Schrödinger Equation and Oscillatory Hilbert Transforms of Second Degree

Journal of Fourier Analysis and Applications

Volumn 4, Issue 3, 1998, Pages 341-356

  Oskolkov, K ^a  

^a UNIVERSITY OF SOUTH CAROLINA   (United States)

Author keywords

Cauchy initial value problem; Continued fraction; Curlicue; Fresnel integral; Gauss' sum; Hubert transform; Schr dinger equation; Selfsimilarity

Indexed keywords

EID: 0032330384     PISSN: 10695869     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF02476032     Document Type: Article

References (28)

1. Arkhipov, G.I. (1984). On the Hubert - Kamke problem, Izv. Akad. Nauk SSSR, Ser. Mat., 48, 3-52; English transl. in: (1985) Math. USSR Izv., 24.
2. Arkhipov, G.I. (1984). On the Hubert - Kamke problem, Izv. Akad. Nauk SSSR, Ser. Mat., 48, 3-52; English transl. in: (1985) Math. USSR Izv., 24.
3. Arkhipov, G.I. and Oskolkov, K.I. (1989). On a special trigonometric series and its applications. Math. USSR Sbornik, 62 (1), 145-155.
4. Bary, N.K. (1964). A Treatise on Trigonometric Series. I. Pergamon Press, New York.
5. Berry, M.V. and Goldberg, J. (1989). Renormalization of curlicues. Nonlinearity, 1, 1-26.
6. Carleson, L. (1966). On convergence and growth of partial sums of Fourier series. Acta Math., 116, 133-157.
7. Cherbit, G. Ed. (1991). Fractals. Non-Integral Dimensions and Applications. John Wiley & Sons, New York.
8. DeVore, R.A. and Lorentz, G.G. (1993). Constructive Approximation. Springer Verlag.
9. Falconer, K.J. (1985). The Geometry of Fractal Sets. Cambridge University Press.
10. Fiedler, H., Jurkat, W., and Koerner, O. (1977). Asymptotic expansions of finite theta series. Acta Arithmetica, 32, 129-146.
11. Hardy, G.H. (1966). Collected Papers of G.H. Hardy, 1, Clarendon Press, Oxford.
12. Kolmogorov, A.N. and Tikhomirov, V.M. (1959). Epsilon entropy and epsilon capacity in functional spaces. Uspekhi Mat. Nauk, 14(2), 1-86; English transl, in: (1961) Am. Math. Soc. Transl., Series 2, 17, 277-364.
13. Kolmogorov, A.N. and Tikhomirov, V.M. (1959). Epsilon entropy and epsilon capacity in functional spaces. Uspekhi Mat. Nauk, 14(2), 1-86; English transl, in: (1961) Am. Math. Soc. Transl., Series 2, 17, 277-364.
14. Karacuba, A.A. and Voronin, S.M. (1994). Riemann's zeta-function. Fizmatlit, Moscow (in Russian).
15. [131 Lehmer, D.H. (1976). Incomplete Gauss sums. Mathematica, 23, 125-135.
16. 2 integrable functions. IV. Acta Math. Acad. Sci. Hungary, 20, 383-391.
17. McCualey, J.L. (1993). Chaos, Dynamics and Fractals. An Algorithmic Approach to Deterministic Chaos, Cambridge University Press.
18. Moore, R.P. and van der Poorten, A.J. (1989). On the thermodynamics of curves and other curlicues. Proc. Center for Math. Analysis, 22. Australian National University, 82-108.
19. Oskolkov, K. (1991). A class of I.M. Vinogradov's series and its applications in Harmonic analysis. Progress in Approximation Theory (An International Prospective). Springer Series in Computational Mathematics 19, 353-402, Editors: Gonchar, A.A. and Saff, E.B. Eds., Springer Verlag.
20. Oskolkov, K. (1991). Vinogradov series in the Cauchy problem for equations of Schrödinger type. Proc. Steklov Inst. Math., 200, 265-288; English transl, in: (1993) Proc. Steklov Inst. Math., 2, 291-315.
21. Oskolkov, K. (1991). Vinogradov series in the Cauchy problem for equations of Schrödinger type. Proc. Steklov Inst. Math., 200, 265-288; English transl, in: (1993) Proc. Steklov Inst. Math., 2, 291-315.
22. Oskolkov, K. (1989). Series and integrals of I.M. Vinogradov and their applications. Proc. Steklov Inst. Math., 190, 186-221; English transl. in: (1992) Proc. Steklov Inst. Math., 1, 193-229.
23. Oskolkov, K. (1989). Series and integrals of I.M. Vinogradov and their applications. Proc. Steklov Inst. Math., 190, 186-221; English transl. in: (1992) Proc. Steklov Inst. Math., 1, 193-229.
24. Oskolkov, K. (1992). On functional properties of incomplete Gaussian sums. Can. J. Math., 43, 182-212.
25. Pontrjagin, L. and Schnirelman, I. (1938). Sur une propriété métrique de la dimension. Ann. Math., 33, 156-162.
26. Stein, E. and Wainger, S. (1990). Discrete analogues of singular Radon transforms. Bull. Amer. Math. Sac., 23, 537-544.
27. Vinogradov, I.M. (1980). The Method of Trigonometric Sums in Number Theory, 2nd ed., Nauka, Moscow; English transi, in: (1985) Selected Works, Springer Verlag.
28. Vinogradov, I.M. (1980). The Method of Trigonometric Sums in Number Theory, 2nd ed., Nauka, Moscow; English transi, in: (1985) Selected Works, Springer Verlag.