Dualizing the Poisson summation formula.

If f(x) and g(x) are a Fourier cosine transform pair, then the Poisson summation formula can be written as 2sumfrominfinityn = 1g(n) + g(0) = 2sumfrominfinityn = 1f(n) + f(0). The concepts of linear transformation theory lead to the following dual of this classical relation. Let phi(x) and gamma(x) = phi(1/x)/x have absolutely convergent integrals over the positive real line. Let F(x) = sumfrominfinityn = 1phi(n/x)/x - integralinfinity0phi(t)dt and G(x) = sumfrominfinityn = 1gamma (n/x)/x - integralinfinity0 gamma(t)dt. Then F(x) and G(x) are a Fourier cosine transform pair. We term F(x) the "discrepancy" of phi because it is the error in estimating the integral phi of by its Riemann sum with the constant mesh spacing 1/x.

The Markoff-Duffin-Schaeffer inequalities abstracted.

The kth Markoff-Duffin-Schaeffer inequality provides a bound for the maximum, over the interval -1

Rubel's universal differential equation.

Fourth-order differential equations such as 16y'(m)y'(2) - 32y(m)y(n)y' + 17y(0(3) ) = 0 are developed. It is shown that the equation is "universal" in the sense that any continuous function can be approximated with arbitrary accuracy over the whole x axis by a solution y(x) of the equation. This solution is a piecewise polynomial of degree 9 and of class C(4).

Bounds for the rth characteristic frequency of a beaded string or of an electrical filter.

The fundamental mode of vibration of a beaded string has a shape without change of sign. The rth higher normal mode of vibration has r changes of sign. Given any virtual shape of the string with r changes of sign, an algorithm is found that gives upper and lower bounds for the rth characteristic frequency as a function of the virtual shape. By making a certain transformation it is found that this algorithm holds for the characteristic frequencies of an inductor-capacitor network. Other transformations show that it applies to the rth eigenvalue of a Hermitian matrix.

Clark's Theorem on linear programs holds for convex programs.

Given a linear minimization program, then there is an associated linear maximization program termed the dual. F. E. Clark proved the following theorem. "If the set of feasible points of one program is bounded, then the set of feasible points of the other program is unbounded." A convex program is the minimization of a convex function subject to the constraint that a number of other convex functions be nonpositive. As is well known, a dual maximization problem can be defined in terms of the Lagrange function. The dual objection function is the infimum of the Lagrange function. The feasible Lagrange multipliers are those satisfying: (i) the multipliers are nonnegative and (ii) the dual objective function is not negative infinity. It is found that Clark's Theorem applies unchanged to dual convex programs. Moreover, the programs have equal values.

Convex programs having some linear constraints.

The problem of concern is the minimization of a convex function over a normed space (such as a Hilbert space) subject to the constraints that a number of other convex functions are not positive. As is well known, there is a dual maximization problem involving Lagrange multipliers. Some of the constraint functions are linear, and so the Uzawa, Stoer, and Witzgall form of the Slater constraint qualifications is appropriate. A short elementary proof is given that the infimum of the first problem is equal to the supremum of the second problem.

Hilbert transforms in yukawan potential theory.

If H denotes the classical Hilbert transform and Hu(x) = v(x), then the functions u(x) and v(x) are the values on the real axis of a pair of conjugate functions, harmonic in the upper half-plane. This note gives a generalization of the above concepts in which the Laplace equation Deltau = 0 is replaced by the Yukawa equation Deltau = mu(2)u and in which the Cauchy-Riemann equations have a corresponding generalization. This leads to a generalized Hilbert transform H(mu). The kernel function of this new transform is expressable in terms of the Bessel function K(0). The transform is of convolution type.

Parallel subtraction of matrices.

A new Hermitian semidefinite matrix operation is studied. This operation-called parallel subtraction-is developed from the theory of parallel addition. Since the theory of parallel addition is motivated by the analysis of interconnected electrical networks, parallel subraction may be interpreted in terms of the synthesis of electrical networks. The idea of subtraction is also extended to hybrid addition.

Geometric programming, chemical equilibrium, and the anti-entropy function.

THE CULMINATION OF THIS PAPER IS THE FOLLOWING DUALITY PRINCIPLE OF THERMODYNAMICS: maximum S = minimum S(*). (1) The left side of relation (1) is the classical characterization of equilibrium. It says to maximize the entropy function S with respect to extensive variables which are subject to certain constraints. The right side of (1) is a new characterization of equilibrium and concerns minimization of an anti-entropy function S(*) with respect to intensive variables. Relation (1) is applied to the chemical equilibrium of a mixture of gases at constant temperature and volume. Then (1) specializes to minimum F = maximum F(*), (2) where F is the Helmholtz function for free energy and F(*) is an anti-Helmholtz function. The right-side of (2) is an unconstrained maximization problem and gives a simplified practical procedure for calculating equilibrium concentrations. We also give a direct proof of (2) by the duality theorem of geometric programming. The duality theorem of geometric programming states that minimum cost = maximum anti-cost. (30).