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1973

Volume 4, Issue 4, pp. 557-695


A Result on Differential Inequalities and Its Application to Higher Order Trajectory Derivatives

Robert W. Gunderson

SIAM J. Math. Anal. 4, pp. 557-560 (4 pages)

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A result on differential inequalities is obtained by considering the adjoint differential equation of the variational equation of the right side of the inequality. The main theorem is proved using basic results on differentiability of solutions with respect to initial conditions. The result is then applied to the problem of determining solution behavior using comparison techniques.

On the Solution of a Volterra Integral Equation with a Weakly Singular Kernel

Frank de Hoog and Richard Weiss

SIAM J. Math. Anal. 4, pp. 561-573 (13 pages) | Cited 2 times

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The solution $x(t)$ of the Volterra integral equation of the second kind $x(t) = f_1 (t) + \sqrt t f_2 (t) + \int _0^t g(t,s,x(s))(t - s)^{ - {1 / 2}} ds$ is examined. It is shown that $x(t) = u(t) + \sqrt t v(t)$, where $u(t)$ and $v(t)$ are smooth under appropriate smoothness conditions on $f_1 (t)$, $f_2 (t)$ and $g(t,s,x)$ and satisfy a system of Volterra integral equations of the second kind.

Analysis of Walsh Transforms Using Integration by Parts

C. K. Yuen

SIAM J. Math. Anal. 4, pp. 574-584 (11 pages) | Cited 2 times

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Each Walsh transform of a smooth function can be expressed as a weighted average of a derivative of the function. The weighting function is an integral of the Walsh function. Its mean and maximum values can be found easily so that we can estimate the size of Walsh transforms without having to actually compute them.

Inequalities Involving a Function and Its Inverse

R. P. Boas, Jr. and M. B. Marcus

SIAM J. Math. Anal. 4, pp. 585-591 (7 pages) | Cited 2 times

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The paper presents a simple technique for establishing a class of inequalities, some of which arise in connection with $\varepsilon $-entropy and its applications in probability, and which include a eneralization of Young’s inequality.

Imbedding a Class of Linear Integral Equations Through the First Critical Point

Alan Schumitzky and Tom Wenska

SIAM J. Math. Anal. 4, pp. 592-608 (17 pages) | Cited 1 time

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In this paper, we investigate the continuation of an imbedded solution $x(t) = x(t,\alpha )$ of \[x(t) = f(t) + \int_0^\alpha {k(t,s)x(s)ds = f(t) + (K_\alpha x)(t)} \] through its first “critical point” $\alpha = c$. Under the assumption that the Fredholm resolvent $\Gamma _\alpha = (I - K_\alpha )^{ - 1} - I$ has a simple pole in its meromorphic expansion about $\alpha = c$ we obtain a simple eigenspace corresponding to$\lambda = 1$ for the operator $K_c $ ; and in accordance with the redholm alter-native, we have an imbedded solution for $\alpha > c$ for forcing functions orthogonal to the one-dimensional eigenspace of the adjoint operator $K_c^ * $. The principal technique is the explicit solving of the Bartle-Schmidt bifurcation equation.

Linear Integral Equations of the Third Kind

G. R. Bart and R. L. Warnock

SIAM J. Math. Anal. 4, pp. 609-622 (14 pages) | Cited 17 times

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Linear integral equations of the third kind are studied as equations in two different spaces of generalized functions. In the first space $D_\tau $, which consists of linear combinations of delta functions and continuous functions, the equation of the third kind has properties similar to those of the Fredholm equation of the second kind. The second space $P_\tau $ is comprised of linear combinations of delta functions and functions continuous except for poles, integration over the poles being defined by Cauchy’s principal value. $\ln P_\tau $ the behavior of the third-kind equation is essentially different from that of second-kind Fredholm equations. Solutions in both $D_\tau $ and $P_\tau $ may be constructed explicitly via Fredholm theory. Examples showing the suitability of these spaces in physical problems are cited, and earlier literature on third-kind equations is surveyed briefly.

Singular Differential Equations in Hilbert Space

John Lagnese

SIAM J. Math. Anal. 4, pp. 623-637 (15 pages) | Cited 3 times

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A singular differential equation in a Hilbert space $H$ is one of the form \[ A\frac{{du(t)}} {{dt}} - Bu(t) = f(t), \] where $A$ and $B$ are linear operators in $H$ which may be unbounded and $A$ may have zero as a spectral point. In the current paper we obtain necessary and sufficient conditions for solvability and uniqueness of solution of such equations. Representation formulas and eigenfunction expansions for the solutions are also obtained. We show that our results apply to a large class of boundary value problems for certain nonclassical partial differential equations. This class contains in particular equations which occur in various physical problems such as fluid flow through a fissured rock, shear in second order fluids, soil mechanics and thermodynamics.

Lie Theory and the Appell Functions $F_1 $

Willard Miller, Jr.

SIAM J. Math. Anal. 4, pp. 638-655 (18 pages) | Cited 2 times

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It is shown that $sl(5,\mathbb{C})$ is the dynamical symmetry algebra of the Appell functions $F_1 $ . This permits use of representation theory and the techniques of Weisner and Vilenkin to derive systematically a variety of addition theorems, generating functions and Mellin–Barnes integrals for the $F_1 $.

Hilbert Transforms, Plemelj Relations, and Fourier Transforms of Distributions

Marion Orton

SIAM J. Math. Anal. 4, pp. 656-670 (15 pages) | Cited 11 times

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Using the theory of analytic representations, it is shown that a generalized Hilbert transformation may be defined on the space $\mathcal{D}'$ of Schwartz distributions. It yields extended Plemelj and dispersion relations for all distributions in $\mathcal{D}'$. Distributions which are limits of functions analytic in the upper or lower half-plane are seen to be completely characterized by a certain Fourier transform property. Corresponding results were previously obtained for distributions in the subspaces $\mathcal{S}'$ and $O'_\alpha $ of $\mathcal{D}'$.

$A$-Stable Methods and Padé Approximations to the Exponential

Byron L. Ehle

SIAM J. Math. Anal. 4, pp. 671-680 (10 pages) | Cited 31 times

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The set of Padé approximations to the exponential function is studied. It is shown that all entries on the first and second subdiagonal of the Padé table are analytic and bounded by 1 in the entire left half-plane. These results are then applied to the problem of producing $A$-stable numerical methods for solving initial value problems. It is shown that they easily permit one to generate several classes of methods of arbitrarily high order which are $A$-stable.

Spherical Summability of Conjugate Multiple Fourier Series and Integrals at the Critical Index

G. E. Lippman

SIAM J. Math. Anal. 4, pp. 681-695 (15 pages) | Cited 2 times

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In Euclidean $k$-space, $k \geqq 2$, for Bochner–Riesz summability at the critical index $\gamma = {{(k - 1)} / 2}$, we obtain a localization theorem for Fourier integrals conjugate with respect to spherical harmonic kernels. It is also shown that this result is best possible with respect to the index of sum-mability and that localization does not hold at the critical index for conjugate multiple Fourier series.
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