EXTENSIONS OF THE HEISENBERG-WEYL INEQUALITY

In this paper a number of generalizations of the classical Heisenberg-Weyl uncertainty inequality are given. We prove the n-dimensional Hirschman entropy in- equality (Theorem 2.1) from the optimal form of the Hausdorff-Young theorem and deduce a higher dimensional uncertainty inequality (Theorem 2.2). From a general weighted f.orm of the Hausdorff-Young theorem, a one-dimensional weighted entropy inequality is proved and some weighted forms of the Heisenberg-Weyl inequalities are given.


INTRODUCTION.
Let be the Fourier transform of f defined by (x) f e-2ixyf(y)dy, x .
If f e L2()with L 2 -norm ilfll2 i, then by Plancherel's theorem III12 I, so that If(x) 2 and l(y) 2 are probabilzty frequency functions.The variance of a ptcbability frequency Junction g is defined by V[g] f (x-m)2g(x)dx where m / xg(x)dx is the mean.With these notations, the Heisenberg uncertainty principle of quantum mechanics can be stated in terms of the Fourier transform by the inequality V[Ifl2]V[lI 2] (162) -I (1.1)In the sequel, we assume without loss of generality that the mean m O.If g is a probability frequency function, then the entropy of g is defined by E[g] / g(x)log g(x)dx.
With f as above, Hirschman [i] Droved that E[Ifl2] + e[l12] EH (1.2) with E H O, and suggested that (1.2) holds with E H log 2-i.If E H has that form, then by an inequality of Shannon and Weaver I it follows that (1.2) implies (I.I).
The purpose of this paper is to give extensions of the Heisenberg-Weyl inequality (1.1).In the next section a new proof of the entropy inequality (1.2) for functions on Nn is given and an n-dimensional Heisenberg-Weyl inequality is deduced.The n- dimensional generalization of inequality (1.4) is also given in the next section.The two inequaiities are quite different, even in the case p 2, but depend strongly on the sharp Hausdorff-Young inequality.In the third section a weighted form of the Heisenberg-Weyl inequality in one dimension is obtained from a weighted form of the Hausdorff-Young inequality ([5][6][7] [8]).Unlike the constant A(p) in (1.3) the con- stant of the weighted Hausdorff-Young inequality (3.3) of (Theorem 3.1) is far from sharp.If the constant is not too large, then a weighted form of Hirschman's entropy inequality can also be given, from which another uncertainty inequality is deduced.
Throughout, p' p/(p-1), with p' if p I, is the conjugate index of p, and similarly for other letters.S( is the Schwartz class of slowly increasing functions r on n.We say g is in the weighted L w -space with weight w, if wg a L r and norm IIgllr, w llwgllr.If x e n, then x (Xl,X 2 Xn) and dx dXl...dxn the n- dimensional Lebesgue measure, fi(x), x n denotes the partial derivative of with respect to the i th component and fij (fi)j The letter C denotes a constant which may be different at different occurrences, but is independent of f.

THE HIRSCHMAN INEQUALITY.
The Fourier transform of f on n is given by (x) n e-2ix'yf(y)dy x e n, x'y xlY +...+ XnYn.
and the entropy of a function on n is defined as before with replaced by n.We shall need the following well known result (c.f.[9; 13.32 ii]): If f d l, then X lim (f IflPd)I/P exp / log Ifld.
(2.1) p/o+ X X Using this fact we obtain easily the n-dimensional form of Hirschman's inequality (1.2) THEOREM 2.1.If f e I 2 (An) such that llfl12 II}I12 whenever the left side has meaning.
If f e L 2 the result is obtained as in [l] only now one takes for mT, me(x e -e]X[2' and for T' ge (y) e-n/2e-lYl2/e."We omit the details.
If Igl L2() is a probability frequency function, then the relation between entropy and variance is expressed by The n-dimensional form of this inequality is given in the following lemma: Using the lemma and Theorem 2.1, we easily establish an n-dimensional extension of the Heisenberg-Weyl inequality.
If we denote the bracketed terms above by D[Ifl 2] and D[I121, the discrepancy of Schwarz's inequality, or the difference between variance and covarlance of Ifl 2 and II 2, then the two dimensional Heisenberg-Weyl inequality shows that the dis- crepancies of II 2 and II 2 cannot both be small; D[Ifl 2] D[II 2] > (1672) -2 A different generalization of (I.I) may be obtained along the lines of Theorem I.I.
THEOREM 2.3.Let f g s(n), such that f(xl,x 2 x n) 0, whenever x i 0 for some i.
Recall that if g is a Lebesgue measurable function on , then the equi-measurable decreasing rearrangement of g is defined by g (t) inf{y > O: l{x g : Ig(x) Y}I t}, where y 0 and IEI denotes Lebesgue measure of the set E. Clearly, if g is an even function on , decreasing on (0,), then for t O, g (t) g(t/2).We shall use this fact below.DEFINITION 3.1.Let u and v be locally integrable functions of .We write (u,v) e F p 6 p 6 q < =, if ,q' sup (IS[u*(t)]qdt)I/q(fl/s[(I/v)*(t)]P'dt)i/P'< , (3.1) 0 0 where in the case p the second integral is replaced by the essential supremum of (i/v)*(t) over (0, i/s).
If u and I/v are even and decreasing on (0, =) then (3.1) is equivalent to sup (/s/2 [u(x) qdx) i/q (/i/(2s) s>o 0 0 and in this case we write (u, v) e Fp,q.
; Theorem 1.1]).Suppose (u v) p,q, 4 p 4 q < and f e L p.v (i) If lim [ifn fllp, v 0 for a sequence of simple functions, then {n con- n/ verges in L q to a function L q.
is independent of the sequence {n and is called u u the Fourier transform of f.
(ii) there is a constant B 0 such that for all f L p (iii) If g e L?/u, q I, then Parseval's formula f (y)g(y)dy f f(t)(t)dt (3.3)   holds.
We note ([5], [6], [8]) that Theorem 3.1 is sharp in the sense that if u and v are even and satisfy (3.3), then (u, v) satisfies (3.2).The constant B in (3.3) is not sharp, however it is of the form B k.C where k k(p,q) is independent of u and v and C is the supremum of (3.1), and in the case u, I/v decreasing and even the supremum (3.2).
Note that the case p also holds, provided the second integral in the F P,q , condition is interpreted as the essential supremum of (I/v) over (0, I/s).
The same result holds also if we take (i/u, v) g Fp,q.
Observe also that the case u v E and q p', < p < 2 reduces to (1.4), but with a different constant.
Weighted inequalities of the form (3.6) were also obtained by Cowling and Price [3] but by quite different methods.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning THEOREM 2.2.Let f e e2(n) with [If[]2 [[[[2 and fn xixj [f(x)[2dx, ij f YiYj [(Y) 12dy' bij i,j 1,2 n; be the entries of the matrices B and respectively, then (det B)(det ) (16 2)-n.

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation