# Operational Properties of Two Integral Transforms of Fourier Type and their Convolutions

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33 citations

### Cites background from "Operational Properties of Two Integ..."

...It is worth saying that for any integral transform, in general, an inversion formula is very important (see [2, 3, 6, 7, 10, 12, 14, 15, 16, 17, 22, 23])....

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### Cites background from "Operational Properties of Two Integ..."

...– Equations (16) and (17) in [23, Theorem 1] are in fact generalized convolution and product theorems (see [27,28])....

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...Namely, a convolution transform, mathematically, is diagonalized by another transform; and in the new (momentum) representation a convolution turns into an operator of multiplication by a function (see [27,28])....

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17 citations

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### Cites background from "Operational Properties of Two Integ..."

...Generalized convolution with weight-function is a nice idea based on the so-called factorizationidentity(see[ 11 ,13,16]).Thefollowingtheorempresentsthegeneralized convolutions with Hermitian weight-function....

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##### References

4,736 citations

### "Operational Properties of Two Integ..." refers background in this paper

...We mention interesting properties of the transforms Fc ,F s (see [ 1 , 15, 18]):...

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3,025 citations

### "Operational Properties of Two Integ..." refers background in this paper

..., (1971) Convolution Equations and Projection Methods for their Solutions, , Nauka, Moscow, (in Russian) Hochstadt, H., (1973) Integral Equations, , John Wiley & Sons, N. Y Hörmander, L., (1983) The Analysis of Linear Partial Differential Operators I, , Springer-Verlag, Berlin Kakichev, V.A., On the convolution for integral transforms (1967) Izv....

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..., (1971) Convolution Equations and Projection Methods for their Solutions, , Nauka, Moscow, (in Russian) Hochstadt, H., (1973) Integral Equations, , John Wiley & Sons, N. Y Hörmander, L., (1983) The Analysis of Linear Partial Differential Operators I, , Springer-Verlag, Berlin Kakichev, V.A., On the convolution for integral transforms (1967) Izv. ANBSSR, Ser. Fiz. Mat., (2), pp. 48-57. , (in Russian) Kakichev, V.A., Thao, N.X., Tuan, V.K., On the generalized convolutions for Fourier cosine and sine transforms (1998) EastWest Jour. Math., 1 (1), pp. 85-90 Naimark, M.A., (1959) Normed Rings, , P. Noordhoff Ltd., Groningen, Netherlands Rudin, W., (1991) Functional Analysis, , McGraw-Hill, N. Y Smith, S.W., (2002) Digital Signal Processing: A Practical Guide for Engineers and Scientists, , ISBN 0-7506-7444-X (e-book) Sneddon, I., (1951) Fourier Transforms, , McGraw-Hill, New York-Toronto-London Thao, N....

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..., (1971) Convolution Equations and Projection Methods for their Solutions, , Nauka, Moscow, (in Russian) Hochstadt, H., (1973) Integral Equations, , John Wiley & Sons, N....

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..., (1971) Convolution Equations and Projection Methods for their Solutions, , Nauka, Moscow, (in Russian) Hochstadt, H., (1973) Integral Equations, , John Wiley & Sons, N. Y Hörmander, L., (1983) The Analysis of Linear Partial Differential Operators I, , Springer-Verlag, Berlin Kakichev, V.A., On the convolution for integral transforms (1967) Izv. ANBSSR, Ser. Fiz. Mat., (2), pp. 48-57. , (in Russian) Kakichev, V.A., Thao, N.X., Tuan, V.K., On the generalized convolutions for Fourier cosine and sine transforms (1998) EastWest Jour. Math., 1 (1), pp. 85-90 Naimark, M.A., (1959) Normed Rings, , P. Noordhoff Ltd., Groningen, Netherlands Rudin, W., (1991) Functional Analysis, , McGraw-Hill, N....

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..., (1971) Convolution Equations and Projection Methods for their Solutions, , Nauka, Moscow, (in Russian) Hochstadt, H., (1973) Integral Equations, , John Wiley & Sons, N. Y Hörmander, L., (1983) The Analysis of Linear Partial Differential Operators I, , Springer-Verlag, Berlin Kakichev, V.A., On the convolution for integral transforms (1967) Izv. ANBSSR, Ser. Fiz. Mat., (2), pp. 48-57. , (in Russian) Kakichev, V.A., Thao, N.X., Tuan, V.K., On the generalized convolutions for Fourier cosine and sine transforms (1998) EastWest Jour. Math., 1 (1), pp. 85-90 Naimark, M.A., (1959) Normed Rings, , P. Noordhoff Ltd., Groningen, Netherlands Rudin, W., (1991) Functional Analysis, , McGraw-Hill, N. Y Smith, S.W., (2002) Digital Signal Processing: A Practical Guide for Engineers and Scientists, , ISBN 0-7506-7444-X (e-book) Sneddon, I....

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2,577 citations

### "Operational Properties of Two Integ..." refers background in this paper

...and the corresponding Hermite function φn by φn(x) = (−1)e 1 2x 2 ( d dx )n e−x 2 (see [18])....

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...1, Theorem 57 in [18] and the factorization identities of the convolutions, we obtain Tk(φ1 ∗ φ0) = γ2(−φ1)φ0 = −γ2φ0φ1, Tk(φ0 ∗ φ1) = γ2φ0(iφ1) = iγ2φ0φ1....

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...Moreover, Fc, Fs are isometric operators in L2[0,+∞) satisfying the identities: F 2 c = I, F 2 s = I (see [2, 18])....

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...2) (see Sneddon [15], Titchmarsh [18])....

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...These transforms and the Fourier integral transform have been studied for a long time, and applied to many fields of mathematics (see Hörmander [9], Rudin [13], or [18])....

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