HARMONIC INTERPOLATING WAVELETS IN NEUMANN BOUNDARY VALUE PROBLEM IN A CIRCLE

: The Neumann boundary value problem (BVP) in a unit circle is discussed. For the solution of the Neumann BVP, we built a method employing series representation of given 2 π -periodic continuous boundary function by interpolating wavelets consisting of trigonometric polynomials. It is convenient to use the method due to the fact that such series is easy to extend to harmonic polynomials inside a circle. Moreover, coeﬃcients of the series have an easy-to-calculate form. The representation by the interpolating wavelets is constructed by using an interpolation projection to subspaces of a multiresolution analysis with basis 2 π -periodic scaling functions (more exactly, their binary rational compressions and shifts). That functions were developed by Subbotin and Chernykh on the basis of Meyer-type wavelets. We will use three kinds of such functions, where two out of the three generates systems, which are orthogonal and simultaneous interpolating on uniform grids of the corresponding scale and the last one generates only interpolating on the same uniform grids system. As a result, using the interpolation property of wavelets mentioned above, we obtain the exact representation of the solution for the Neumann BVP by series of that wavelets and numerical bound of the approximation of solution by partial sum of such series.


Introduction
Subbotin and Chernykh [1] constructed real 2π-periodic orthogonal wavelets and applied them to represent and analyze solutions of Dirichlet, Neumann, and Poisson boundary value problems for harmonic and biharmonic functions.In [2] the Dirichlet BVP in a unit circle was solved by means of interpolating-orthogonal periodic wavelets from [3].In the present paper, we propose to use the same wavelets for solving the Neumann BVP in a unit circle.Moreover, our main interest is the exact representation of the solution for the Neumann BVP by series of wavelet bases and behavior of partial sums of such series.For the sake of convenience, we give the reader an adequate background for further study and partially repeat sections with interpolating and interpolating-orthogonal 2πperiodic wavelet construction from [1,3].

Preliminaries
Consideration of autocorrelation functions for orthonormal scaling functions instead of orthonormal scaling functions is commonly used construction technique for interpolating wavelets in R. It is equivalent to replacement of scaling ϕ(x) function by function, which Fourier transform coincides with | ϕ(ω)| 2 .
It is easy to see, that for n ∈ Z So the sequence of spaces (1.3) has only 2 j distinct linearly independent terms.Hence, we can assume in the following discussion that k = 0, 2 j − 1.
Define system of spaces {V j s := span{Φ j,k s (x) : k = 0, 2 j − 1} : j ∈ Z}.As follows from ∆ j ε ∩ Z = {0} for j ≤ 0 and ϕ s (0) = 1, we see that i.e., for all integers such that j ≤ 0 and for all k ∈ Z relation Φ j,k s (x) = Φ j,0 s (x) = const holds and thus we can consider the system of spaces {V j s } only for j ∈ N ∪ {0}.Further, for j ∈ N ∪ {0} define spaces W j s as direct complement of V j s to V j s+1 with the interpolation system {Ψ j,k s (x) : Using interpolating condition of basis {Φ j+1,k s (x) : k=0, 2 j+1 − 1} on the grid {x l j+1 : l=0, 2 j+1 − 1} and assuming x := 2πl/2 j+1 in (1.4), we find the coefficients b n (n = 0, 2 j+1 − 1): In view of b n obtained, the sum on the right side of the expression (1.4) may be written as two sums over even and odd indices As a result, we have and it implies that In view of definitions of spaces V j s and W j s , for all j ∈ N ∪ {0} and for all k = 0, 2 j − 1 relation holds.

With (1.3), (1.7) and preceding expression the following relation holds for a function
The definition of W j s imply V j s = V 0 s ⊕ (⊕ j−1 l=0 W l s ).Then S s,2 j (x; f ) is the partial sum of order 2 j for (1.7) and from (1.6) series (1.7) converges uniformly.Thus for

Application to the solution of the Neumann BVP in a circle
Setting of the Neumann BVP in the unit circle K 1 (see, for example, [6]): where re ix (0 ≤ r < 1, 0 ≤ x < 2π) are points of the unit circle K 1 centered at the origin of the polar coordinate system.It has been well known that necessary condition of solvability of the Neumann problem is and the problem have a unique solution up to an additive constant.Define harmonic in the unit circle polynomials Φ j,k s (r, x): and consider series Since U (r, x) is a harmonic in the unit circle function with continuous boundary value U (1, x), it follows that the above series converges uniformly on the boundary of K 1 by taking into account (1.8) and (1.9) (where for f (x) we take U (1, x)).Because of maximum principle for harmonic functions, we obtain the following representation for U (r, x) in form of uniformly convergent in K 1 series Using (1.8), we have the following representation for function g 1 (x) ∈ C 2π in form of uniformly convergent in K 1 series We may extend terms of the series into the interior of the unit circle to harmonic polynomials c j,k (g 1 )Φ j,k s (r, x) and, consequently, we may extend the series into the interior of the unit circle to harmonic in K 1 and in continuous K 1 function.
(2.4) Because of series in (2.3) converges uniformly, we can perform a term-by-term differentiation with respect to r and multiplication by r and as result we get As is easy to see that this function is harmonic in K 1 .In view of setting of the Neumann BVP, we have ∂U ∂r (r, x) r=1 = g 1 (x), this implies that for 0 ≤ r < 1 the equality r ∂U ∂r (r, x) = g 1 (r, x) holds as equality of two harmonic functions which are equal at the boundary of K 1 .Hence In consequence of (2.2), we also have 2π 0 g 1 (r, x)dx = 0. (2.5) Indeed if we expand function g 1 (r, x) = r ∂U ∂r (r, x) in a series by system {r |n| e inx : n ∈ Z} (for instance, with the use of Poisson kernel), then we get for 0 ≤ r < 1 Interchanging of integration and summation and using (2.2), we arrive at resulting in (2.5).Thus, using (2.5) and taking into account ϕ s (0) = 1, we obtain and numerical series on the left side of the equality converges.Consequently, the following equality holds Therefore, by setting and partial sum S s,2 J (r, x; U, Ψ −1 ) in the form Note that the following representations hold (2.7) It follows from where N + ε,j = ⌈2 j (1 + ε)⌉ and the second equality holds in view of Let S s,2 J (r, x; g 1 ) be a partial sum of series in (2.where the first equality follows from (2.7), the second equality follows from Parseval's identity, the second inequality follows from Hölder's inequality and the last inequality follows from Theorem in [2].As the final result we have U (r, x) − S s,2 J (r, x; U, Ψ −1 ) C(K 1 ) ≤ π √ 3 1 + S s,2 J E N − ε,J (g 1 ) C 2π .

Conclusion
Theorem 1 gives the solution (2.6) (up to an additive constant) of the problem (2.1) in form of uniformly convergent in K 1 series of harmonic interpolating 2π-periodic wavelets.In this case, coefficients of series in (2.6) have an easy-to-calculate form in preference to calculating coefficients (integrals) in case of implementing orthogonal 2π-periodic wavelets.This useful fact simplify the numerical implementation of the suggested method.