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From Integral Reciprocity to Spectral Transform Algebras

Abstract

Integral transforms with product kernels

\[ (\mathcal K f)(y)=\int_0^\infty f(x)K(xy)\,dx \]

possess an elementary reciprocity inherited from the symmetry \(xy=yx\). That observation alone, however, does not describe their operator structure. After unitary Mellin transformation, every such transform becomes a reflected multiplier,

\[ \mathcal M\mathcal K\mathcal M^{-1}=M_qJ, \qquad (Jg)(t)=g(-t), \]

where \(q\) is the Mellin symbol of \(K\). This representation gives exact criteria for boundedness, adjoints, normality, self-adjointness, unitarity, involutivity, invertibility, and Plancherel identities.

More importantly, product-kernel transforms are not closed under composition: the product of two reflected multipliers is an ordinary Mellin multiplier. The smallest operator algebra containing both classes is

\[ \mathfrak A = \{M_p+M_qJ:p,q\in L^\infty(\mathbb R)\}, \]

a \(\mathbb Z_2\)-graded algebra naturally identified with the reflection crossed product

\[ L^\infty(\mathbb R)\rtimes\mathbb Z_2. \]

Its multiplication law encodes both reciprocal integral transforms and Mellin convolution operators in a single spectral calculus. A direct-integral \(2\times2\) matrix representation yields explicit spectral and functional-calculus formulae. Plancherel transforms arise precisely from unimodular reflected symbols.

The resulting framework separates three levels that are often conflated: elementary integral reciprocity, algebraic transform composition, and spectral unitarity.

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1. Introduction

Consider the product-kernel integral transform

\[ (\mathcal K_K f)(y) = \int_0^\infty f(x)K(xy)\,dx. \tag{1.1} \]

For suitable \(f,g,K\),

\[ \int_0^\infty f(x)(\mathcal K_Kg)(x)\,dx = \int_0^\infty g(y)(\mathcal K_Kf)(y)\,dy, \tag{1.2} \]

because both sides equal

\[ \int_0^\infty\int_0^\infty f(x)g(y)K(xy)\,dx\,dy. \tag{1.3} \]

This is the basic integral reciprocity.

It is structurally informative but mathematically incomplete. In particular, it does not answer the following questions:

\[ \begin{aligned} &\text{When is }\mathcal K_K\text{ bounded on }L^2(0,\infty)?\\ &\text{When is it self-adjoint?}\\ &\text{When is it normal?}\\ &\text{When does }\mathcal K_K^2=I?\\ &\text{When does it satisfy a Plancherel identity?}\\ &\text{What operators arise by composing two such transforms?}\\ &\text{What is the smallest algebra closed under those compositions?} \end{aligned} \]

The correct spectral coordinate system is supplied by the Mellin transform.

The central observation of this paper is that product-kernel transforms occupy only the odd component of a larger \(\mathbb Z_2\)-graded transform algebra. Under the Mellin transform,

\[ \boxed{ \mathcal K_K \longleftrightarrow M_qJ, } \tag{1.4} \]

where \(M_q\) is multiplication by the Mellin symbol \(q\) and \(J\) is reflection.

Composition then immediately gives

\[ (M_qJ)(M_rJ) = M_{q\,r^\vee}, \qquad r^\vee(t):=r(-t). \tag{1.5} \]

Thus two product-kernel transforms compose not to another product-kernel transform, but to a Mellin multiplier.

This forces the larger algebra

\[ \boxed{ \mathfrak A = \{M_p+M_qJ\}. } \tag{1.6} \]

The algebra itself, rather than a single reciprocity identity, is the natural mathematical object.

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2. The unitary Mellin transform

Let

\[ H=L^2((0,\infty),dx). \]

Define

\[ (\mathcal Mf)(t) = \frac{1}{\sqrt{2\pi}} \int_0^\infty f(x)x^{-1/2-it}\,dx. \tag{2.1} \]

Under the logarithmic substitution \(x=e^u\), define

\[ g(u)=e^{u/2}f(e^u). \]

Then

\[ \|g\|_{L^2(\mathbb R)} = \|f\|_{L^2(0,\infty)}, \]

and (2.1) becomes the ordinary unitary Fourier transform of \(g\).

Consequently,

\[ \boxed{ \mathcal M: L^2(0,\infty) \longrightarrow L^2(\mathbb R) } \tag{2.2} \]

is unitary.

Let

\[ (Jh)(t)=h(-t). \tag{2.3} \]

Then

\[ J^2=I, \qquad J^*=J. \tag{2.4} \]

For \(q\in L^\infty(\mathbb R)\), write

\[ (M_qh)(t)=q(t)h(t). \tag{2.5} \]

Reflection acts on multipliers according to

\[ JM_q=M_{q^\vee}J, \qquad q^\vee(t)=q(-t). \tag{2.6} \]

Equation (2.6) will generate the entire transform algebra.

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3. Spectral representation of product-kernel transforms

Let

\[ (\mathcal K_Kf)(y) = \int_0^\infty f(x)K(xy)\,dx \tag{3.1} \]

initially for compactly supported test functions.

Define the Mellin symbol

\[ q_K(t) = \int_0^\infty K(u)u^{-1/2-it}\,du \tag{3.2} \]

whenever this integral is meaningful.

Equivalently,

\[ q_K(t) = \sqrt{2\pi}\,(\mathcal MK)(t). \tag{3.3} \]

Theorem 3.1. Mellin reflection representation

Under conditions justifying Fubini's theorem,

\[ \boxed{ \mathcal M(\mathcal K_Kf)(t) = q_K(t)(\mathcal Mf)(-t). } \tag{3.4} \]

Hence

\[ \boxed{ \mathcal M\mathcal K_K\mathcal M^{-1} = M_{q_K}J. } \tag{3.5} \]

Proof

From (3.1),

\[ \begin{aligned} (\mathcal M\mathcal K_Kf)(t) &= \frac1{\sqrt{2\pi}} \int_0^\infty \int_0^\infty f(x)K(xy)\,dx\, y^{-1/2-it}\,dy. \end{aligned} \]

Set

\[ u=xy, \qquad y=\frac ux, \qquad dy=\frac{du}{x}. \]

Then

\[ y^{-1/2-it} = u^{-1/2-it}x^{1/2+it}, \]

and therefore

\[ \begin{aligned} (\mathcal M\mathcal K_Kf)(t) &= \frac1{\sqrt{2\pi}} \int_0^\infty f(x)x^{-1/2+it}\,dx \int_0^\infty K(u)u^{-1/2-it}\,du\\ &= q_K(t)(\mathcal Mf)(-t). \end{aligned} \]

This proves (3.4) and (3.5). \(\square\)

---

Corollary 3.2. Boundedness

If

\[ q_K\in L^\infty(\mathbb R), \]

then \(\mathcal K_K\) extends uniquely to a bounded operator on \(L^2(0,\infty)\), and

\[ \boxed{ \|\mathcal K_K\| = \|q_K\|_\infty. } \tag{3.6} \]

Thus the analytic boundedness problem for the integral transform becomes the elementary multiplier problem

\[ q_K\in L^\infty. \]

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4. Reciprocity versus Hilbert-space symmetry

The elementary reciprocity relation

\[ \int f(x)(\mathcal K_Kg)(x)\,dx = \int g(x)(\mathcal K_Kf)(x)\,dx \tag{4.1} \]

is bilinear.

It should not be confused with self-adjointness in the complex Hilbert space \(L^2\).

From (3.5),

\[ (M_qJ)^* = JM_{\overline q} = M_{\overline{q^\vee}}J. \tag{4.2} \]

Therefore:

Proposition 4.1. Adjoint

\[ \boxed{ \mathcal K_K^* \longleftrightarrow M_{\overline{q^\vee}}J. } \tag{4.3} \]

Consequently,

\[ \boxed{ \mathcal K_K=\mathcal K_K^* \iff q(t)=\overline{q(-t)} \quad\text{a.e.} } \tag{4.4} \]

For ordinary kernels, a real-valued \(K\) gives precisely this Mellin symmetry.

Thus:

\[ \boxed{ \text{integral reciprocity} \neq \text{self-adjointness}. } \]

The former comes from \(xy=yx\); the latter imposes an additional conjugation condition.

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5. Composition and the failure of closure

Let

\[ A_q=M_qJ. \tag{5.1} \]

Then

\[ \begin{aligned} A_qA_r &= M_qJM_rJ\\ &= M_qM_{r^\vee}J^2\\ &= M_{q\,r^\vee}. \end{aligned} \]

Hence:

Theorem 5.1. Two-transform composition

\[ \boxed{ A_qA_r=M_{q\,r^\vee}. } \tag{5.2} \]

Translated back to \(L^2(0,\infty)\),

\[ \boxed{ \mathcal K_K\mathcal K_L } \]

is spectrally a Mellin multiplier.

It is generally not another product-kernel transform.

This is the first structural reason to enlarge the theory.

Product-kernel transforms do not form an operator algebra.

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6. Mellin multipliers and product-kernel transforms

Let

\[ \mathfrak A_0 = \{M_p:p\in L^\infty(\mathbb R)\}, \tag{6.1} \]

and

\[ \mathfrak A_1 = \{M_qJ:q\in L^\infty(\mathbb R)\}. \tag{6.2} \]

Then

\[ \mathfrak A_0\mathfrak A_0 \subseteq \mathfrak A_0, \]\[ \mathfrak A_0\mathfrak A_1 \subseteq \mathfrak A_1, \]\[ \mathfrak A_1\mathfrak A_0 \subseteq \mathfrak A_1, \]

and

\[ \mathfrak A_1\mathfrak A_1 \subseteq \mathfrak A_0. \]

Therefore

\[ \boxed{ \mathfrak A = \mathfrak A_0\oplus\mathfrak A_1 } \tag{6.3} \]

is naturally \(\mathbb Z_2\)-graded.

Explicitly,

\[ \boxed{ \mathfrak A = \{M_p+M_qJ: p,q\in L^\infty(\mathbb R)\}. } \tag{6.4} \]

We call \(\mathfrak A\) the spectral transform algebra associated with multiplicative reciprocity.

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7. Algebra law

For any measurable function \(h\), write

\[ h^\vee(t)=h(-t). \]

Let

\[ A(p,q)=M_p+M_qJ. \]

Then

\[ \begin{aligned} A(p,q)A(r,s) &= M_{pr} + M_{ps}J + M_qJM_r + M_qJM_sJ\\ &= M_{pr} + M_{ps}J + M_{q r^\vee}J + M_{q s^\vee}. \end{aligned} \]

Hence:

Theorem 7.1. Spectral transform product

\[ \boxed{ A(p,q)A(r,s) = A\!\left( pr+qs^\vee, \; ps+qr^\vee \right). } \tag{7.1} \]

The involution is

\[ \boxed{ A(p,q)^* = A\!\left( \overline p, \overline{q^\vee} \right). } \tag{7.2} \]

Thus \(\mathfrak A\) is a \(*\)-algebra.

The reflection action

\[ \alpha(p)=p^\vee \tag{7.3} \]

is an action of \(\mathbb Z_2\) on \(L^\infty(\mathbb R)\).

Consequently the algebra is naturally identified with the crossed product

\[ \boxed{ \mathfrak A \cong L^\infty(\mathbb R)\rtimes_\alpha\mathbb Z_2. } \tag{7.4} \]

This is the correct closure of the transform family.

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8. Direct-integral matrix representation

The algebra becomes still more explicit by pairing the spectral points \(t\) and \(-t\).

Define

\[ Uh(t) = \begin{pmatrix} h(t)\\ h(-t) \end{pmatrix}, \qquad t>0. \tag{8.1} \]

Then

\[ L^2(\mathbb R) \cong L^2\bigl((0,\infty),\mathbb C^2\bigr). \tag{8.2} \]

Under this identification,

\[ A(p,q) \]

acts fiberwise by

\[ \boxed{ \mathbf A_{p,q}(t) = \begin{pmatrix} p(t) & q(t)\\ q(-t) & p(-t) \end{pmatrix}. } \tag{8.3} \]

Thus the infinite-dimensional operator algebra decomposes into a measurable field of \(2\times2\) matrices.

This produces an immediate spectral calculus.

For the pure product-kernel transform,

\[ A_q=M_qJ, \]

the fiber is

\[ \boxed{ \mathbf A_q(t) = \begin{pmatrix} 0&q(t)\\ q(-t)&0 \end{pmatrix}. } \tag{8.4} \]

Its square is

\[ \mathbf A_q(t)^2 = q(t)q(-t)I_2. \tag{8.5} \]

Therefore its fiber eigenvalues satisfy

\[ \boxed{ \lambda_\pm(t) = \pm\sqrt{q(t)q(-t)}. } \tag{8.6} \]

This formula is the spectral origin of involutive transform theory.

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9. Classification of a product-kernel transform

Let

\[ A_q=M_qJ. \]

Several operator properties now follow exactly.

Theorem 9.1. Square

\[ \boxed{ A_q^2 = M_{q q^\vee}. } \tag{9.1} \]

Therefore

\[ \boxed{ A_q^2=I \iff q(t)q(-t)=1 \quad\text{a.e.} } \tag{9.2} \]

This is the complete involutivity criterion.

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Theorem 9.2. Adjoint products

Since

\[ A_q^*=M_{\overline{q^\vee}}J, \]

we obtain

\[ \boxed{ A_qA_q^* = M_{|q|^2}, } \tag{9.3} \]

and

\[ \boxed{ A_q^*A_q = M_{|q^\vee|^2}. } \tag{9.4} \]

Hence:

Normality

\[ \boxed{ A_q\text{ normal} \iff |q(t)|=|q(-t)| \quad\text{a.e.} } \tag{9.5} \]

Unitarity

\[ \boxed{ A_q\text{ unitary} \iff |q(t)|=1 \quad\text{a.e.} } \tag{9.6} \]

Self-adjointness

\[ \boxed{ A_q\text{ self-adjoint} \iff q(t)=\overline{q(-t)} \quad\text{a.e.} } \tag{9.7} \]

Self-adjoint unitary involution

The following are then equivalent:

\[ A_q=A_q^*, \qquad A_q^2=I, \]

and

\[ \boxed{ q(-t)=\overline{q(t)}, \qquad |q(t)|=1. } \tag{9.8} \]

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10. Plancherel transforms

The Plancherel condition is now exact.

Because \(\mathcal M\) is unitary,

\[ \|\mathcal K_Kf\|_2 = \|M_qJ\mathcal Mf\|_2. \]

Thus

\[ \|\mathcal K_Kf\|_2^2 = \int_{\mathbb R} |q(t)|^2 |(\mathcal Mf)(-t)|^2\,dt. \tag{10.1} \]

Hence:

Theorem 10.1. Plancherel criterion

A bounded product-kernel transform satisfies

\[ \boxed{ \|\mathcal K_Kf\|_2 = \|f\|_2 \quad\text{for all }f } \tag{10.2} \]

if and only if

\[ \boxed{ |q_K(t)|=1 \quad\text{for almost every }t. } \tag{10.3} \]

Thus Plancherel is not a consequence of reciprocity alone.

It is spectral rigidity:

\[ \boxed{ \text{reciprocity} + \text{unimodular Mellin symbol} = \text{Plancherel transform}. } \tag{10.4} \]

This distinguishes the three principal levels:

\[ \boxed{ \begin{array}{c} K(xy)=K(yx) \\[1mm] \Downarrow\\ \text{bilinear reciprocity} \\[2mm] q(-t)=\overline{q(t)} \\[1mm] \Downarrow\\ \text{self-adjointness} \\[2mm] |q(t)|=1 \\[1mm] \Downarrow\\ \text{Plancherel / unitarity}. \end{array} } \tag{10.5} \]

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11. Invertibility

The same representation gives an exact inversion criterion.

Suppose

\[ \operatorname*{ess\,inf}_{t\in\mathbb R}|q(t)|>0. \tag{11.1} \]

Define

\[ r(t)=\frac1{q(-t)}. \tag{11.2} \]

Then

\[ A_qA_r = M_{q r^\vee} = I, \]

and similarly

\[ A_rA_q=I. \]

Therefore:

Proposition 11.1

\[ \boxed{ A_q^{-1} = A_{1/q^\vee} } \tag{11.3} \]

whenever \(q\) is essentially bounded away from zero.

Thus inversion of the integral transform is reduced to pointwise inversion of the reflected Mellin symbol.

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12. Self-adjoint spectral theory

Assume

\[ q(-t)=\overline{q(t)}. \tag{12.1} \]

Then \(A_q\) is self-adjoint, and

\[ q(t)q(-t)=|q(t)|^2. \]

Therefore the fiber matrix is

\[ \begin{pmatrix} 0&q(t)\\ \overline{q(t)}&0 \end{pmatrix} \tag{12.2} \]

with eigenvalues

\[ \boxed{ \pm |q(t)|. } \tag{12.3} \]

Consequently,

\[ \boxed{ \sigma(A_q) = \operatorname{ess\,cl} \left( \{\pm |q(t)|:t\in\mathbb R\} \right). } \tag{12.4} \]

Here \(\operatorname{ess\,cl}\) denotes essential closure.

In particular,

\[ \|A_q\| = \operatorname*{ess\,sup}|q|. \]

If \(A_q\) is additionally unitary,

\[ |q|=1, \]

then

\[ \boxed{ \sigma(A_q)\subseteq\{-1,+1\}. } \tag{12.5} \]

Thus a self-adjoint Plancherel product-kernel transform is automatically an involution.

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13. Explicit functional calculus

For a self-adjoint \(A_q\), let

\[ r(t)=|q(t)|. \]

Then

\[ A_q^2=M_{r^2}. \tag{13.1} \]

Because \(r(t)=r(-t)\), the multiplier \(M_r\) commutes with \(A_q\).

For a bounded Borel function \(\Phi\), define

\[ \Phi_{\rm even}(r) = \frac{\Phi(r)+\Phi(-r)}2, \]

and

\[ \Phi_{\rm odd}(r) = \frac{\Phi(r)-\Phi(-r)}2. \]

Then the functional calculus gives

\[ \boxed{ \Phi(A_q) = M_{\Phi_{\rm even}(r)} + M_{\frac{\Phi_{\rm odd}(r)}{r}}A_q, } \tag{13.2} \]

with the second term interpreted continuously at \(r=0\) whenever appropriate.

Thus every bounded spectral function of the integral transform again lies inside the two-component transform algebra.

This is one reason the larger algebra \(\mathfrak A\) is more natural than the original product-kernel family.

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14. A classical Plancherel example

Consider the normalized cosine transform

\[ (\mathcal Cf)(y) = \sqrt{\frac2\pi} \int_0^\infty f(x)\cos(xy)\,dx. \tag{14.1} \]

Its kernel is

\[ K(u) = \sqrt{\frac2\pi}\cos u. \]

The Mellin symbol is

\[ q_C(t) = \sqrt{\frac2\pi} \Gamma\!\left(\frac12-it\right) \cos\!\left( \frac\pi2\left(\frac12-it\right) \right). \tag{14.2} \]

Classical Gamma-function identities imply

\[ q_C(-t)=\overline{q_C(t)} \tag{14.3} \]

and

\[ |q_C(t)|=1. \tag{14.4} \]

Therefore the abstract criteria immediately yield

\[ \mathcal C^*=\mathcal C, \]\[ \mathcal C^2=I, \]

and

\[ \|\mathcal Cf\|_2=\|f\|_2. \]

Thus the classical cosine transform appears as a self-adjoint unitary element of the odd component of the spectral transform algebra.

The same analysis applies to other Mellin-diagonalizable integral transforms.

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15. Why reciprocity alone is insufficient

The elementary identity

\[ \iint f(x)g(y)K(xy)\,dx\,dy = \iint g(y)f(x)K(yx)\,dy\,dx \]

contains no information about the magnitude of the Mellin symbol.

Consequently it cannot distinguish among:

\[ |q(t)|<1, \]\[ |q(t)|>1, \]\[ |q(t)|=1, \]

or an unbounded \(q\).

These correspond to fundamentally different operator classes.

Thus

\[ \boxed{ \text{reciprocity is geometric/algebraic;} \qquad \text{Plancherel is spectral}. } \tag{15.1} \]

The Mellin representation is precisely what exposes the difference.

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16. Transform algebra versus individual transforms

The point of the construction is not merely that

\[ M_qJ \]

is easy to analyze.

The deeper structure is the alternation

\[ \boxed{ \text{multiplier} \leftrightarrow \text{reflected multiplier}. } \]

Specifically,

\[ \mathfrak A_0\mathfrak A_0 \subset\mathfrak A_0, \]\[ \mathfrak A_0\mathfrak A_1 \subset\mathfrak A_1, \]\[ \mathfrak A_1\mathfrak A_0 \subset\mathfrak A_1, \]\[ \mathfrak A_1\mathfrak A_1 \subset\mathfrak A_0. \]

Thus composition carries parity.

Even products of product-kernel transforms become Mellin multipliers.

Odd products become product-kernel transforms again.

For example,

\[ A_{q_1}A_{q_2}A_{q_3} = M_{q_1q_2^\vee q_3}J, \tag{16.1} \]

while

\[ A_{q_1}A_{q_2}A_{q_3}A_{q_4} = M_{q_1q_2^\vee q_3q_4^\vee}. \tag{16.2} \]

More generally,

\[ A_{q_1}\cdots A_{q_{2m}} = M_{ q_1q_2^\vee q_3q_4^\vee \cdots q_{2m-1}q_{2m}^\vee }, \tag{16.3} \]

whereas

\[ A_{q_1}\cdots A_{q_{2m+1}} = M_{ q_1q_2^\vee \cdots q_{2m-1}q_{2m}^\vee q_{2m+1} }J. \tag{16.4} \]

This gives an exact calculus for arbitrary finite transform chains.

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17. Commutators

Two product-kernel transforms need not commute.

Indeed,

\[ A_qA_r=M_{qr^\vee}, \]

while

\[ A_rA_q=M_{rq^\vee}. \]

Therefore

\[ \boxed{ [A_q,A_r] = M_{qr^\vee-rq^\vee}. } \tag{17.1} \]

Hence

\[ \boxed{ A_qA_r=A_rA_q \iff q(t)r(-t)=r(t)q(-t) \quad\text{a.e.} } \tag{17.2} \]

This is considerably more informative than imposing commutativity at the outset.

It identifies the exact spectral obstruction.

If both symbols are nowhere zero, the condition becomes

\[ \frac{q(t)}{q(-t)} = \frac{r(t)}{r(-t)}. \tag{17.3} \]

Thus commuting reciprocal transforms share the same reflection ratio.

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18. Anticommutation

Similarly,

\[ A_qA_r+A_rA_q = M_{qr^\vee+rq^\vee}. \]

Therefore

\[ \boxed{ \{A_q,A_r\}=0 \iff qr^\vee=-rq^\vee. } \tag{18.1} \]

The transform algebra therefore naturally accommodates both commuting and anticommuting integral transforms.

This becomes especially transparent in the \(2\times2\) fiber representation, where these identities reduce to matrix commutator and anticommutator relations at each spectral point.

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19. Projections and idempotents

For a general element

\[ A(p,q)=M_p+M_qJ, \]

the idempotency equation

\[ A(p,q)^2=A(p,q) \tag{19.1} \]

becomes, by (7.1),

\[ p^2+qq^\vee=p, \tag{19.2} \]

and

\[ q(p+p^\vee)=q. \tag{19.3} \]

Thus projection-like transforms correspond to pointwise algebraic constraints on their spectral coefficients.

If self-adjointness is imposed as well,

\[ p=\overline p, \qquad q=\overline{q^\vee}, \]

the problem reduces to classifying measurable \(2\times2\) projection fields.

The spectral transform algebra therefore contains not only reciprocal and Plancherel transforms, but also projection operators and their associated decompositions.

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20. Resolvents

The \(2\times2\) representation also makes resolvents explicit.

For the pure transform \(A_q\),

\[ \lambda I-A_q \]

has fiber matrix

\[ \begin{pmatrix} \lambda&-q(t)\\ -q(-t)&\lambda \end{pmatrix}. \]

Its determinant is

\[ \lambda^2-q(t)q(-t). \]

Whenever this is essentially bounded away from zero,

\[ \boxed{ (\lambda I-A_q)^{-1} = \frac{ \lambda I+A_q }{ \lambda^2-M_{qq^\vee} }. } \tag{20.1} \]

More explicitly,

\[ (\lambda I-A_q)^{-1} = M_{ \frac{\lambda} {\lambda^2-qq^\vee} } + M_{ \frac{q} {\lambda^2-qq^\vee} }J. \tag{20.2} \]

Thus the resolvent remains inside the spectral transform algebra.

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21. Spectral transform algebra as the natural closure

The preceding results establish the following structural hierarchy:

\[ \boxed{ \begin{array}{c} \text{product kernel }K(xy) \\[2mm] \Downarrow \\ \text{reflected Mellin multiplier }M_qJ \\[2mm] \Downarrow \\ \text{composition produces ordinary multipliers} \\[2mm] \Downarrow \\ \mathfrak A = L^\infty(\mathbb R) \oplus L^\infty(\mathbb R)J \\[2mm] \Downarrow \\ L^\infty(\mathbb R)\rtimes\mathbb Z_2 \\[2mm] \Downarrow \\ \text{fiberwise }2\times2\text{ spectral calculus}. \end{array} } \tag{21.1} \]

Integral reciprocity is therefore the entry point, not the terminal theorem.

The natural endpoint is an operator algebra in which reciprocity, Plancherel theory, involutivity, spectral decomposition, commutation, inversion, and functional calculus become aspects of one object.

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22. Main theorem

The preceding development may be summarized as follows.

Theorem 22.1. Spectral transform algebra theorem

Let

\[ H=L^2(0,\infty), \]

and let \(\mathcal M:H\to L^2(\mathbb R)\) be the unitary Mellin transform.

For every product-kernel transform

\[ (\mathcal K_Kf)(y) = \int_0^\infty f(x)K(xy)\,dx \]

whose Mellin symbol

\[ q_K(t) = \int_0^\infty K(u)u^{-1/2-it}\,du \]

belongs to \(L^\infty(\mathbb R)\),

\[ \mathcal M\mathcal K_K\mathcal M^{-1} = M_{q_K}J. \]

The norm-closed operator algebra generated by all Mellin multipliers and all such product-kernel transforms is unitarily equivalent to

\[ \boxed{ L^\infty(\mathbb R)\rtimes_\alpha\mathbb Z_2, \qquad \alpha(q)=q^\vee. } \tag{22.1} \]

Under the direct-integral decomposition over \(t>0\), every element is represented by

\[ \boxed{ \begin{pmatrix} p(t)&q(t)\\ q(-t)&p(-t) \end{pmatrix}. } \tag{22.2} \]

Accordingly:

\[ \mathcal K_K^2=I \iff q_Kq_K^\vee=1, \]\[ \mathcal K_K^*=\mathcal K_K \iff q_K=\overline{q_K^\vee}, \]\[ \mathcal K_K\text{ is normal} \iff |q_K|=|q_K^\vee|, \]

and

\[ \boxed{ \mathcal K_K\text{ satisfies Plancherel} \iff |q_K|=1 \quad\text{a.e.} } \tag{22.3} \]

For two transforms,

\[ [\mathcal K_K,\mathcal K_L] \]

is Mellin multiplication by

\[ q_Kq_L^\vee-q_Lq_K^\vee. \tag{22.4} \]

Thus all principal algebraic and spectral properties of the transform family reduce to pointwise relations between Mellin symbols.

\(\square\)

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23. Discussion

The framework exposes a distinction between three notions.

First is reciprocity:

\[ B_K(f,g)=B_K(g,f). \]

This is inherited directly from the product structure \(xy\).

Second is algebraic closure. Product-kernel transforms alone are not closed under composition. Their natural closure necessarily contains ordinary Mellin multipliers and forms a \(\mathbb Z_2\)-graded crossed-product algebra.

Third is spectral rigidity. Plancherel, involutivity, and self-adjointness require additional pointwise constraints on the Mellin symbol.

In particular,

\[ \boxed{ \text{reciprocity does not imply Plancherel}. } \]

Instead,

\[ \boxed{ \text{Plancherel} = \text{unimodularity of the Mellin symbol}. } \]

Likewise,

\[ \boxed{ \text{involutivity} = q(t)q(-t)=1, } \]

and

\[ \boxed{ \text{self-adjointness} = q(t)=\overline{q(-t)}. } \]

The spectral transform algebra provides a common setting in which these conditions can be compared rather than treated as unrelated transform identities.

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24. Further directions

Several extensions arise naturally.

Weighted Mellin spaces

Replacing \(L^2(dx)\) by weighted spaces shifts the Mellin spectral line and should produce analogous reflection algebras around a general line

\[ \Re s=c. \]

The reflection becomes

\[ s\mapsto 2c-s. \]

Matrix-valued kernels

For

\[ K(u)\in M_n(\mathbb C), \]

the Mellin symbol becomes matrix-valued, and the transform algebra becomes a matrix-valued crossed product. Noncommutativity then occurs both in the reflection action and internally in the symbol algebra.

Operator-valued kernels

The same mechanism extends formally to operator-valued symbols, where the scalar criteria

\[ |q|=1 \]

are replaced by pointwise unitarity.

Spectral factorization

Given a desired operator property, one may reverse the construction.

For example, classify all symbols satisfying

\[ q(t)q(-t)=1 \]

and then recover corresponding kernels through inverse Mellin transformation.

This converts transform discovery into a spectral-design problem.

Approximate Plancherel theory

If

\[ \bigl||q(t)|-1\bigr| \]

is small, the transform is quantitatively close to an isometry. This provides a natural perturbative theory.

Multivariate multiplicative actions

For kernels

\[ K(x_1y_1,\ldots,x_ny_n), \]

the Mellin spectrum lives on \(\mathbb R^n\), with reflection

\[ t\mapsto -t. \]

The same crossed-product structure persists with substantially richer spectral geometry.

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25. Conclusion

A product-kernel integral transform is spectrally not merely an integral operator.

It is a reflected Mellin multiplier:

\[ \boxed{ \mathcal K_K \sim M_qJ. } \]

That single representation separates elementary reciprocity from the stronger operator-theoretic properties of the transform.

It also reveals that product-kernel transforms cannot be understood as an isolated family, because

\[ (M_qJ)(M_rJ) = M_{qr^\vee}. \]

Their natural closure is therefore

\[ \boxed{ \mathfrak A = L^\infty(\mathbb R) \rtimes \mathbb Z_2, } \]

with a concrete \(2\times2\) fiber representation.

Inside this algebra,

\[ \text{boundedness}, \quad \text{adjoints}, \quad \text{normality}, \quad \text{involution}, \quad \text{unitarity}, \quad \text{Plancherel}, \quad \text{commutation}, \quad \text{inversion}, \]

and

\[ \text{functional calculus} \]

all reduce to explicit pointwise equations for the Mellin symbol.

The progression is therefore

\[ \boxed{ \text{integral reciprocity} \longrightarrow \text{Mellin reflection} \longrightarrow \text{transform algebra} \longrightarrow \text{spectral classification}. } \]

 From Integral Reciprocity to Spectral Transform Algebras.

1. PART I — SOURCE RESET: THE CROSSED PRODUCT AS OBJECT

   1. A = L∞(R) ⋊α Z2 as new source.
   2. Erase integral-kernel privilege.
   3. Even/odd grading as constitutive structure.
   4. Reflection action α(q)=q∨.
   5. Source invariants versus Mellin representation artifacts.

2. PART II — GRM BOUNDARY AUDIT

   6. TBLIND removal of known transform examples.
   7. FB: consequences already executable inside A.
   8. RB: requirements for full spectral classification.
   9. C*: first jointly executable carrier.
   10. CUT* / FNA*: first unresolved algebraic/spectral dependencies.

3. PART III — FIBER GEOMETRY

   11. Orbit decomposition {t,-t}.
   12. Measurable 2×2 matrix fields.
   13. Fiber eigenstructure and degeneracy.
   14. Fixed point t=0 as exceptional orbit.
   15. Measurable selection and reconstruction.

4. PART IV — INTERNAL ALGEBRAIC STRUCTURE

   16. Center and commutant.
   17. Ideals and invariant subalgebras.
   18. Projections, partial isometries, polar decomposition.
   19. Commutator/anticommutator geometry.
   20. Generated subalgebras from transform families.

5. PART V — SPECTRAL CLASSIFICATION

   21. General spectrum of A(p,q).
   22. Normal, unitary, self-adjoint and involutive strata.
   23. Resolvent geometry.
   24. Functional calculus beyond pure odd elements.
   25. Spectral multiplicity and invariant decomposition.

6. PART VI — INVERSE DESIGN / KREQ

   26. Property → symbol constraints.
   27. Symbol constraints → admissible kernel classes.
   28. Existence versus constructor.
   29. Nonuniqueness and kernel equivalence classes.
   30. Exact inverse-Mellin liftback.

7. PART VII — REPRESENTATION SUCCESSORS

   31. Weighted Mellin lines Re(s)=c.
   32. Matrix-valued symbols.
   33. Operator-valued symbols.
   34. Multivariate multiplicative groups.
   35. Replace Z2 reflection by larger symmetry groups.

8. PART VIII — PERTURBATION AND STABILITY

   36. Approximate unitarity.
   37. Stability of spectral gaps.
   38. Perturbation of involutions.
   39. Defect/residue propagation.
   40. Quantitative distance to transform strata.

9. PART IX — GRM SUCCESSOR THEORY

   41. When crossed-product representation itself fails.
   42. Typed obstruction extraction.
   43. Carrier mutation forced by failure.
   44. New symmetry/action discovery.
   45. Higher-arity composition and non-pairwise closure.

10. PART X — LIFTBACK / REPLAY / CLOSURE

11. Spectral object → integral realization.

12. Exact ancestry of reconstructed kernels.

13. Representation-independent invariants.

14. Replay across alternative carriers.

15. Frontier: which structures remain invisible even to the crossed product.