“Schur test”的意思、由来-开放百科全书

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Schur test

释义

Common usage and Young's inequality
Proof
See also
References

In mathematical analysis, the Schur test, named after German mathematician Issai Schur, is a bound on the operator norm of an integral operator in terms of its Schwartz kernel (see Schwartz kernel theorem).

Here is one version.^[1] Let be two measurable spaces (such as ). Let be an integral operator with the non-negative Schwartz kernel , , :

If there exist real functions and and numbers such that

for almost all and

for almost all , then extends to a continuous operator with the operator norm

Such functions , are called the Schur test functions.

In the original version, is a matrix and .^[2]

Common usage and Young's inequality

A common usage of the Schur test is to take Then we get:

This inequality is valid no matter whether the Schwartz kernel is non-negative or not.

A similar statement about operator norms is known as Young's inequality for integral operators:^[3]

where satisfies , for some , then the operator extends to a continuous operator , with

Proof

Using the Cauchy–Schwarz inequality and the inequality (1), we get:

Integrating the above relation in , using Fubini's Theorem, and applying the inequality (2), we get:

It follows that for any .

References

1. ^Paul Richard Halmos and Viakalathur Shankar Sunder, Bounded integral operators on spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (Results in Mathematics and Related Areas), vol. 96., Springer-Verlag, Berlin, 1978. Theorem 5.2.
2. ^I. Schur, Bemerkungen zur Theorie der Beschränkten Bilinearformen mit unendlich vielen Veränderlichen, J. reine angew. Math. 140 (1911), 1–28.
3. ^Theorem 0.3.1 in: C. D. Sogge, Fourier integral operators in classical analysis, Cambridge University Press, 1993. {{ISBN|0-521-43464-5}}

1 : Inequalities

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Common usage and Young's inequality

Proof

See also

References