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Riesz-Thorin theorem
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In mathematics, the Riesz-Thorin theorem, often referred to as the Riesz-Thorin Interpolation Theorem or the Riesz-Thorin Convexity Theorem is a result about interpolation of operators. This should not be confused with somewhat different mathematical procedure of interpolation
of functions. It is named after Marcel Riesz and his student G. Olof Thorin.
This theorem deals with linear maps acting between
Lp spaces.
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Encyclopedia
In mathematics, the Riesz-Thorin theorem, often referred to as the Riesz-Thorin Interpolation Theorem or the Riesz-Thorin Convexity Theorem is a result about interpolation of operators. This should not be confused with somewhat different mathematical procedure of interpolation
of functions. It is named after Marcel Riesz and his student G. Olof Thorin.
This theorem deals with linear maps acting between
Lp spaces. Its usefulness stems from the fact that some of these spaces have rather simpler structure than others. Usually that refers to which is a Hilbert space, or to and L∞ (see examples below). Therefore one may prove theorems about the more complicated cases by proving them in two simple cases and then using the Riesz-Thorin theorem to pass from the simple cases to the complicated cases. A related approach is to use the Marcinkiewicz theorem.
Definition
A slightly informal version of the theorem can be stated as follows:
Theorem: Assume T is a bounded linear operator from to and at the same time from to . Then it is also a bounded operator from to for any r between p and q.
This is informal because an operator cannot formally be defined on two different spaces at the same time. To formalize it we need to say: let T be a linear operator defined on a family F of functions which is dense in both and (for example, the family of all simple functions). And assume that Tf is in both and for any f in F, and that T is bounded in both norms. Then for any r between p and q we have that F is dense in , that Tf is in for any f in F and that T is bounded in the norm. These three ensure that T can be extended to an operator from to .
In addition an inequality for the norms holds, namely
A version of this theorem exists also when the domain and range of T are not identical. In this case, if T is bounded from to then one should draw the point in the unit square. The two q-s give a second point. Connect them with a straight line segment and you get the r-s for which T is bounded. Here is again the almost formal version
Theorem: Assume T is a bounded linear operator from to and at the same time from to . Then it is also a bounded operator from to where
and t is any number between 0 and 1.
The perfect formalization is done as in the simpler case.
One last generalization is that the theorem holds for for any measure space Ω. In particular it holds for the spaces.
Convexity
Another more general form of the theorem is as follows . Suppose that μ1 and μ2 are two measures on possibly different measure spaces. Let T be a linear mapping from the space of μ1-integrable functions into the space of μ2-measurable functions, and for 1 = p,q = 8, define to be the operator norm of a continuous extension of T to
if such an extension exists, and 8 otherwise. Then the theorem asserts that the function
is convex in the rectangle (a,b) ∈ [0,1]×[0,1].
Application examples
Hausdorff-Young inequality
We consider the Fourier operator, namely let T be the operator that takes a function on the unit circle and outputs
the sequence of its Fourier coefficients
-
Parseval's theorem shows that T is bounded from to with norm 1. On the other hand, clearly,
so T is bounded from to with norm 1. Therefore we may invoke the Riesz-Thorin theorem to get, for any 1 < p < 2 that T, as an operator from to , is bounded with norm 1, where
In a short formula, this says that
This is the well known Hausdorff-Young inequality. It might be interesting to note that for p > 2 the natural extrapolation of this inequality fails, and the fact that a function belongs to , does not give any additional information on the order of growth of its Fourier series beyond the fact that it is in .
Convolution operators
Let f be a fixed integrable function and let T be the operator of convolution with f, i.e., for each function g we have
It is well known that T is bounded from to and it is trivial that it is bounded from L∞ to L∞ (both bounds are by ). Therefore the Riesz-Thorin theorem gives
We take this inequality and switch the role of the operator and the operand, or in other words, we think of S as the operator of convolution with g, and get that S is bounded from to . Further, since g is in we get,
in view of Hölder's inequality, that S is bounded from to L∞, where again . So interpolating we get
where the connection between p, r and s is
Thorin's contribution
The original proof of this theorem, published in 1926 by Marcel Riesz, was a long and difficult calculation. Riesz' student G. Olof Thorin subsequently discovered a far more elegant proof and published it in 1939. The English mathematician J. E. Littlewood once enthusiastically referred to Thorin's proof as "the most impudent idea in mathematics".
Here is a brief sketch of that proof:
One of its main ingredients is the following rather well known result about analytic functions. Suppose that is a bounded analytic function on the two lines
and and on the strip between these two lines.
Suppose also that at every point on those two lines.
Then, by applying the Phragmén-Lindelöf principle (a kind of maximum principle for infinite domains) one gets that
at every point between these two lines, and in particular at the point .
Thorin ingeniously defined a special analytic function connected with the operator . He used the fact that is bounded on
to deduce that on the line
, and, analogously, he used the boundedness of on
to deduce that on the line
. Then, after using the result mentioned above to give that
, he was able to show that this implies that
is bounded on
.
Thorin obtained this function with the help of a generalized notion of an analytic function whose values are elements of spaces instead of being complex numbers.
In the 1960's Alberto Calderón adapted and generalized Thorin's ideas to develop
the method of complex interpolation.
Suppose that and are two Banach spaces which are continuously contained in some suitable larger space.
Calderon's method enables one to construct a family of new Banach spaces ,
for each with which are ``between"
and and have the ``interpolation" property that
every linear operator which is bounded on and on is also bounded on each of the complex interpolation spaces .
Calderon's spaces have many applications. See for example Sobolev space.
Mityagin's theorem
B.Mityagin extended the Riesz-Thorin theorem; we formulate the extension
in the special case of spaces of sequences with unconditional bases (cf. below).
Assume , . Then for any unconditional Banach space of sequences (that is, for any and any , ).
The proof is based on the Krein-Milman theorem.
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