Classification of solutions of a critical Hardy Sobolev by Mancini G., Fabbri I., Sandeep K.

By Mancini G., Fabbri I., Sandeep K.

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The mean power P of the current i(t ) over one period can be found by the simple formula P = i(t ) = 1 2π +π ∫ u (t )i(t )dt . The Ohm law −π u (t ) = u (t ) is valid, since the circuit is active. Then r P= 1 2π +π ∫π [u(t )] 2 dt = u (t ) 2 L2 [ − π ,π ] . − If the voltage u (t ) is represented by a Fourier series with coefficients ck , then the Parseval formula gives P = ∑ ( ck ) . 2 k ∈ℤ Thus, the mean power of current over one period is expressed through coefficients of the Fourier series for the current.

But these atoms are not continuous functions. It could be more convenient to do this by using continuous atoms. This was done when the continuous wavelets were developed. Fact H2. Suppose that the function is not continuous over the interval [0;1] and belongs to C [ 0;1] , that is, it is continuous and has the continuous first derivative. Then the approximation by step-wise functions is still more inadequate. Despite these critical facts, the Haar system is now adapted just to describe continuous and square integrable functions over [ 0;1] or, more 39 Wavelet Analysis abstractedly, to functions with regularity index close to zero.

This is fixed by the classical theorem. 9 If the function f ( x) : 1. is complex-valued and defined over the real axis; 2. satisfies the Dirichlet conditions over each finite interval [ −l , l ] , 3. is absolutely integrable, then its Fourier integral converges to f ( x) at each point of continuity and to (1 2 ) [ f ( x + 0) + f ( x − 0) ] at the points of discontinuity. It must be noted, however that, the notion of convergence must be defined anew. The integral F (ω ) = 1 2π +∞ ∫ f (ξ )e − iωξ d ξ is called the convergent one in −∞ 2 the space L ( ℝ ) , if there exists a function g (ω ) ∈ L2 ( ℝ ) , such that lim ∫ N →∞ R 1 g (ω ) − 2π 2 +N ∫ f (ξ )e − iωξ dξ dω = 0 .

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