On Fourier transforms
HTML articles powered by AMS MathViewer
- by C. Nasim PDF
- Trans. Amer. Math. Soc. 191 (1974), 45-51 Request permission
Abstract:
If $f(x)$ and $g(x)$ satisfy the equations \[ g(x) = \frac {d}{{dx}}\int _0^\infty \frac {1}{t}f(t){k_1}(xt)dt,\quad f(x) = \frac {d}{{dx}}\int _0^\infty \frac {1}{t}g(t){k_1}(xt)dt,\] then we call f and g a pair of ${k_1}$-transforms, where \[ k_1 = \frac {1}{2\pi i} \int _{1/2 - i\infty }^{1/2 + i\infty } \frac {K(s)}{1 - s} x^{1-s} ds. \] In this paper alternative sets of conditions are established for f and g to be ${k_1}$-transform provided $K(s)$ is decomposable in a special way. These conditions involve simpler functions, which replace the kernel ${k_1}(x)$. Results are proved for the function spaces ${L^2}$. The necessary and sufficient conditions are established for the two functions to be self-reciprocal. Conditions are given for generating pairs of transforms for a given kernel. Two examples are given at the end to illustrate the methods and the advantage of the results.References
-
A. Erdélyi et al., Tables of integral transforms. Vol. 1, McGraw-Hill, New York, 1954. MR 15, 868.
—, Tables of integral transforms. Vol. 2, McGraw-Hill, New York, 1954. MR 16, 468.
- Charles Fox, Chain transforms, Proc. Amer. Math. Soc. 5 (1954), 677–688. MR 63478, DOI 10.1090/S0002-9939-1954-0063478-3
- A. P. Guinand, Reciprocal convergence classes for Fourier series and integrals, Canadian J. Math. 13 (1961), 19–36. MR 123143, DOI 10.4153/CJM-1961-002-3
- C. Nasim, On the summation formula of Voronoi, Trans. Amer. Math. Soc. 163 (1972), 35–45. MR 284410, DOI 10.1090/S0002-9947-1972-0284410-9
- T. L. Pearson, Note on the Hardy-Landau summation formula, Canad. Math. Bull. 8 (1965), 717–720. MR 194406, DOI 10.4153/CMB-1965-053-8
- O. P. Sharma, The $H$-functions as kernels in chain transforms, Proc. Nat. Inst. Sci. India Part A 34 (1968), 320–325. MR 249965 E. C. Titchmarsh, Introduction to the theory of Fourier Integrals, Clarendon Press, Oxford, 1937.
- David Vernon Widder, The Laplace Transform, Princeton Mathematical Series, vol. 6, Princeton University Press, Princeton, N. J., 1941. MR 0005923
Additional Information
- © Copyright 1974 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 191 (1974), 45-51
- MSC: Primary 44A05
- DOI: https://doi.org/10.1090/S0002-9947-1974-0342964-X
- MathSciNet review: 0342964