Positive-definiteness, Integral Equations and Fourier Transforms
نویسنده
چکیده
We show that positive definite kernel functions k(x, y), if continuous and integrable along the main diagonal, coincide with kernels of positive integral operators in L2(R). Such an operator is shown to be compact; under the further assumption k(x, x) → 0 as |x| → ∞ it is also trace class and the corresponding bilinear series converges absolutely and uniformly. If k1/2(x, x) ∈ L1(R), all these results are carried through to a ‘rotated’ Fourier transform: k̂(ν1,−ν2) is the kernel of a compact positive operator and is represented by the absolutely and uniformly convergent series of Fourier transforms of eigenfunctions. The trace of the operator is an invariant under Fourier transforms.
منابع مشابه
Numerical Solution of Volterra-Fredholm Integral Equations with The Help of Inverse and Direct Discrete Fuzzy Transforms and Collocation Technique
متن کامل
Applications of Generalized Convolutions Associated with the Fourier and Hartley Transforms
In this paper we present new generalized convolutions with weight-function associated with the Fourier and Hartley transforms, and consider applications. Namely, using the generalized convolutions, we construct normed rings on the space L(R), provide the sufficient and necessary condition for the solvability of a class of integral equations of convolution type, and receive the explicit solution...
متن کاملOn Some Quantum and Analytical Properties of Fractional Fourier Transforms
Fractional Fourier transforms (FrFT) are a natural one-parameter family of unitary transforms that have the ordinary Fourier transform embedded as a special case. In this paper, following the efforts of several authors, we explore the theory and applications of FrFT, from the standpoints of both quantum mechanics and analysis. These include the phase plane interpretation of FrFT, FrFT’s role in...
متن کاملInfluences of magnetic field in viscoelastic fluid
This communication influences on magnetohydrodynamic flow of viscoelastic fluid with magnetic field induced by oscillating plate. General solutions have been found out for velocity and shear stress profiles using mathematical transformations (Integral transforms). The governing partial differential equations have been solved analytically under boundary conditions u(0,t)=A_0 H(t)sinΩt and u(0,t)...
متن کاملSolving a class of nonlinear two-dimensional Volterra integral equations by using two-dimensional triangular orthogonal functions
In this paper, the two-dimensional triangular orthogonal functions (2D-TFs) are applied for solving a class of nonlinear two-dimensional Volterra integral equations. 2D-TFs method transforms these integral equations into a system of linear algebraic equations. The high accuracy of this method is verified through a numerical example and comparison of the results with the other numerical methods.
متن کامل