Abstract
The numerical algorithms for finding the lines of branching and branching-off solutions of nonlinear problem on mean-square approximation of a real finite nonnegative function with respect to two variables by the modulus of double discrete Fourier transform dependent on two parameters, are constructed and justified.
Highlights
The mean-square approximation of real finite nonnegative function with respect to two variables by the modulus of double discrete Fourier transform dependent on physical parameters, is widely used, in particular, at modeling and solution of the synthesis problems of different types of antenna arrays, signal processing etc. [1,2,3]
The methods of investigation and numerical finding the solutions of one-parametric spectral problems at presence of discrete spectrum [4,5,6,7,8] are most well-developed
It justifies the application of implicit functions methods to multiparametric spectral problems [9]
Summary
The mean-square approximation of real finite nonnegative function with respect to two variables by the modulus of double discrete Fourier transform dependent on physical parameters, is widely used, in particular, at modeling and solution of the synthesis problems of different types of antenna arrays, signal processing etc. [1,2,3]. Operator D is positive on nonnegative functions cone K of the space C(Ω) [14] According to it D leaves ( ) ing this equality as A I∗ − A∗ (F exp(i arg(AI∗ ))) = 0 invariant the cone K , i.e. DK ⊂ K. From the Theorem 1 follows satisfaction of conditions of the Schauder principle [16] according to which the operator B = (B1, B2 )T has a fixed point f∗ = (u∗, v∗ )T belonging to the set SR. This point is a solution of a system of Equation (14) and Equation (10), respectively. Substituting (20) and (22) into (14) and taking ( ) into account that f0 Q′,c(0) solves the system (14) we obtain a system of nonlinear equations with respect to small solutions w , ω :
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