Integral equations Fredholm equations and Fredholm theory Lemma

Hadamard's inequality in Fredholm theory

Builds on Hadamard's inequality Continuous kernel Fredholm determinant Fredholm minors Entire function

Statement

Let $a < b$ , let $K \colon [a, b] \times [a, b] \to \C$ be a continuous kernel, and let $M = \max \abs{K}$ . Then for every $n \ge 1$ and all $x_1, \dots, x_n, y_1, \dots, y_n \in [a, b]$ , $\abs{\det[K(x_i, y_j)]_{i,j=1}^{n}} \le n^{n/2} M^n.$ Consequently, the $n$ -th coefficient $c_n = \frac{1}{n!} \int_{[a, b]^n} \det[K(x_i, x_j)] \, dx$ of the Fredholm determinant satisfies $\abs{c_n} \le \frac{n^{n/2} \, M^n (b - a)^n}{n!},$ and the series $\sum_n (-1)^n c_n \lambda^n$ converges for every $\lambda \in \C$ .

Remarks

The first bound is Hadamard's inequality for the matrix $[K(x_i, y_j)]$ : each column has $n$ entries of modulus at most $M$ , so its Euclidean norm is at most $\sqrt{n} M$ . Integrating over $[a, b]^n$ gives the bound on $c_n$ . By Stirling's formula $n! \ge (n/e)^n$ , so $\abs{c_n \lambda^n} \le \left(e M (b - a) \abs{\lambda}\right)^n / n^{n/2}$ , and the $n$ -th root of this tends to $0$ . Hence the radius of convergence is infinite, and $D$ is an entire function.

The same argument, applied to matrices of size $n + k$ , shows that the series defining the Fredholm minors converge for every $\lambda$ , uniformly in the fixed points.

The crude bound $\abs{\det} \le n! \, M^n$ , from expanding the determinant into $n!$ products, is not enough: it only gives $\abs{c_n} \le (M (b - a))^n$ and hence convergence for $\abs{\lambda} < 1 / (M (b - a))$ . The factor $n^{n/2}$ instead of $n!$ is what makes the whole theory work for every $\lambda$ .

Going deeper

For a student

Fredholm needed exactly this estimate to show that his determinant and minors are entire functions of $\lambda$ . Without it, his formulas would only work near $\lambda = 0$ , where the Neumann series already does the job.

Reading path

To understand this page from the ground up, read these 19 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Continuous kernel concept
Determinant concept
Entire function concept
Fredholm determinant concept
Fredholm minors concept
Norm concept
Orthogonal vectors concept
Positive semidefinite matrix concept
Vector space definition
Inner product definition
Linear combination definition
Linear subspace definition
Orthogonal vectors definition
Linear independence definition
Linear span definition
Orthonormal family definition
Gram–Schmidt process concept
Hadamard's inequality statement
Hadamard's inequality in Fredholm theory statement

Tag	`9NZ9CXC8FY`
URI	`m2://9NZ9CXC8FY`
Kind	statement, lemma
Writing	W0 stub
MSC	45B05, 15A45
Status	This copy of the site has no build data. Run `m2 status 9NZ9CXC8FY`.
About	Fredholm determinant
Uses	Hadamard's inequality, Continuous kernel, Fredholm determinant, Fredholm minors, Entire function