Stone's theorem and completeness of orthogonal systems

It is well known (e.g. Stone [1]) that the Stone-Weierstrass approximation theorem can be used to prove the completeness of various systems of orthogonal polynomials, e.g. Chebyshev polynomials. In this paper, Stone's theorem is used to prove a more general completeness theorem, which includes as special cases Plancherel's theorem, the corresponding theorem for Hankel transforms, the completeness of various polynomial systems, and certain expansions in Jacobian elliptic functions. The essential feature common to all these systems is a certain algebraic structure — if S is an appropriate vector space spanned by orthogonal functions, then the algebra A generated by S is contained in the closure of S in a suitable norm.

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The aerodynamic forces on an aerofoil in non-uniform unsteady motion in a closed tunnel

Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences ◽

10.1098/rsta.1957.0021 ◽

1957 ◽

Vol 250 (977) ◽

pp. 247-278 ◽

Cited By ~ 1

Keyword(s):

Harmonic Oscillation ◽

Elliptic Functions ◽

Detailed Examination ◽

Unsteady Motion ◽

Physical Plane ◽

Aerodynamic Forces ◽

Jacobian Elliptic Functions ◽

First Order ◽

Inviscid Incompressible Fluid ◽

Special Cases

The two-dimensional unsteady motion of an aerofoil, situated midway between parallel walls, and moving through an inviscid, incompressible fluid, is investigated. A completely general upwash distribution is taken, and expressions are obtained for the pressure on the aerofoil surface and the lift and moment about the mid-chord point. By a conformal transformation involving Jacobian elliptic functions the physical plane is mapped into a rectangle, and the theory is based on a solution of Laplace’s equation satisfying certain given boundary conditions on this rectangle. Special cases are considered in which the upwash is ( a ) sudden upgust, and ( b ) a harmonic oscillation. Detailed examination is made of a rigid-body aerofoil performing translational and rotational harmonic oscillations. The aerodynamic forces are expressed in terms of dimensionless ‘air-load coefficients’, which are then compared with corresponding coefficients for an aerofoil in an infinitely deep stream. The air-load coefficients are obtained in a form which readily enables first-order corrections for wall interference to be evaluated. It is shown that the formulae derived are at variance with corresponding results obtained by other authors using different methods.

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Jacobian Elliptic Functions

Foundations of Space Dynamics ◽

10.1002/9781119455301.app2 ◽

2021 ◽

pp. 345-346

Keyword(s):

Elliptic Functions ◽

Jacobian Elliptic Functions

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Jacobian elliptic functions and a family of bivariate peak polynomials

European Journal of Combinatorics ◽

10.1016/j.ejc.2021.103371 ◽

2021 ◽

Vol 97 ◽

pp. 103371

Author(s):

Shi-Mei Ma ◽

Jun Ma ◽

Yeong-Nan Yeh ◽

Roberta R. Zhou

Keyword(s):

Elliptic Functions ◽

Jacobian Elliptic Functions

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Ultrasonic Waves in Bubbly Liquids: An Analytic Approach

Mathematics ◽

10.3390/math9111309 ◽

2021 ◽

Vol 9 (11) ◽

pp. 1309

Author(s):

P. R. Gordoa ◽

A. Pickering

Keyword(s):

Mathematical Model ◽

Partial Differential Equations ◽

Differential Equations ◽

Acoustic Waves ◽

Elliptic Functions ◽

Coupled System ◽

Ultrasonic Waves ◽

Partial Differential ◽

Special Cases ◽

Spherical Bubbles

We consider the problem of the propagation of high-intensity acoustic waves in a bubble layer consisting of spherical bubbles of identical size with a uniform distribution. The mathematical model is a coupled system of partial differential equations for the acoustic pressure and the instantaneous radius of the bubbles consisting of the wave equation coupled with the Rayleigh–Plesset equation. We perform an analytic analysis based on the study of Lie symmetries for this system of equations, concentrating our attention on the traveling wave case. We then consider mappings of the resulting reductions onto equations defining elliptic functions, and special cases thereof, for example, solvable in terms of hyperbolic functions. In this way, we construct exact solutions of the system of partial differential equations under consideration. We believe this to be the first analytic study of this particular mathematical model.

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ANALYTICAL STUDY ON NONLINEAR DIFFERENTIAL–DIFFERENCE EQUATIONS VIA A NEW METHOD

Modern Physics Letters B ◽

10.1142/s0217984910022846 ◽

2010 ◽

Vol 24 (08) ◽

pp. 761-773

Author(s):

HONG ZHAO

Keyword(s):

Difference Equations ◽

Elliptic Function ◽

Analytical Study ◽

Elliptic Functions ◽

Expansion Method ◽

Periodic Wave ◽

Jacobian Elliptic Function ◽

Jacobian Elliptic Functions ◽

Periodic Wave Solutions ◽

Trigonometric Function Solutions

Based on the computerized symbolic computation, a new rational expansion method using the Jacobian elliptic function was presented by means of a new general ansätz and the relations among the Jacobian elliptic functions. The results demonstrated an effective direction in terms of a uniformed construction of the new exact periodic solutions for nonlinear differential–difference equations, where two representative examples were chosen to illustrate the applications. Various periodic wave solutions, including Jacobian elliptic sine function, Jacobian elliptic cosine function and the third elliptic function solutions, were obtained. Furthermore, the solitonic solutions and trigonometric function solutions were also obtained within the limit conditions in this paper.

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