Integral Transforms Related to a Generalized Convolution

On the Fourier cosine-Kontorovich-Lebedev generalized convolution transforms

“…No application of convolution type transforms of solving integro-differential was presented in recent investigations [3], [4], [13], [15]. In this section, we apply a general class of Fourier cosine and Kontorovich-Lebedev generalized convolution transforms to solve a class of integro-differential problems, which seems to be difficult to solve in closed form by using other techniques.…”

“…Recently, several authors have been interested in the convolution transforms of this type (see [3], [4], [13], [15]). In this paper, we are interested in the transform f → D(h γ * f ), where (h γ * f ) is the generalized convolution (1.11).…”

Some results of Watson and Plancherel‐type integral transforms related to the Hartley, Fourier convolutions and applications

“…Plancherel 21 proved Plancherel's theorem in general form for the Fourier transform on $$$ {L}_2\left(\mathbb{R}\right) $$$ and confirmed that this transformation is unitary and this result is also still true for Hartley transforms. In Section 4, we give the Plancherel‐type theorem by using the techniques in Al‐Musallam and Tuan 4 and study the approximation in the norm of space $$$ {L}_2\left(\mathbb{R}\right) $$$ for the operator $$$ \left({T}_k\right) $$$ constructed in formula (). Specifically, if suppose that the image function $$$ \Psi (x)=\left({T}_kf\right)(x) $$$ and the original function $$$ f(x) $$$, then it can approximate to sequences of functions in the space $$$ {L}_2\left(\mathbb{R}\right) $$$ that converge normally in $$$ {L}_2\left(\mathbb{R}\right) $$$ to some arbitrary function which also belongs to $$$ {L}_2\left(\mathbb{R}\right).$…”

“…Plancherel 21 proved Plancherel's theorem in general form for the Fourier transform on L 2 (R) and confirmed that this transformation is unitary and this result is also still true for Hartley transforms. In Section 4, we give the Plancherel-type theorem by using the techniques in Al-Musallam and Tuan 4 and study the approximation in the norm of space L 2 (R) for the operator (T k ) constructed in formula (3.13). Specifically, if suppose that the image function Ψ(x) = (T k 𝑓 ) (x) and the original function 𝑓 (x), then it can approximate to sequences of functions in the space L 2 (R) that converge normally in L 2 (R) to some arbitrary function which also belongs to L 2 (R).…”

Integral Transforms of Fourier Cosine and Sine Generalized Convolution Type

“…Recently, a class of integral transforms that is related to the generalized convolution (1.11) has been introduced and investigated in [12]. In this paper, we will consider a class of integral transform which has a connection with the generalized convolution (1.13), namely, the transforms of the form…”