Fractional functions and their representations. (English) Zbl 1285.46024
Summary: For arbitrary non-identically zero functions \(f\), we introduce some natural fractional functions \(f{_1}\) having \(f\) as denominators and we consider their representations \(f{_1}\) by appropriate numerator functions within a reproducing kernel Hilbert spaces framework. That is, in the present work we would like to introduce very general fractional functions (e.g., having the possibility of admitting zeros in their denominators) by means of the theory of reproducing kernels.
MSC:
| 46E22 | Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) |
| 42A38 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
| 47B32 | Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces) |
| 30C40 | Kernel functions in one complex variable and applications |
Keywords:
fractional function; best approximation; Moore-Penrose generalized inverse; Hilbert space; linear transform; reproducing kernel; linear mapping; convolution; norm inequality; Tikhonov regularizationReferences:
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