## Final Report Summary - SPEVOTAP (Special voice transforms and applications)

The project was progressing as outlined in the proposal. I focused my research on theoretical and applied problems.

Spherical functions have many applications in physics. During the project, a weighted minimum problem on the sphere was formulated and it was shown that the spherical functions can be localised on the sphere so that the solution of this problem is the simplest possible. This localisation is connected to the discrete orthogonality of the spherical functions which was proved by M. Pap and F. Schipp. Using these points, a tight frame and a wavelet system on the sphere were constructed and the properties of these systems were studied.

The voice transform generated by a representation of the Blaschke group on the Hardy space of the unit disc can be related to the Zernike functions frequently used in the optical tests. This connection with the Blaschke group permitted to prove the addition formula regarding to the Zernike functions. Zernike series expansion is an important tool in expressing the wavefront data in optical tests. The aberrations of the human eyes are characterised by Zernike coefficients. Based on the discrete orthogonality of the Zernike functions (proved earlier by M.Pap and F. Schipp), a reconstruction algorithm of some test corneal like surfaces was proposed (joint with F. Schipp, A. Soumelidie, Z. Fazakas). Computer implementations of experimental results on artificial corneal-like surfaces were made. Also comparison of precision with the measurement methods used by conventional topographers proved that the approximations of the coefficients using the discrete orthogonality are the best and the reconstruction of the corneal like surface has minimal error.

Another important topic was a description of a new multiresolution analysis in the Hardy space over the unit disc. The construction is an analogy with the discrete affine wavelets and in fact is the discretisation of the continuous voice transform generated by a representation of the Blaschke group. The constructed discretisation scheme gives an opportunity of practical realisation of the hyperbolic wavelet representation of signals with transfer functions in the Hardy space over the unit disc. Furthermore, convergence properties of the hyperbolic wavelet representation were studied.

It was studied the properties of the voice transform generated by a representation of the Blaschke group on the weighted Bergman spaces outlined by the general theory developed by Feichtinger and Groechenig. It was necessary to adapt this general theory for the Blaschke group, where the Q-density from the right has an easier geometric interpretation. Applying the adapted theory it could be proved that every function from the minimal Mobius invariant space of the unit disc will generate an atomic decomposition in the weighted Bergman spaces. This result is more general than the usual atomic decompositions obtained by complex methods, which is just a particular case of this result. Another benefit from the new atomic decomposition is the fact that the coefficients of the atomic decomposition are determined exactly in terms of the voice transform. In the earlier results, just the existence of the coefficients was proved.

Non periodic analytic signals are boundary functions of elements of the Hardy space for the upper half plane. The construction of analytic wavelets for the analysis of these signals is an old problem formulated by Y. Meyer. Joint with H. G. Feichtinger we could give a multiresolution analysis for the Hardy space of the upper half plane. The resolution levels can be described by the Malmquist Takenaka system for the upper half plane with a special localisation of the poles, which is a new system of rational analytic wavelets for the upper half plane. In addition, a new reconstruction algorithm was proposed. Convergence properties of the hyperbolic wavelet representation were also studied.

In the representation of non periodic signals, the use of the Malmquist-Takenaka system for the upper half plane is more efficient. From the point of view of the computer implementations of the multiresolution approximation described before, the discrete orthogonality of this system occurred as a new question. The discrete orthogonality of the Malmquist-Takenaka system for the upper half plane was proved. Based on the discretisation, it was introduced a new rational interpolation operator and it was studied the properties of this operator. A finite sampling theorem for a special subset of non periodic analytic signals was presented.

A multiresolution construction was proposed in the Bergman space of the unit disc in the first year of the project. In this period, it was finalised the exposition of the results and now is ready for submission.

It was introduced a new example of sampling set for the Bergman space which is a discrete subset of the Blaschke group, correspondingly a new frame. Using this set, a multiresolution analysis in the Bergman space can be generated. The different resolution spaces are defined first by a nonorthogonal basis. It was proved that the levels of the multiresolutions can be spanned by rational orthogonal basis. This system is the analogue of the Mamquist-Takenaka system, possesses similar properties and is connected to the contractive zero divisors of a finite set in the Bergman space of the unit disc.

Two papers are under preparation:

A survey paper for the conference volume of HCAA2012 joint with H. G. Feichtinger, K. H. Groechenig, L. D. Abrau, M. Pap, Applications of the coorbit space theory and atomic decompositions.

H. G. Feichtinger and K. H. Grochenig described a unified approach to atomic decomposition through integrable group representations in Banach spaces.

They gave also examples for the abstract theory. Even in those situations where atomic decompositions are known to exists the general approach can provide new insights. Due to the flexibility of this theory, the class of possible atoms is much larger than it was supposed to be in concrete cases.

It is considered a localisation operator defined by tight Gabor frames. An explicit formula for the boundary form is proved. The boundary form expresses quantitatively the asymptotic interaction between the generating function of the Gabor tight frame and the oriented boundary of the lattice localisation domain from the point of of the projection functional, which measures to what degree a given trace class operator fails to be orthogonal projection. It provides an evaluation framework for finding best asymptotic matching between generating functions and lattice domains.

I acknowledge this Marie Curie fellowship which created ideal conditions for my research. During this period, I could buy new laptop, new monographs related to my research. The Marie Curie project offered an ideal support to work in stimulating environment, surrounded by an outstanding group, and gave the material support to participate on international conferences and to meet the most prestigious sciences of my research field.

I am thankful to Prof. Dr Hans Georg Feichtinger head of NuHAG for stimulating me and for giving me inspiration in my work.

Spherical functions have many applications in physics. During the project, a weighted minimum problem on the sphere was formulated and it was shown that the spherical functions can be localised on the sphere so that the solution of this problem is the simplest possible. This localisation is connected to the discrete orthogonality of the spherical functions which was proved by M. Pap and F. Schipp. Using these points, a tight frame and a wavelet system on the sphere were constructed and the properties of these systems were studied.

The voice transform generated by a representation of the Blaschke group on the Hardy space of the unit disc can be related to the Zernike functions frequently used in the optical tests. This connection with the Blaschke group permitted to prove the addition formula regarding to the Zernike functions. Zernike series expansion is an important tool in expressing the wavefront data in optical tests. The aberrations of the human eyes are characterised by Zernike coefficients. Based on the discrete orthogonality of the Zernike functions (proved earlier by M.Pap and F. Schipp), a reconstruction algorithm of some test corneal like surfaces was proposed (joint with F. Schipp, A. Soumelidie, Z. Fazakas). Computer implementations of experimental results on artificial corneal-like surfaces were made. Also comparison of precision with the measurement methods used by conventional topographers proved that the approximations of the coefficients using the discrete orthogonality are the best and the reconstruction of the corneal like surface has minimal error.

Another important topic was a description of a new multiresolution analysis in the Hardy space over the unit disc. The construction is an analogy with the discrete affine wavelets and in fact is the discretisation of the continuous voice transform generated by a representation of the Blaschke group. The constructed discretisation scheme gives an opportunity of practical realisation of the hyperbolic wavelet representation of signals with transfer functions in the Hardy space over the unit disc. Furthermore, convergence properties of the hyperbolic wavelet representation were studied.

It was studied the properties of the voice transform generated by a representation of the Blaschke group on the weighted Bergman spaces outlined by the general theory developed by Feichtinger and Groechenig. It was necessary to adapt this general theory for the Blaschke group, where the Q-density from the right has an easier geometric interpretation. Applying the adapted theory it could be proved that every function from the minimal Mobius invariant space of the unit disc will generate an atomic decomposition in the weighted Bergman spaces. This result is more general than the usual atomic decompositions obtained by complex methods, which is just a particular case of this result. Another benefit from the new atomic decomposition is the fact that the coefficients of the atomic decomposition are determined exactly in terms of the voice transform. In the earlier results, just the existence of the coefficients was proved.

Non periodic analytic signals are boundary functions of elements of the Hardy space for the upper half plane. The construction of analytic wavelets for the analysis of these signals is an old problem formulated by Y. Meyer. Joint with H. G. Feichtinger we could give a multiresolution analysis for the Hardy space of the upper half plane. The resolution levels can be described by the Malmquist Takenaka system for the upper half plane with a special localisation of the poles, which is a new system of rational analytic wavelets for the upper half plane. In addition, a new reconstruction algorithm was proposed. Convergence properties of the hyperbolic wavelet representation were also studied.

In the representation of non periodic signals, the use of the Malmquist-Takenaka system for the upper half plane is more efficient. From the point of view of the computer implementations of the multiresolution approximation described before, the discrete orthogonality of this system occurred as a new question. The discrete orthogonality of the Malmquist-Takenaka system for the upper half plane was proved. Based on the discretisation, it was introduced a new rational interpolation operator and it was studied the properties of this operator. A finite sampling theorem for a special subset of non periodic analytic signals was presented.

A multiresolution construction was proposed in the Bergman space of the unit disc in the first year of the project. In this period, it was finalised the exposition of the results and now is ready for submission.

It was introduced a new example of sampling set for the Bergman space which is a discrete subset of the Blaschke group, correspondingly a new frame. Using this set, a multiresolution analysis in the Bergman space can be generated. The different resolution spaces are defined first by a nonorthogonal basis. It was proved that the levels of the multiresolutions can be spanned by rational orthogonal basis. This system is the analogue of the Mamquist-Takenaka system, possesses similar properties and is connected to the contractive zero divisors of a finite set in the Bergman space of the unit disc.

Two papers are under preparation:

A survey paper for the conference volume of HCAA2012 joint with H. G. Feichtinger, K. H. Groechenig, L. D. Abrau, M. Pap, Applications of the coorbit space theory and atomic decompositions.

H. G. Feichtinger and K. H. Grochenig described a unified approach to atomic decomposition through integrable group representations in Banach spaces.

They gave also examples for the abstract theory. Even in those situations where atomic decompositions are known to exists the general approach can provide new insights. Due to the flexibility of this theory, the class of possible atoms is much larger than it was supposed to be in concrete cases.

It is considered a localisation operator defined by tight Gabor frames. An explicit formula for the boundary form is proved. The boundary form expresses quantitatively the asymptotic interaction between the generating function of the Gabor tight frame and the oriented boundary of the lattice localisation domain from the point of of the projection functional, which measures to what degree a given trace class operator fails to be orthogonal projection. It provides an evaluation framework for finding best asymptotic matching between generating functions and lattice domains.

I acknowledge this Marie Curie fellowship which created ideal conditions for my research. During this period, I could buy new laptop, new monographs related to my research. The Marie Curie project offered an ideal support to work in stimulating environment, surrounded by an outstanding group, and gave the material support to participate on international conferences and to meet the most prestigious sciences of my research field.

I am thankful to Prof. Dr Hans Georg Feichtinger head of NuHAG for stimulating me and for giving me inspiration in my work.