×

zbMATH — the first resource for mathematics

A direct approach to the Mellin transform. (English) Zbl 0885.44004
The authors describe a systematic historical survey of the well-known Mellin transform and present a new approach to the Mellin transform that is fully independent of Laplace or Fourier transform theory in a unified form containing basic properties and major results under natural, minimal hypotheses upon the functions involved therein.
The basis of the approach are two definitions of the transform, a local and global Mellin transform, the Mellin translation and convolution structure. In particular approximation-theoretical methods connected with the Mellin convolution singular integral enable one to establish the Mellin inversion theory. Of special interest are the Mellin operators of differentiation and integration, more correctly of anti-differentiation, enabling one to establish the fundamental theorem of the differential and integral calculus in the Mellin frame. These operators are different from those considered so far and more general and are of particular importance in solving differential and integral equations.
As applications, the wave equation on \(\mathbb{R}_+\times \mathbb{R}_+\) and the heat equation in a semi-infinite rod are solved in detail.

MSC:
44A15 Special integral transforms (Legendre, Hilbert, etc.)
44-03 History of integral transforms
35K05 Heat equation
35L05 Wave equation
35A22 Transform methods (e.g., integral transforms) applied to PDEs
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Apostol, T.M. (1976).Introduction to Analytic Number Theory, Springer, New York. · Zbl 0335.10001
[2] Apostol, T.M. (1992).Modular Functions and Dirichlet Series in Number Theory, Springer, New York.
[3] Bertrand, J., Bertrand, P., and Ovarlez J.P. (1996). The Mellin Transform,The Transforms and Applications. Handbook (A.D. Poularkas, ed.). CRC Press, Boca Raton, FL, 829–885.
[4] Bohr, H. and Cramér, H. (1923–1927) Die neuere Entwicklung der analytischen Zahlentheorie,Enzyklopädie der Mathematischen Wissenschaften, Band II, 3. Teil, 2, Hälfte, 722–849.
[5] Bryckhov, Yu.A., Glaeske, H.J., Prudnikov, A.P., and Vu Kim Tuan. (1992). Multidimensional Integral Transformations, Gordon and Breach, Philadelphia. · Zbl 0752.44004
[6] Butzer, P.L. (1987). Dirichlet and his role in the founding of mathematical physics.Arch. Internat. Hist. Sci 37, 49–82. · Zbl 0654.01012
[7] Butzer, P.L. and Berens, H. (1967). Semi-groups of Operators and Approximation,Grundlehren Math. Wiss. 145, Springer, Berlin. · Zbl 0164.43702
[8] Butzer, P.L. and Engels, W. (1989). Background to an extension of Gibbs differentation in Walsh analysis.Theory and Applications of Gibbs Derivatives, (P.L. Butzer and R.S. Stanković, eds.). Proc. 1st Internat. Workshop on Gibbs Derivatives, Kupari-Dubrovnik, Yugoslavia, Math. Inst., Beograd, 19–57.
[9] Butzer, P.L. and Gessinger, A. (1995). Ergodic theorems for semigroups and cosine operator functions at zero and infinity with rates; applications to partial differential equations.Mathematical Analysis, Wavelets, and Signal Processing, (M.E.H. Ismail et al., eds.). Proc. Internat. Conf., Cairo,Contemp. Math. 190,Am. Math. Soc., Providence, 67–94. · Zbl 0843.47004
[10] Butzer, P.L. and Hauss, M. (1991). Stirling functions of first and second kind; some new applications.Approximation, Interpolation and Summability (Proc. Conf. in honor of Prof. Jakimovski, Research Inst. of Math. Science, Bar Ilan University, Tel Aviv), Weizmann Press, Israel, 4, 89–108. · Zbl 0809.39009
[11] Butzer, P.L., Hauss, M., and Stens, R.L. (1991). The sampling theorem and its unique role in various branches of mathematics.Mitt. Math. Ges. Hamburg,12, 523–547. · Zbl 0824.94007
[12] Butzer, P.L. and Jansche, S. (1997). The discrete Mellin transform, Mellin-Fourier series, and the Poisson summation formula.Functional Analysis and Approximation Theory (F. Altomare et al., eds.).Proc. 3rd. Int. Conf., Maratea,Rend. Circ. Mat. Palermo (2), to appear. · Zbl 0885.44004
[13] Butzer, P.L. and Jansche, S. (1997). The exponential sampling theorem of signal analysis,Atti Sem. Mat. Fis. Univ. Modena (C. Bardaro et al, eds.).Proc. Conferenze in onore di Calogero Vinti, Perugia. To appear. · Zbl 0885.44004
[14] Butzer, P.L, Jansche, S., and Stens, R.L. (1992). Functional analytic methods in the solution of the fundamental theorems on best algebraic approximation,Approximation Theory, (G.A. Anastassiou, ed.). Proc. 6th Southeastern Approximation Theorists Annual Conf., Memphis. Lecture Notes inPure Appl. Math. 138, 151–205, Dekker, New York. · Zbl 0782.41030
[15] Butzer, P.L. and Nessel, R.J. (1971). Fourier Analysis and Approximation.Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften, Mathematische Reihe 40. Birkhäuser, Basel, Academic Press, New York. · Zbl 0217.42603
[16] Butzer, P.L., Schmidt, M., and Stark, E.L. (1988). Observations on the history of central B-splines.Arch. Hist. Exact Sci. 39, 137–156. · Zbl 0765.01007
[17] Butzer, P.L., Splettstösser, W., and Stens, R.L. (1988). The sampling theorem and linear prediction in signal analysis.Jahresber. Deutsch. Math.-Verein. 90, 1–70. · Zbl 0633.94002
[18] Butzer, P.L. and Stark, E.L. (1986). Riemann’s example of a continuous nondifferentiable function in the light of two letters (1865) of Christoffel to Prym.Bull. Soc. Math. Belg. Sér. A 38, 45–73. · Zbl 0629.01013
[19] Butzer, P.L. and Stens, R.L. (1977). The operational properties of the Chebyshev transform. I: General properties.Funct. Approx. Comment. Math. 5, 129–160. · Zbl 0375.44003
[20] Butzer, P.L. and Wagner, H.J. (1972). Approximation by Walsh polynomials and the concept of a derivative.Applications of Walsh Functions, (R.W. Zeek and A.E. Showalter, eds.).Proc. Symp., Catholic University of America, Washington, D.C., 388–392.
[21] Butzer, P.L. and Westphal, U. (1975). An access to fractional differentiation via fractional difference quotients,Fractional Calculus and its Applications, (B. Ross, ed.).Proc. Conf., New Haven, Lecture Notes inMath. 457, 116–145, Springer, Heidelberg. · Zbl 0307.26006
[22] Cahen, E. (1894). Sur la fonction{\(\zeta\)}(s) de Riemann et sur des fonctions analogues.Ann. Sci. École Norm. Sup. (3) 11, 75–164. · JFM 25.0702.01
[23] Clausen, T. (1858). Beweis des von Herrn Schlömilch aufgestellten Lehrsatzes.Ark. Mat. Phys. 30, 166–170.
[24] Colombo, S. (1959). Les transformations de Mellin et de Hankel, Centre National de la Recherche Scientifique, Paris. · Zbl 0123.30106
[25] Colombo, S. and Lavoine, J. (1959). Transformations de Laplace et de Mellin, Formulaires, Mode d’utilisation,Memorial Sci. Math. 169, Centre National de la Recherche Scientifique, Paris. · Zbl 0246.44001
[26] Comtet, L. (1974).Advanced Combinatorics. Reidel Publishing, Dordrecht. · Zbl 0283.05001
[27] Davies, B. (1984).Integral Transforms and their Applications. Springer, New York. · Zbl 0996.44001
[28] Doetsch, G. (1937).Theorie und Anwendungen der Laplace-Transformation. Springer, Berlin. (Reprint, Dover, New York, 1943). · JFM 63.0368.01
[29] Doetsch, G. (1950).Handbuch der Laplace-Transformation. vol. 1, Birkhäuser, Basel. · Zbl 0040.05901
[30] Edwards, H.M. (1974). Riemann’s Zeta Function.Pure Appl. Math., Academic Press, New York. · Zbl 0315.10035
[31] Elfving, G. (1981).The History of Mathematics in Finland 1828–1918. Frenckell, Helsinki. · Zbl 0476.01003
[32] Erhardt, B.A. (1994).Die kontinuierliche und diskrete Mellin-Transformation mit Anwendungen in der Zahlen- und Signaltheorie. Diplomarbeit, RWTH Aachen, Aachen.
[33] Euler, L. (1761). Remarques sur un beau rapport entre les séries des puissances taut directes que rèciproques.Mem. Acad. Sci. Berlin,17, 83–106. (=Opera (1), 15, 70–90).
[34] Hadamard, J. (1896). Sur la distribution des zéros de la fonction{\(\zeta\)}(s) et ses conséquences arithmetiques.Bull. Soc. Math. France 24, 199–220. · JFM 27.0154.01
[35] Hamburger, H. (1922). Über einige Beziehungen, die mit der Funktionalgleichung der Riemannschen{\(\zeta\)}-Funktion equivalent sind.Math. Ann. 85, 129–140. · JFM 48.1214.01
[36] Hardy, G.H. and Littlewood, J.E. (1915). New proofs of the prime-number theorem and similar theorems.Quart. J. Math. Oxford 46, 215–219. · JFM 45.1252.02
[37] Hardy, G.H. and Littlewood, J.E. (1918). Contributions to the theory of the Riemann zeta-function and the theory of the distribution of primes.Acta Math. 41, 119–196. · JFM 46.0498.01
[38] Hauss, M. (1995). Verallgemeinerte Stirling, Bernoulli und Euler Zahlen, deren Anwendungen und schnell konvergente Reihen für Zeta Funktionen. Dissertation, RWTH Aachen, Aachen. · Zbl 0867.11010
[39] Hewitt, E. and Stromberg, K. (1965). Real and Abstract Analysis. Graduate Texts in Math. 25, Springer, New York. · Zbl 0137.03202
[40] Ivić, A. (1985).The Riemann Zeta-Function. Wiley-Interscience, New York. · Zbl 0556.10026
[41] Jansche, S. (1991). Beste Approximation in linearen normierten Räumen mit Anwendungen auf die Approximation durch Algebraische Polynome. Diplomarbeit, RWTH Aachen, Aachen.
[42] Jordan, C. (1965).Calculus of Finite Differences. Chelsea Publishing, New York. · Zbl 0154.33901
[43] Kloosterman, H.D. (1922). Een integraal voor de{\(\zeta\)}-functie van Riemann.Christian Huygens Math. Tijdschrift 2, 172–177. · JFM 48.0347.01
[44] Kolbe, W. and Nessel, R.J. (1972). Saturation theory in connection with Mellin transform methods,SIAM J. Math. Anal. 3, 246–262. · Zbl 0235.41011
[45] Landau, E. (1906/07). Euler und die Funktionalgleichung der Riemannschen Zetafunktion.Bibl. Math. (3)7, 69–79. · JFM 37.0043.01
[46] Lindelöf, E. (1933). Robert Hjalmar Mellin,Acta Math. 61, I-VI. · JFM 59.0034.02
[47] Magnus, W., Oberhettinger, F., and Soni, R.P. (1966). Formulas and Theorems for the Special Functions of Mathematical Physics. Grundlehren Math. Wiss. 52, Springer, Berlin. · Zbl 0143.08502
[48] Malmstén, C.J. (1849). De integralibus quibusdam definitis, seriebusque infinitis.J. Reine Angew. Math. 38, 1–39. · ERAM 038.1035cj
[49] Mellin, Hj. (1895). Über die fundamentale Wichtigkeit des Satzes von Cauchy für die Theorien der Gamma- und hypergeometrischen Functionen.Acta Soc. Sci. Fennicae. 20, 1–115. · JFM 28.0382.03
[50] Mellin, Hj. (1898). Über eine Verallgemeinerung der Riemannschen Funktion{\(\zeta\)}(s).Acta Soc. Sci. Fennicae. 24, 1–50.
[51] Mellin, Hj. (1901). Eine Formel für den Logarithmus transcendenter Functionen von endlichem Geschlecht.Acta Math. 25, 165–183. · JFM 32.0398.02
[52] Mellin, Hj. (1901). Über den Zusammenhang zwischen den linearen Differential- und Differenzengleichungen.Acta Math. 25, 139–164. · JFM 32.0348.02
[53] Mellin, Hj. (1902). Die Dirichletschen Rheien, die zahlentheoretischen Funktionen und die unendlichen Produkte vom endlichen Geschlecht.Acta Soc. Sci. Fennicae. 31, 1–48.
[54] Mellin, Hj. (1903). Die Dirichlet’schen Reihen, die zahlentheoretischen Funktionen und die unendlichen Produkte von endlichem Geschlecht,Acta Math. 28, 37–64. · JFM 35.0218.02
[55] Mellin, Hj. (1910). Abriß einer einheitlichen Theorie der Gamma-und der hypergeometrischen Funktionen.Math. Ann. 68, 305–337. · JFM 41.0500.04
[56] Mellin, Hj. (1917). Bemerkungen in Anschluß an den Beweis eines Satzes von Hardy über die Zetafunktion.Ann. Acad. Sci. Fenn. Ser. A I Math. 11. · JFM 47.0288.03
[57] Oberhettinger F. (1974).Tables of Mellin Transforms. Springer, Berlin. · Zbl 0289.44003
[58] Patterson, S.J. (1988). An Introduction to the Theory of the Riemann-Zeta-Function.Cambridge Studies Advanced Math. 14, Cambridge University Press, Cambridge. · Zbl 0641.10029
[59] Perron, O. (1908). Zur Theorie der Dirichletschen Reihen.J. Reine Angew. Math. 134, 95–143. · JFM 39.0328.02
[60] Pincherle, S. (1888). Sulle funzioni ipergeometriche generallizate,Rend. Accad. Naz. Lincei 4, 792–799. · JFM 20.0432.01
[61] Platonov, M.L. (1964). On the numbers of a combinatorial structure,Sibirsk. Mat. Zh. 5, 1317–1325. (Russian).
[62] Prössdorf, S. and Silbermann, B. (1991).Numerical Analysis for Integral and Related Operator Equations. Akademie Verlag, Berlin. · Zbl 0763.65102
[63] Rademacher, H. (1973).Topics in Analytic Number Theory. Grundlehren Math. Wiss. 169, Springer, Berlin. · Zbl 0253.10002
[64] Riemann, B. (1990). Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse, Gesammelte Mathematische Werke, Wissenschaftlicher Nachlaß und Nachträge,Collected Papers (nach der Ausgabe von H. Weber und R. Dedekind, neu herausgegeben von R. Narasimhan, ed.). Springer, Berlin, Teubner, Leipzig, 177–185.
[65] Samko, S.G., Kilbas, A.A., and Marichev, O.I. (1993).Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach, Amsterdam. · Zbl 0818.26003
[66] Sasiela, R.J. (1994).Electromagnetic Wave Propagation in Turbulence: Evaluation and Application of Mellin Transforms. Springer, Berlin. · Zbl 0863.76002
[67] Schlömilch, O., (1849). Lehrsatz.Ark. Math. Phys. 12, 415.
[68] Schlömilch, O. (1858). Ueber eine Eigenschaft gewisser Reihen.Zeit. fuer Math. und Phys. 3, 130–132.
[69] Sneddon, I.N. (1955).Functional Analysis, Encyclopedia of Physics, Mathematical Methods. (S. Flügge, ed.). Vol. II. Springer, Berlin, 198–348.
[70] Sneddon, I.N. (1972).The Use of Integral Transforms. McGraw-Hill, New York. · Zbl 0237.44001
[71] Stens, R.L. and Wehrens, M. (1979). Legendre transform methods and best algebraic approximation.Comment. Math. Prace Mat. 21, 351–380. · Zbl 0433.41010
[72] Szmydt, Z. and Ziemian, B. (1992).The Mellin Transformation and Fuchsian Type Purtial Differential Equations. Kluwer, Dordrecht. · Zbl 0771.35002
[73] Titchmarsh, E.C. (1948).Introduction to the Theory of Fourier Integrals. Oxford University Press, Oxford. · Zbl 0031.03202
[74] Titchmarsh, E.C. (1986).The Theory of the Riemann Zeta-function. Clarendon Press, Oxford. · Zbl 0601.10026
[75] Uflyand, Y.A. (1963).Integral Transforms in Problems of the Theory of Elasticity. Izdat, Akad. Nauk SSSR (in Russian). · Zbl 0126.19901
[76] de la Vallée Poussin, C. (1896). Recherches analytiques sur la théorie des nombres premiers.Ann. Soc. Sci. Bruxelles.20, 183–256, 281–297. · JFM 27.0155.03
[77] Weber, H. (1900).Die partiellen Differentialgleichungen der mathematischen Physik, nach Riemann’s Vorlesungen. Vieweg, Braunschweig. · JFM 31.0745.01
[78] Weil, A. (1975). (ed.). Essais historiques sur la théorie des nombres.Monographie No. 22 de l’Enseignement Mathématique, Genève.
[79] Widder, D.V (1946).The Laplace Transform. University of Princeton Press, Princeton, NJ. · Zbl 0060.24801
[80] Widder, D.V. (1971). (ed.). An Introduction to Transform Theory.Pure Appl. Math. 42, Academic Press, New York. · Zbl 0219.44001
[81] Zemanian, A.H., (1968).Generalized Integral Transformation. Interscience, New York. · Zbl 0181.12701
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.