Abstract
In this paper, we show that arbitrary Hermite function or appropriate linear combination of those functions is a weight-function of four explicit generalized convolutions for the Fourier cosine and sine transforms. With respect to applications, normed rings on \({L^1(\mathbb{R}^d)}\) are constructed, and sufficient and necessary conditions for the solvability and explicit solutions in \({L^1(\mathbb{R}^d)}\) of the integral equations of convolution type are provided by using the constructed convolutions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Böttcher, A., Gohberg, I.L., Karlovich, Y., Krupnik, Y.N., Roch, S., Silbermann, B., Spitkovski, I.: Banach algebras generated by N idempotents and applications, singular integral equations and related topics. In: Operator Theory: Advances and Applications, vol. 90, pp. 19-54. Birkhäuser Verlag (1996)
Bracewell, R.N.: The Hartley Transform. Oxford University Press, Oxford (1986)
Bracewell R.N.: Aspects of the Hartley transform. Proc. IEEE 82, 381–387 (1994)
Britvina L.E.: A class of integral transforms related to the Fourier cosine convolution. Integral Transforms Spec. Funct. 16, 379–389 (2005)
Brown, J.W., Churchill, R.V.: Fourier Series and Boundary Value Problems. McGraw-Hill, N. Y. (2006)
Cho P.S., Kuterdem H.G., Marks R.J. II: A spherical dose model for radio surgery plan optimization. Phys. Med. Bio. 43, 3145–3148 (1998)
Feldman I., Gohberg I., Krupnik N.: Convolution equations on finite intervals and factorization of matrix functions. Integral Equ. Oper. Theory 36, 201–211 (2000)
Garcia-Vicente F., Delgado J.M., Peraza C.: Experimental determination of the convolution kernel for the study of the spatial response of a detector. Med. Phys. 25, 202–207 (1998)
Garcia-Vicente F., Delgado J.M., Rodriguez C.: Exact analytical solution of the convolution integral equation for a general profile fitting function and Gaussian detector kernel. Phys. Med. Bio. 45, 645–650 (2000)
Giang, B.T., Mau, N.V., Tuan, N.M.: Convolutions for the Fourier transforms with geometric variables and applications. Math. Nachr. (accepted)
Giang B.T., Mau N.V., Tuan N.M.: Operational properties of two integral transforms of Fourier type and their convolutions. Integral Equ. Oper. Theory 65, 363–386 (2009)
Giang, B.T., Tuan, N.M.: Generalized convolutions and the integral equations of the convolution type. Complex Var. Elliptic Equ. (2009) (99999:1, doi:10.1080/17476930902998886)
Giang B.T., Tuan N.M.: Generalized convolutions for the integral transforms of Fourier type and applications. Fract. Calc. Appl. Anal. 12, 253–268 (2009)
Glaeske H.J., Tuan V.K.: Mapping properties and composition structure of multidimensional integral transform. Math. Nachr. 152, 179–190 (1991)
Hochstadt, H.: Integral equations. Wiley, N. Y. (1973)
Kakichev, V.A.: On the convolution for integral transforms. Izv. ANBSSR Ser. Fiz. Mat. 2, 48–57 (1967) (in Russian)
Kakichev V.A., Thao N.X., Tuan V.K.: On the generalized convolutions for Fourier cosine and sine transforms. East-West J. Math. 1, 85–90 (1998)
Kirillov, A.A.: Elements of the theory of representations. Nauka, Moscow, 1972 (in Russian)
Naimark, M.A.: Normed Rings. P. Noordhoff Ltd., Groningen, Netherlands (1959)
Nha N.D.V., Duc D.T., Tuan V.K.: Weighted l p -norm inequalities for various convolution type transformations and their applications. Armen. J. Math. 1, 1–18 (2008)
Olejniczak, K.J.: The Hartley transform. In: Poularikas, A.D. (eds.) The Transforms and Applications Handbook, The Electrical Engineering Handbook Series, part 4, 2nd edn, pp. 341–401. CRC Press with IEEE Press, Florida (2000)
Rösler M.: Generalized Hermite polynomials and the heat equations for Dunkl operator. Comm. Math. Phys. 192, 519–542 (1998)
Saitoh S.: The Weierstrass transform and an isometry in the heat equation. Appl. Anal. 16, 1–6 (1983)
Saitoh S.: One approach to some general integral transforms and its applications. Integral Transform. Spec. Funct. 3, 49–84 (1995)
Sawano Y., Fujiwara H., Saitoh S.: Real inversion formulas of the Laplace transform on weighted function spaces. Complex Anal. Oper. Theory 2, 511–521 (2008)
Stein, E.M., Shakarchi, R.: Princeton Lectures in Analysis: Fourier Analysis, An Introduction. Princeton University Press, Princeton and Oxford (2007)
Thao N.X., Khoa N.M.: On the generalized convolution with a weight function for the Fourier sine and cosine transforms. Integral Transforms Spec. Funct. 17, 673–685 (2006)
Thao N.X., Tuan V.K., Hong N.T.: Generalized convolution transforms and Toeplitz plus Hankel integral equations. Frac. Calc. App. Anal. 11, 153–174 (2008)
Titchmarsh, E.C.: Introduction to the Theory of Fourier Integrals. Chelsea, New York (1986)
Tomovski Z., Tuan V.K.: On Fourier transforms and summation formulas of generalized Mathieu series. Math. Sci. Res. J. 13, 1–10 (2009)
Tuan N.M., Tuan P.D.: Generalized convolutions relative to the Hartley transforms with applications. Sci. Math. Jpn. 70, 77–89 (2009)
Yakubovich, S.B., Luchko, Y.: The Hypergeometric Approach to Integral Transforms and Convolutions. Mathematics and its Applications, vol. 287. Kluwer Academic Publishers, Dordrecht/Boston/London (1994)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Saburou Saitoh.
Rights and permissions
About this article
Cite this article
Tuan, N.M., Huyen, N.T.T. The Hermite Functions Related to Infinite Series of Generalized Convolutions and Applications. Complex Anal. Oper. Theory 6, 219–236 (2012). https://doi.org/10.1007/s11785-010-0053-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11785-010-0053-x