi\|_\infty\le|\lambda|M(b-a)\|\psi\|_\infty\), so \(\psi=0\).

Compare Volterra’s bound in the variable-limit case, \(|K_n|\le M^n(x-y)^{n-1}/(n-1)!\). The factorial makes Volterra’s series converge for every \(\lambda\). With fixed limits there is no factorial, and for good reason: the series must diverge at the first value of \(\lambda\) where \(\varphi=\lambda K\varphi\) has a nonzero solution. These values, the characteristic values of \(K\), are reciprocals of eigenvalues of the operator \(K\). The problem facing Fredholm was to describe the solution for all \(\lambda\), including near and at these points.

From matrices to kernels: the determinant

For the general reader. To solve a system of linear equations one can use determinants: the system has a unique solution exactly when a single number, the determinant, is nonzero, and the solution is a ratio of determinants. Fredholm replaced the sum in each equation by an integral, let the number of unknowns go to infinity, and watched what happened to the determinant. It became a convergent series, an entire function of the parameter.

The heuristic

Divide \([a,b]\) into \(n\) pieces of length \(h\) with points \(x_i\), and replace the equation by \(\varphi_i-\lambda h\sum_jK(x_i,x_j)\varphi_j=f_i\). For a square matrix \(A\), the determinant of \(I-\lambda A\) expands in principal minors:

where the inner sum runs over all ordered \(k\)-tuples (repeated indices give zero). With \(A_{ij}=hK(x_i,x_j)\), the inner sum is a Riemann sum for a \(k\)-fold integral. Letting \(n\to\infty\) suggests the following definitions.

Convergence: Hadamard’s inequality

Hadamard’s inequality says that \(|\det A|\le\prod_p\|a_p\|\), the product of the Euclidean lengths of the rows. If \(|K|\le M\), each row of the \(k\times k\) matrix \([K(s_p,s_q)]\) has length at most \(M\sqrt k\), so

Since \(k!\ge(k/e)^k\), the terms are bounded by \((e|\lambda|M(b-a))^k/k^{k/2}\), whose \(k\)th root tends to 0. So the series defines an entire function. This is where the growth of the determinant is tamed: \(k^{k/2}\) is much smaller than the \(k!\) that a naive bound \(k!\,M^k\) would give.

The identities

Expand the \((k+1)\times(k+1)\) determinant \(K\binom{x,s_1,\dots,s_k}{y,s_1,\dots,s_k}\) along its first row. The entry \(K(x,y)\) multiplies the principal minor \(K\binom{s}{s}\); each other entry \(K(x,s_j)\) multiplies a minor that, after integration over the \(s\)’s and relabelling, equals \(-\int K(x,z)\,(\cdots)\,dz\) with \((\cdots)\) the integrand of the previous term of \(D(z,y;\lambda)\). Collecting powers of \(\lambda\) gives the first identity. The derivative formula comes from differentiating the \(k\)-fold integral term by term and using symmetry in the variables. Once the identity holds, \(R=D(x,y;\lambda)/D(\lambda)\) satisfies \(R-\lambda KR=K\), which is exactly the resolvent equation \((I-\lambda K)(I+\lambda R)=I\).

Fredholm’s theorems

For the general reader. A square system of linear equations has a simple dichotomy. Either it has exactly one solution for every right-hand side, or the corresponding homogeneous system has nonzero solutions, and then the equations are solvable only for special right-hand sides. Fredholm proved that integral equations of the second kind obey exactly the same rule. This dichotomy is now called the Fredholm alternative.

Write \(K^{\mathsf T}(x,y)=K(y,x)\) for the transposed kernel.

Sketch of the proof

Part 1 is Lesson 05. For parts 2 and 3, Fredholm introduced higher minors \(D\binom{x_1,\dots,x_r}{y_1,\dots,y_r};\lambda\big)\), built like the first minor with \(r\) rows and columns fixed. If \(\lambda_0\) is a zero of \(D\) of order \(m\), then the minor of order \(r\) does not vanish identically for some \(r\le m\). Take the smallest such \(r\). Expanding along a row shows that, as a function of \(x_1\), this minor solves the homogeneous equation, and choosing the other variables suitably produces \(r\) independent solutions. As a function of \(y_1\), it solves the transposed equation, which gives the equality of dimensions. The solvability condition in part 3 is necessary because \(\int(\varphi-\lambda K\varphi)\psi=\int\varphi(\psi-\lambda K^{\mathsf T}\psi)=0\); Fredholm showed it is sufficient by writing a solution in terms of the minors of order \(r\) and \(r+1\).

In modern language: \(K\) is a compact operator on \(C[a,b]\) or \(L^2[a,b]\), and \(I-\lambda K\) is a Fredholm operator of index zero (Lesson 09). Frigyes Riesz proved in 1918, without determinants, that this structure holds for every compact (in his words, completely continuous) operator, by an argument that works in any normed space. The multiplicity \(m\) of a zero of \(D\) equals the algebraic multiplicity of the eigenvalue \(1/\lambda_0\) of \(K\), which may exceed the geometric multiplicity \(r\), just as for matrices.

The Dirichlet problem solved

For the general reader. Heat a metal plate along its edge in a fixed way and wait. The temperature inside settles to a harmonic function determined by the edge values. Proving that such a function always exists was a central problem of nineteenth-century analysis. Fredholm turned it into an integral equation on the boundary and used his alternative: because the solution is unique, it must exist.

Double-layer potentials

Let \(\Omega\subset\mathbb R^2\) be a bounded region whose boundary \(\Gamma\) is a closed \(C^2\) curve, with outward unit normal \(\nu\). Let \(\Phi(x,y)=\frac{1}{2\pi}\ln\frac{1}{|x-y|}\) be the fundamental solution of the Laplacian. For a continuous density \(\mu\) on \(\Gamma\), the double-layer potential

is harmonic off \(\Gamma\). For \(x\in\Gamma\) the kernel \(k(x,y)=\partial\Phi(x,y)/\partial\nu(y)\) stays bounded as \(y\to x\), because \((y-x)\cdot\nu(y)=O(|x-y|^2)\) on a \(C^2\) curve. So the boundary integral operator \((T\mu)(x)=\int_\Gamma k(x,y)\mu(y)\,ds(y)\) has a continuous kernel, exactly the setting of Fredholm’s theory.

For \(\mu\equiv1\) this is Gauss’s lemma: \(\int_\Gamma\partial\Phi/\partial\nu\,ds\) equals \(-1\) inside, \(-\tfrac12\) on \(\Gamma\) and \(0\) outside, since it measures the angle \(\Gamma\) subtends at \(x\), divided by \(-2\pi\).

Existence from uniqueness

To solve \(\Delta u=0\) in \(\Omega\), \(u=f\) on \(\Gamma\), seek \(u\) as a double layer. By the jump relation the density must satisfy

a Fredholm equation of the second kind with \(\lambda=2\).

Proof. By Fredholm’s alternative it suffices to show that \(\mu-2T\mu=0\) forces \(\mu=0\). If so, the double layer \(u\) has interior boundary values 0, hence \(u\equiv0\) in \(\Omega\) by the maximum principle. The normal derivative of a double layer is continuous across \(\Gamma\) (a theorem of Lyapunov), so the exterior function has zero normal derivative on \(\Gamma\). It is harmonic outside and decays like \(1/|x|\) at infinity, so by Green’s identity \(\int_{\text{ext}}|\nabla u|^2=0\) and \(u\equiv0\) outside too. The jump relation gives \(\mu=u_{\mathrm{ext}}-u_{\mathrm{int}}=0\).

The argument needs no convexity and no small parameter. Neumann’s iteration had required convex regions; Poincaré had needed delicate estimates. Fredholm replaced both by a single structural fact. The same method handles the Neumann problem, the exterior problems and, with more care, regions with corners. Boundary integral equations of this kind remain the basis of numerical methods for elliptic problems.

Elastic crystals and fundamental solutions

For the general reader. A fundamental solution describes the response of a medium to a force concentrated at one point. For an isotropic elastic solid it was found in the nineteenth century. A crystal is harder: its stiffness depends on direction. Fredholm’s thesis found the point-force solution for such anisotropic bodies, and in 1908 he extended the method to general elliptic equations with constant coefficients.

The equations

In linear elasticity the displacement \(u=(u_1,u_2,u_3)\) of a body in equilibrium under a body force \(F\) satisfies

where the elastic constants \(c_{ijkl}\) have the symmetries \(c_{ijkl}=c_{jikl}=c_{klij}\) and give a positive definite energy. For a homogeneous crystal they are constant. The system is elliptic: the symbol matrix \(A(\xi)_{ik}=\sum_{j,l}c_{ijkl}\xi_j\xi_l\) is positive definite for \(\xi\ne0\).

The second formula, an average of the inverse symbol over the great circle perpendicular to \(x\), is usually associated with Ilya Lifshitz and Lev Rozentsveig (1947). Fredholm’s thesis had already constructed the fundamental solution of the anisotropic system, in his own operator notation and without Fourier transforms, which were not yet a standard tool for such problems. With \(G\) in hand, the displacement due to forces and boundary data can be written as potentials, and boundary problems reduce to integral equations on the surface, the route that led him to Lesson 07. In 1908 he treated the fundamental solutions of general elliptic equations with constant coefficients by related methods.

Fredholm operators and the index

For the general reader. For a square matrix, the number of independent solutions of the homogeneous system always equals the number of conditions needed to solve the inhomogeneous one. Fredholm proved the same balance for his integral equations. Later mathematicians studied operators where the two numbers differ, and found that their difference, the index, is remarkably stable. It cannot change under small or compact perturbations, and it can be computed from topology.

Index theory began with singular integral equations on curves, where Fritz Noether (1921) found that the dimensions of the two kernels need not agree and that their difference is given by a winding number. Its modern form is a statement about Toeplitz operators. Let \(H^2\subset L^2(\mathbb T)\) be the Hardy space of functions with only nonnegative Fourier coefficients, \(P\) the orthogonal projection onto it, and for continuous \(f\) on the circle let \(T_f h=P(fh)\).

This is the simplest case of the Atiyah–Singer index theorem (1963), which computes the index of any elliptic differential operator on a compact manifold from topological data. Fredholm modules, a central notion of Alain Connes’s noncommutative geometry and Gennadi Kasparov’s KK-theory, abstract exactly this structure.

Fredholm determinants today

For the general reader. Fredholm built his determinant to solve equations. A century later the same object answers questions about randomness. The probability that a large random matrix has no eigenvalue in an interval, or that the largest eigenvalue lies below a threshold, is a Fredholm determinant. These laws now describe phenomena from the zeros of the Riemann zeta function to the growth of crystals.

Trace-class operators

A compact operator \(A\) on a Hilbert space is trace class if \(\sum_ns_n(A)<\infty\), where \(s_n\) are its singular values. For such \(A\) the determinant \(\det(I+A)\) is defined by continuity from finite rank and agrees with Fredholm’s series when \(A\) is an integral operator with a sufficiently regular kernel. Alexander Grothendieck extended the theory to Banach spaces in the 1950s.

For Fredholm’s equation this means \(D(\lambda)=\prod_j(1-\lambda\mu_j)=\prod_j(1-\lambda/\lambda_j)\), where \(\lambda_j=1/\mu_j\) are the characteristic values.

Random matrices

Let \(H\) be a large random Hermitian matrix from the Gaussian unitary ensemble, with eigenvalues rescaled to have mean spacing 1 near the centre of the spectrum. Michel Gaudin and Madan Lal Mehta showed around 1961 that the probability of finding no eigenvalue in an interval of length \(s\) tends to

In 1980 Michio Jimbo, Tetsuji Miwa, Yasuko Môri and Mikio Sato showed that \(\log E\) satisfies a Painlevé V equation. In 1994 Craig Tracy and Harold Widom showed that the largest eigenvalue, suitably rescaled, has distribution \(F_2(s)=\det(I-K_{\mathrm{Ai}}|_{L^2(s,\infty)})\), with the Airy kernel \(K_{\mathrm{Ai}}(x,y)=\frac{\mathrm{Ai}(x)\mathrm{Ai}'(y)-\mathrm{Ai}'(x)\mathrm{Ai}(y)}{x-y}\). The Tracy–Widom law has since appeared in the longest increasing subsequence of a random permutation and in random growth models.

Professor, actuary, violin maker, 1903–1927

Recognition came quickly. The Royal Swedish Academy of Sciences awarded Fredholm the Wallmark Prize in 1903, and the Paris Academy gave him the Poncelet Prize for 1908. Leipzig made him an honorary doctor in 1909, and Uppsala awarded him the Björkén Prize in 1910. He was elected to the Royal Swedish Academy of Sciences in 1914, to the Finnish Society of Sciences and Letters as a foreign member in 1922, and to the Accademia dei Lincei.

His career ran on two tracks. In 1902 he joined the new Swedish state insurance office, and from 1904 he also worked as an actuary for the Skandia insurance company; he proposed a formula for the surrender value of a life insurance policy. On 28 September 1906 he was appointed professor of rational mechanics and mathematical physics at Stockholms Högskola, which he remained until his death. He served the Högskola as pro-dean and dean around 1909–1910, sat on government committees and was a member of the International Committee for Weights and Measures.

His research after 1903 was spare. In 1908 he published the work on fundamental solutions described in Lesson 08. He taught the theory of integral equations to a small circle; his students included Nils Zeilon, who took his doctorate in 1914 and worked on fundamental solutions of partial differential equations, and Carl-Gustaf Rossby, who completed a degree under him in 1925 and went on to found the theory of planetary waves in the atmosphere. On 31 May 1911, aged 45, he married Agnes Maria Liljeblad, who had worked as Mittag-Leffler’s secretary. Their son Bengt was born in 1912.

Fredholm’s workshop habits never left him. He built a mechanical device for solving differential equations, and in his last years he worked on the acoustics of the violin. The unfinished manuscripts have proved hard to interpret.

Death and legacy

Ivar Fredholm died on 17 August 1927 at Mörby, in Danderyd just north of Stockholm, aged 61. His collected works, with a biographical notice by his student Nils Zeilon, appeared in 1955. A lunar crater and the asteroid 21659 Fredholm carry his name.

Legacy

• Spectral theory. Hilbert’s six communications on integral equations (1904–1910) began from Fredholm’s paper. Erhard Schmidt’s geometry of function spaces, Frigyes Riesz’s compact operators and the Riesz–Fischer theorem, and John von Neumann’s Hilbert space all grew from that start.
• Operator theory. Fredholm operators, the index, and Atkinson’s and Riesz’s theorems are basic tools in analysis, geometry and mathematical physics, and lead to the Atiyah–Singer index theorem and K-theory.
• Partial differential equations. Boundary integral equations and fundamental solutions are standard methods for elliptic problems, in theory and in numerical computation.
• Determinants. Fredholm determinants govern random matrix statistics, integrable systems and the Tracy–Widom laws.

Few mathematicians are known for so short a list of papers. Hilbert recognized the importance of the 1903 theory as soon as Holmgren described it, and Göttingen made it the starting point of a decade of work. Its method, solving an infinite problem by passing to the limit in a finite one with enough care, is now so familiar that it is easy to forget how bold it was.

Connections

People linked to Ivar Fredholm across this collection. A name without a link has no page here yet. See Ivar Fredholm in the network of all 337 lives →

Teachers and advisors

• Gösta Mittag-Leffler 1888-1898

Students

• Carl-Gustaf Rossby 1925
• Nils Zeilon 1914

Influenced by

• Henri Poincaré 1899
• Jacques Hadamard
• Sofia Kovalevskaya
• Vito Volterra
• Helge von Koch

Influenced

• David Hilbert
• Frigyes Riesz
• Erhard Schmidt

Colleagues and collaborators

• Émile Picard 1899
• Erik Holmgren 1901

Also mentioned in this life

• Niels Henrik Abel
• George Biddell Airy
• Michael Atiyah
• Stefan Banach
• Peter Gustav Lejeune Dirichlet
• Leonhard Euler
• Joseph Fourier
• Carl Friedrich Gauss
• Alexander Grothendieck
• G. H. Hardy
• William Thomson, Lord Kelvin
• Pierre-Simon Laplace
• Joseph Liouville
• Aleksandr Lyapunov
• Siméon Denis Poisson
• Jean-Victor Poncelet
• Bernhard Riemann
• John von Neumann
• Karl Weierstrass

Fields

• partial differential equations 8 lives
• mathematical physics 45 lives

On CoursesHub

• ODEs / PDEs / Green Functions · Physical Mathematics

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Erik Ivar Fredholm, MacTutor History of Mathematics.
2. Erik Ivar Fredholm, Wikipedia.
3. M. Bernkopf, Fredholm, Erik Ivar, Dictionary of Scientific Biography, reprinted at MacTutor.
4. Ivar Fredholm, Henri Poincaré Papers (correspondence, 1899).
5. Erik Ivar Fredholm, Mathematics Genealogy Project.
6. N. Zeilon, biographical notice in I. Fredholm, Œuvres complètes, 1955.

Mathematics

1. I. Fredholm, Sur une nouvelle méthode pour la résolution du problème de Dirichlet, Öfversigt af Kongl. Vetenskaps-Akademiens Förhandlingar 57 (1900).
2. I. Fredholm, Sur les équations de l’équilibre d’un corps solide élastique, Acta Mathematica 23 (1900).
3. I. Fredholm, Sur une classe d’équations fonctionnelles, Acta Mathematica 27 (1903), 365–390.
4. D. Hilbert, Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen, Teubner, 1912.
5. F. Riesz, Über lineare Funktionalgleichungen, Acta Mathematica 41 (1918).
6. R. Kress, Linear Integral Equations, 3rd ed., Springer, 2014 (Chapters 3–6 for the Fredholm alternative and potential theory).
7. B. Simon, Trace Ideals and Their Applications, 2nd ed., AMS, 2005.
8. F. Bornemann, On the numerical evaluation of Fredholm determinants, arXiv:0804.2543.
9. C. A. Tracy and H. Widom, Level-spacing distributions and the Airy kernel, arXiv:hep-th/9211141.
10. J. Dieudonné, History of Functional Analysis, North-Holland, 1981.