Lectures on Integral Transforms

Akhiezer, N. I.

doi:10.1090/mmono/070

Cited by 55 publications

(56 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…, 2n − 1 such that µ s (n, ψ) > 0. The latter equivalence is a consequence of the following properties: (1) the functions of this family with numbers s ∈ Z and s + 2nm, m ∈ Z, are identical; (2) …”

Section: Sharp Integral Inequalities For Periodic Functionsmentioning

confidence: 99%

“…84, p. 189]). If, in addition, the function tJ ′ (t) is strictly increasing on (0, +∞), then, in inequalities (6.1) and (6.2) for 1 ≤ p < ∞, n ∈ N, r ≥ 1, and β ∈ R, there are no other extremal polynomials at least in the following cases (see [3,Corollary 6], [5,Theorems 1,2]): (1) in the case of the usual derivative of order r ∈ N; (2) n = 1, r ≥ 1, and β ∈ R or n ≥ 2, r ≥ ln(2n)/ ln(n/(n − 1)), and β ∈ R.…”

Section: Generalization Of Bernstein-szegő Inequalitiesmentioning

confidence: 99%

“…The proof of this theorem can be found, for instance, in [2,7,19,23,24]. As a direct consequence, we obtain the following criterion of positive definiteness in terms of nonnegativity of the Fourier transform: if f ∈ C(R) ∩ L 1 (R), then f ∈ Φ(R) ⇐⇒ f (t) ≥ 0, t ∈ R, where f (t) := R e −itx f (x)dx, t ∈ R.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Positive Definite Functions and Sharp Inequalities for Periodic Functions

Zastavnyi

2017

UMJ

View full text Add to dashboard Cite

Let ϕ be a positive definite and continuous function on R, and let µ be the corresponding Bochner measure. For fixed ε, τ ∈ R, ε = 0, we consider a linear operator Aε,τ generated by the function ϕ:Let J be a convex and nondecreasing function on [0, +∞). In this paper, we prove the inequalitiesand f ∈ C(T) and obtain criteria of extremal function. We study in more detail the case in which ε = 1/n, n ∈ N, τ = 1, and ϕ(x) ≡ e iβx ψ(x), where β ∈ R and the function ψ is 2-periodic and positive definite. In turn, we consider in more detail the case where the 2-periodic function ψ is constructed by means of a finite positive definite function g. As a particular case, we obtain the Bernstein-Szegő inequality for the derivative in the Weyl-Nagy sense of trigonometric polynomials. In one of our results, we consider the case of the family of functions g 1/n,h (x) := hg(x) + (1 − 1/n − h)g(nx), where n ∈ N, n ≥ 2, −1/n ≤ h ≤ 1 − 1/n, and the function g ∈ C(R) is even, nonnegative, decreasing, and convex on (0, +∞) with supp g ⊂ [−1, 1]. This case is related to the positive definiteness of piecewise linear functions. We also obtain some general interpolation formulas for periodic functions and trigonometric polynomials which include the known interpolation formulas of M. Riesz, of G. Szegő, and of A.I. Kozko for trigonometric polynomials.

show abstract

Section: Sharp Integral Inequalities For Periodic Functionsmentioning

confidence: 99%

Section: Generalization Of Bernstein-szegő Inequalitiesmentioning

confidence: 99%

See 1 more Smart Citation

Positive Definite Functions and Sharp Inequalities for Periodic Functions

Zastavnyi

2017

UMJ

View full text Add to dashboard Cite

show abstract

“…To satisfy the radiation conditions, we choose γ (λ) from the conditions Re γ (λ) ≥ 0 and Im γ (λ) ≤ 0 . As shown in [1,[4][5][6][7][8], the boundary-value problem (1) (with regard for all imposed conditions) is reduced to the following two dual integral equations:…”

Section: Statement Of the Problem And Dual Integral Equationsmentioning

confidence: 99%

Discrete Mathematical Model of the Problem of Diffraction for E -Polarized Waves on Slots in the Impedance Plane

Nesvit

2015

J Math Sci

View full text Add to dashboard Cite

We consider the problem of diffraction for E -polarized plane monochromatic waves on slots in the impedance plane. A discrete mathematical model of the boundary integral equations of this problem is constructed. We also perform a numerical experiment based on the use of the efficient numerical method of discrete singularities.A stationary electromagnetic field (its time dependence is specified by the factor e −iωt ) is described by the Maxwell equations. In the two-dimensional case, these equations lead to two independent boundary-value problems for the steady-state wave equation. The two-dimensional diffraction problems for monochromatic electromagnetic waves on perfectly conducting structures lead to the Dirichlet and Neumann two-dimensional external boundary-value problems for the Helmholtz equation.In the present paper, we construct and perform the numerical analysis of a discrete mathematical model of the problem of diffraction on slots in the impedance plane. This model enables one to find the required characteristics of the electromagnetic fields and, in particular, to construct directional diagrams.The problems of scattering and diffraction of electromagnetic waves by impedance metallic and superconducting thin tapes were investigated in the monograph [5] and paper [4] by the method of discrete singularities. The same problems for the pre-Cantor tapes were studied in [12].The mathematical models of diffraction problems based on hypersingular integral equations [10] were also numerically investigated by the method of discrete singularities. Statement of the Problem and Dual Integral EquationsWe consider a problem of the mathematical theory of diffraction of E -polarized plane monochromatic electromagnetic waves on a finite set of slots in the impedance plane.The surface impedance of the metal is an important physical characteristic specifying the amplitude and phase relationships between the electric and magnetic fields on the surface.The model studied in the present work can be regarded as an approximation of actual fractal [11] antennas for the case where the field depends on two Cartesian coordinates. Fractal antennas are used in various contemporary mobile devices because their compactness and broadband properties made them irreplaceable for the purposes of wireless communications in the Bluetooth, Wi-Fi, and GSM standards. Their theoretical investigation is important from the viewpoint both of the development of mathematical methods capable of the solution of complex boundary-value problems of electrodynamics and of the construction of new physical models as close to the reality as possible.

show abstract

“…It is well known that H is a bounded map in the L p spaces, 1 < p < ∞, and that it is usually considered in the L 2 spaces where π −1 H is an isometric isomorphism [1]. In the space of the continuous functions equipped with the uniform metric, the Hilbert Transform is an unbounded operator.…”

Section: G(x) X − T Dx = Limmentioning

confidence: 99%